Computational Mathematics for Complex Simulation in Science and Engineering

Abstract

We study the integrability to second order of infinitesimal Einstein deformations on compact Riemannian and in particular on Kähler manifolds. We find a new way of expressing the necessary and sufficient condition for integrability to second order, which also gives a very clear and compact way of writing the Koiso obstruction. As an application we consider the Kähler case, where the condition can be further simplified and in complex dimension 3 turns out to be purely algebraic. One of our main results is the complete and explicit description of integrable to second order infinitesimal Einstein deformations on the complex 2-plane Grassmannian, which also has a quaternion Kähler structure. As a striking consequence we find that the symmetric Einstein metric on the Grassmannian $Gr_2(C^n+2)$ for $n$ odd is rigid.

Abstract

This paper presents a study of solution strategies for the Cahn-Hilliard-Biot equations, a complex mathematical model for understanding flow in deformable porous media with changing solid phases. Solving the Cahn-Hilliard-Biot system poses significant challenges due to its coupled, nonlinear and non-convex nature. We explore various solution algorithms, comparing monolithic and splitting strategies, focusing on both their computational efficiency and robustness.

Abstract

The development of reliable mathematical models and numerical discretization methods is important for the understanding of salt precipitation in porous media, which is relevant for environmental problems like soil salinization. Models on the pore scale are necessary to represent local heterogeneities in precipitation and to include the influence of solution-air-solid interfaces. A pore-network model for saturated flow, which includes the precipitation reaction of salt, is presented. It is implemented in the open-source simulator DuMu\$\$^\\backslashtextrm\X\\\$\$. In this paper, we restrict ourselves to one-phase flow as a first step. Since the throat transmissibilities determine the flow behaviour in the pore network, different concepts for the decreasing throat transmissibility due to precipitation are investigated. We consider four concepts for the amount of precipitation in the throats. Three concepts use information from the adjacent pore bodies, and one employs a pore-throat model obtained by averaging the resolved pore-scale model in a thin-tube. They lead to different permeability developments, which are caused by the different distribution of the precipitate between the pore bodies and throats. We additionally apply two different concepts for the calculation of the transmissibility. One obtains the precipitate distribution from analytical assumptions, the other from a geometric minimization principle using a phase-field evolution equation. The two concepts do not show substantial differences for the permeability development as long as simple pore-throat geometries are used. Finally, advantages and disadvantages of the concepts are discussed in the context of the considered physical problem and a reasonable effort for the implementation and computational costs.

Abstract

This dataset includes the code to reproduce the results from the paper titled "Greedy Kernel Methods for Approximating Breakthrough Curves for Reactive Flow from 3D Porous Geometry Data". In this paper we address the challenging application of 3D pore scale reactive flow under varying geometry parameters. We demonstrate that the vectorial kernel orthogonal greedy approximation (VKOGA) procedure with a data-adapted two-layer kernel yields excellent predictors for learning from 3D geometry voxel data via both morphological descriptors or principal component analysis. The numerical experiments from this paper can be reproduced using this code. See the README for more information and installation instructions.

Abstract

In the framework of reduced basis methods, we recently introduced a new certified hierarchical and adaptive surrogate model, which can be used for efficient approximation of input-output maps that are governed by parametrized partial differential equations. This adaptive approach combines a full order model, a reduced order model and a machine-learning model. In this contribution, we extend the approach by leveraging novel kernel models for the machine learning part, especially structured deep kernel networks as well as two layered kernel models. We demonstrate the usability of those enhanced kernel models for the RB-ML-ROM surrogate modeling chain and highlight their benefits in numerical experiments.

Abstract

In order to investigate the emergence of periodic oscillations of rimming flows, we study analytically the stability of steady states for the model of (Benilov, Kopteva, O'Brien, 2005), which describes the dynamics of a thin fluid film coating the inner wall of a rotating cylinder and includes effects of surface tension, gravity, and hydrostatic pressure. We apply multi-parameter perturbation methods to eigenvalues of Fréchet derivatives and prove the transition of a pair of conjugate eigenvalues from the stable to the unstable complex half-plane under appropriate variations of parameters. In order to prove rigorously the corresponding branching of periodic solutions from critical equilibria, we extend the multi-parameter Hopf-bifurcation theory to the case of infinite-dimensional dynamical systems.

Abstract

In this note, we extend the analysis for the residual-based a posteriori error estimators in the energy norm defined for the algebraic flux correction (AFC) schemes (Jha, 2021) to the newly proposed algebraic stabilization schemes (John and Knobloch, 2022; Knobloch, 2023). Numerical simulations on adaptively refined grids are performed in two dimensions showing the higher efficiency of an algebraic stabilization with similar accuracy compared with an AFC scheme

Abstract

In this note it is shown that the famous multiplier absolute stability test of R. O'Shea, G. Zames and P. Falb is necessary and sufficient if the set of Lur'e interconnections is lifted to a Kronecker structure and an explicit method to construct the destabilizing static nonlinearity is presented.

Abstract

The projection lemma (often also referred to as the elimination lemma) is one of the most powerful and useful tools in the context of linear matrix inequalities for system analysis and control. In its traditional formulation, the projection lemma only applies to strict inequalities, however, in many applications we naturally encounter non-strict inequalities. As such, we present, in this note, a non-strict projection lemma that generalizes both its original strict formulation as well as an earlier non-strict version. We demonstrate several applications of our result in robust linear-matrix-inequality-based marginal stability analysis and stabilization, a matrix S-lemma, which is useful in (direct) data-driven control applications, and matrix dilation.

Abstract

We address the challenging application of 3D pore scale reactive flow under varying geometry parameters. The task is to predict time-dependent integral quantities, i.e., breakthrough curves, from the given geometries. As the 3D reactive flow simulation is highly complex and computationally expensive, we are interested in data-based surrogates that can give a rapid prediction of the target quantities of interest. This setting is an example of an application with scarce data, i.e., only having a few available data samples, while the input and output dimensions are high. In this scarce data setting, standard machine learning methods are likely to fail. Therefore, we resort to greedy kernel approximation schemes that have shown to be efficient meshless approximation techniques for multivariate functions. We demonstrate that such methods can efficiently be used in the high-dimensional input/output case under scarce data. Especially, we show that the vectorial kernel orthogonal greedy approximation (VKOGA) procedure with a data-adapted two-layer kernel yields excellent predictors for learning from 3D geometry voxel data via both morphological descriptors or principal component analysis.

Abstract

An interface-preserving moving mesh algorithm in two or higher dimensions is presented. It resolves a moving (d-1)-dimensional manifold directly within the d-dimensional mesh, which means that the interface is represented by a subset of moving mesh cell-surfaces. The underlying mesh is a conforming simplicial partition that fulfills the Delaunay property. The local remeshing algorithms allow for strong interface deformations. We give a proof that the given algorithms preserve the interface after interface deformation and remeshing steps. Originating from various numerical methods, data is attached cell-wise to the mesh. After each remeshing operation, the interface-preserving moving mesh retains valid data by projecting the data to the new mesh cells.An open source implementation of the moving mesh algorithm is available at Reference 1.

Abstract

The aim of this study was to design physics-preserving and precise surrogate models of the nonlinear elastic behaviour of an intervertebral disc (IVD). Based on artificial force-displacement data sets from detailed finite element (FE) disc models, we used greedy kernel and polynomial approximations of second, third and fourth order to train surrogate models for the scalar force-torque -potential. Doing so, the resulting models of the elastic IVD responses ensured the conservation of mechanical energy through their structure. At the same time, they were capable of predicting disc forces in a physiological range of motion and for the coupling of all six degrees of freedom of an intervertebral joint. The performance of all surrogate models for a subject-specific L4\$\$\backslashvert\$\$5 disc geometry was evaluated both on training and test data obtained from uncoupled (one-dimensional), weakly coupled (two-dimensional), and random movement trajectories in the entire six-dimensional (6d) physiological displacement range, as well as on synthetic kinematic data. We observed highest precisions for the kernel surrogate followed by the fourth-order polynomial model. Both clearly outperformed the second-order polynomial model which is equivalent to the commonly used stiffness matrix in neuro-musculoskeletal simulations. Hence, the proposed model architectures have the potential to improve the accuracy and, therewith, validity of load predictions in neuro-musculoskeletal spine models.

Abstract

We study single-phase flow in a fractured porous medium at a macroscopic scale that allows to model fractures individually. The flow is governed by Darcy's law in both fractures and porous matrix. We derive a new mixed-dimensional model, where fractures are represented by $(n-1)$-dimensional interfaces between $n$-dimensional subdomains for $n2$. In particular, we suggest a generalization of the model in~22 by accounting for asymmetric fractures with spatially varying aperture. Thus, the new model is particularly convenient for the description of surface roughness or for modeling curvilinear or winding fractures. The wellposedness of the new model is proven under appropriate conditions. Further, we formulate a discontinuous Galerkin discretization of the new model and validate the model by performing two- and three-dimensional numerical experiments.

Abstract

We propose novel time-domain dynamic integral quadratic constraints with a terminal cost for exponentially weighted slope-restricted gradients of not necessarily convex functions. This extends recent results for subdifferentials of convex function and their link to so-called O'Shea-Zames-Falb multipliers. The benefit of merging time-domain and frequency-domain techniques is demonstrated for linear saturated systems.

Abstract

We introduce a data-driven approach to the modelling and analysis of viscous fluid mechanics. Instead of including constitutive laws for the fluid’s viscosity in the mathematical model, we suggest directly using experimental data. Only a set of differential constraints, derived from first principles, and boundary conditions are kept of the classical PDE model and are combined with a data set. The mathematical framework builds on the recently introduced data-driven approach to solid-mechanics (Kirchdoerfer and Ortiz in Comput Methods Appl Mech Eng 304:81–101, 2016; Conti et al. in Arch Ration Mech Anal 229:79–123, 2018). We construct optimal data-driven solutions that are material model free in the sense that no assumptions on the rheological behaviour of the fluid are made or extrapolated from the data. The differential constraints of fluid mechanics are recast in the language of constant rank differential operators. Adapting abstract results on lower-semicontinuity and -quasiconvexity, we show a -convergence result for the functionals arising in the data-driven fluid mechanical problem. The theory is extended to compact nonlinear perturbations, whence our results apply not only to inertialess fluids but also to fluids with inertia. Data-driven solutions provide a new relaxed solution concept. We prove that the constructed data-driven solutions are consistent with solutions to the classical PDEs of fluid mechanics if the data sets have the form of a monotone constitutive relation.

Abstract

We present a variational approach for the construction of Leray-Hopf solutions to the non-newtonian Navier-Stokes system. Inspired by the work OSS18 on the corresponding Newtonian problem, we minimise certain stabilised Weighted Inertia-Dissipation-Energy (WIDE) functionals and pass to the limit of a vanishing parameter in order to recover a Leray-Hopf solution of the non-newtonian Navier-Stokes equations. It turns out that the results differ depending on the rheology of the fluid. The investigation of the non-newtonian Navier-Stokes system via this variational approach is motivated by the fact that it is particularly well suited to gain insights into weak, respectively strong convergence properties for different flow-behaviour exponents and thus into possibly turbulent behaviour of the fluid flow. ...

Abstract

Accelerating iterative eigenvalue algorithms is often achieved by employing a spectral shifting strategy. Unfortunately, improved shifting typically leads to a smaller eigenvalue for the resulting shifted operator, which in turn results in a high condition number of the underlying solution matrix, posing a major challenge for iterative linear solvers. This paper introduces a two-level domain decomposition preconditioner that addresses this issue for the linear Schrödinger eigenvalue problem, even in the presence of a vanishing eigenvalue gap in non-uniform, expanding domains. Since the quasi-optimal shift, which is already available as the solution to a spectral cell problem, is required for the eigenvalue solver, it is logical to also use its associated eigenfunction as a generator to construct a coarse space. We analyze the resulting two-level additive Schwarz preconditioner and obtain a condition number bound that is independent of the domain's anisotropy, despite the need for only one basis function per subdomain for the coarse solver. Several numerical examples are presented to illustrate its flexibility and efficiency.

Abstract

Coupled systems of porous media and free flow can be modelled by the Stokes equations in the free-flow domain, Darcy's law in the porous medium, and an appropriate set of coupling conditions on the fluid--porous interface. Discretisation of the coupled Stokes--Darcy problem leads to a large, sparse, ill-conditioned, and nonsymmetric linear system. We discretise the system using the MAC scheme, i.e., the finite volume method on staggered grids. To accelerate convergence of the GMRES method, efficient preconditioners are needed. We propose a block diagonal, a block triangular and a constraint preconditioner for the Stokes--Darcy problem with the classical set of coupling conditions based on the Beavers--Joseph condition and the generalised coupling conditions which were developed for arbitrary flow to the interface. We show the robustness and efficiency of the proposed preconditioners in numerical experiments.

Abstract

We study the dynamic behaviour of solutions to a fourth-order quasilinear degenerate parabolic equation for large times arising in fluid dynamical applications. The degeneracy occurs both with respect to the unknown and with respect to the sum of its first and third spatial derivative. The modelling equation appears as a thin-film limit for the interface separating two immiscible viscous fluid films confined between two cylinders rotating at small relative angular velocity. More precisely, the fluid occupying the layer next to the outer cylinder is considered to be Newtonian, i.e. it has constant viscosity, while we assume that the layer next to the inner cylinder is filled by a shear-thinning power-law fluid. Using energy methods, Fourier analysis and suitable regularity estimates for higher-order parabolic equations, we prove global existence of positive weak solutions in the case of low initial energy. Moreover, these global solutions are polynomially stable, in the sense that interfaces which are initially close to a circle, converge at rate 1/t1/β for some β>0 to a circle, as time tends to infinity. In addition, we provide regularity estimates for general nonlinear degenerate parabolic equations of fourth order.

Abstract

Ontologies and knowledge graphs for mathematical algorithms and models are presented, that have been developed by the Mathematical Research Data Initiative. This enables FAIR data handling in mathematics and the applied disciplines. Moreover, challenges of harmonization during the ontology development are discussed.

Abstract

In Germany, renewable energy sources play a crucial role in electricity generation, with wind and photovoltaic (PV) leading the way. In 2022, large wind turbines contributed 24.1% of the electricity generated, while PV accounted for 10.6%. Contrary, small wind power holds a marginal share of less than 0.01%. This is unfortunate as the decentralized nature of small wind power at low-voltage grid level offers benefits like reducing the need for grid expansion or infrastructure upgrades. Although small wind power currently suits locations with favorable wind potential, changing factors such as rising electricity prices, falling battery storage costs, and growing electrification in heating and transport could create new opportunities. Within this work a residential energy supply system consisting of small wind turbine, PV, heat pump, battery storage, and electric vehicle was dimensioned for different sites in Germany and Canada based on detailed simulation models and genetic algorithms. This was carried out for various economic framework conditions. Results indicate that with electricity purchase costs above 0.42€/kWh, combined with a 25% reduction in small wind turbine and battery storage investment expenses, economic viability could be significantly enhanced. This might expand the applicability of small wind power to diverse sites.

Abstract

We expose in a tutorial fashion the mechanisms which underlie the synthesis of optimization algorithms based on dynamic integral quadratic constraints. We reveal how these tools from robust control allow to design accelerated gradient descent algorithms with optimal guaranteed convergence rates by solving small-sized convex semi-definite programs. It is shown that this extends to the design of extremum controllers, with the goal to regulate the output of a general linear closed-loop system to the minimum of an objective function. Numerical experiments illustrate that we can not only recover gradient decent and the triple momentum variant of Nesterov's accelerated first order algorithm, but also automatically synthesize optimal algorithms even if the gradient information is passed through non-trivial dynamics, such as time-delays.

Abstract

We study the excess minimum risk in statistical inference, defined as the difference between the minimum expected loss when estimating a random variable from an observed feature vector and the minimum expected loss when estimating the same random variable from a transformation (statistic) of the feature vector. After characterizing lossless transformations, i.e., transformations for which the excess risk is zero for all loss functions, we construct a partitioning test statistic for the hypothesis that a given transformation is lossless, and we show that for i.i.d. data the test is strongly consistent. More generally, we develop information-theoretic upper bounds on the excess risk that uniformly hold over fairly general classes of loss functions. Based on these bounds, we introduce the notion of a δ-lossless transformation and give sufficient conditions for a given transformation to be universally δ-lossless. Applications to classification, nonparametric regression, portfolio strategies, information bottlenecks, and deep learning are also surveyed.

Abstract

Purpose The progression of deep endometriosis (DE) in women of reproductive age is highly variable. This study aimed to analyze the sonomorphological changes of rectal endometriosis over long periods of time and the influence of hormonal treatment.Methods This retrospective study included premenopausal women with rectal DE treated conservatively between 2002 and 2021. The lesion length and thickness of the nodule were evaluated at regular intervals over time. We created statistical models with mixed effects to identify potential factors influencing lesion progression and regression.Results 38 patients were monitored over a mean period of 7.2 (± 4.2) years with a mean of 3.1 (± 2.1) check-ups within the observation period. We detected a significant increase in lesion length until the end of the fourth decade of life. In addition, we found a substantial decrease in the length and thickness of the nodule depending on the length of hormonal treatment.Conclusion In conservatively managed patients with rectal endometriosis, without hormonal therapy, lesion size can exhibit a moderate increase up to the end of the fourth decade of life, after which it appears to stabilize. This increase does not follow a linear pattern. Hormonal therapy is crucial in impeding further progression, resulting in either a cessation or a regression of lesion growth.

Abstract

<p>The numerical simulation of neuromuscular processes in skeletal muscles typically imposes high computational costs, resulting in the need for techniques to reduce these costs. One submodel in these types of multiphysics multiscale simulations is the electrophysiological model, describing the propagation of an action potential (AP) in a muscle fiber. Involved in the simulation of the propagation are two traveling waves, resulting in difficulties for classical model order reduction (MOR) techniques based on linear subspaces.</p><p>Instead, we apply the nonlinear MOR method shifted Proper Orthogonal Decomposition (sPOD) to construct dynamically transformed reduced basis functions depending on time-dependent paths spanning the adaptive reduced ansatz space.</p><p>Our numerical experiments demonstrate that the constructed reduced ansatz space can accurately capture the dynamics of two fully separated wavefronts and reduce the degrees of freedom of the whole simulation. However, it cannot represent the activation of the AP in the center of the fiber and overlapping wave parts. The constructed reduced order model outperforms the high-dimensional full order model in terms of the computational costs while the accuracy is maintained and reaches speedup factors between 2 and 73 depending on the time discretization.</p>

Abstract

Abstract The viscous flow of two immiscible fluids in a porous medium on the Darcy scale is governed by a system of nonlinear parabolic equations. If infinite mobility of one phase can be assumed (e.g., in soil layers in contact with the atmosphere) the system can be substituted by the scalar Richards model. Thus, the porous medium domain may be partitioned into disjoint subdomains where either the full two-phase or the simplified Richards model dynamics are valid. Extending the previously considered one-model situations we suggest coupling conditions for this hybrid model approach. Based on an Euler implicit discretization, a linear iterative (L-type) domain decomposition scheme is proposed, and proved to be convergent. The theoretical findings are verified by a comparative numerical study that in particular confirms the efficiency of the hybrid ansatz as compared to full two-phase model computations.

Abstract

In this paper, we use the representation theory of the group $$Spin(m)$$to develop aspects of the global symbolic calculus of pseudo-differential operators on $$Spin(3)$$and $$Spin(4)$$in the sense of Ruzhansky–Turunen–Wirth. A detailed study of $$Spin(3)$$and $$Spin(4)$$-representations is made including recurrence relations and natural differential operators acting on matrix coefficients. We establish the calculus of left-invariant differential operators and of difference operators on the group $$Spin(4)$$and apply this to give criteria for the subellipticity and the global hypoellipticity of pseudo-differential operators in terms of their matrix-valued full symbols. Several examples of first and second order globally hypoelliptic differential operators are given, including some that are locally neither invertible nor hypoelliptic. The paper presents a particular case study for higher dimensional spin groups.

Abstract

We study the problem of nonparametric classification with repeated observations. Let $X$ be the $d$ dimensional feature vector and let $Y$ denote the label taking values in $1,łdots, M\rbrace$. In contrast to usual setup with large sample size $n$ and relatively low dimension $d$, this letter deals with the situation, when instead of observing a single feature vector $X$ we are given $t$ repeated feature vectors $V_1,łdots, V_t$. Some simple classification rules are presented such that the conditional error probabilities have exponential rate of convergence as $tınfty$.

Abstract

This work presents a framework to synthesize structured gain-scheduled controllers for structured plants that are affected by time-varying parametric scheduling blocks. Using a so-called lifting approach, we are able to handle several structured gain-scheduling problems arising from a nested inner and outer loop configuration with partial or full dependence on the scheduling block. Our resulting design conditions are formulated in terms of convex linear matrix inequalities and permit to handle multiple performance objectives.

Abstract

This work presents a framework to synthesize structured gain-scheduled controllers for structured plants whose dynamics change according to time-varying scheduling parameters. Both the system and the controller are assumed to admit descriptions in terms of a linear time-invariant system in feedback with so-called scheduling blocks, which collect all scheduling parameters into a static system. We show that such linear fractional representations permit to exploit a so-called lifting technique in order to handle several structured gain-scheduling design problems. These could arise from a nested inner and outer loop control configuration with partial or full dependence on the scheduling variables. Our design conditions are formulated in terms of convex linear matrix inequalities and permit to handle multiple performance objectives.

Abstract

Developing machine learning-based interatomic potentials from ab initio electronic structure methods remains a challenging task for computational chemistry and materials science. This work studies the capability of transfer learning, in particular discriminative fine-tuning, for efficiently generating chemically accurate interatomic neural network potentials on organic molecules from the MD17 and ANI data sets. We show that pre-training the network parameters on data obtained from density functional calculations considerably improves the sample efficiency of models trained on more accurate ab initio data. Additionally, we show that fine-tuning with energy labels alone can suffice to obtain accurate atomic forces and run large-scale atomistic simulations, provided a well-designed fine-tuning data set. We also investigate possible limitations of transfer learning, especially regarding the design and size of the pre-training and fine-tuning data sets. Finally, we provide GM-NN potentials pre-trained and fine-tuned on the ANI-1x and ANI-1ccx data sets, which can easily be fine-tuned on and applied to organic molecules.

Abstract

Data-dependent greedy algorithms in kernel spaces are known to provide fast converging interpolants, while being extremely easy to implement and efficient to run. Despite this experimental evidence, no detailed theory has yet been presented. This situation is unsatisfactory, especially when compared to the case of the data-independent P-greedy algorithm, for which optimal convergence rates are available, despite its performances being usually inferior to the ones of target data-dependent algorithms. In this work, we fill this gap by first defining a new scale of greedy algorithms for interpolation that comprises all the existing ones in a unique analysis, where the degree of dependency of the selection criterion on the functional data is quantified by a real parameter. We then prove new convergence rates where this degree is taken into account, and we show that, possibly up to a logarithmic factor, target data-dependent selection strategies provide faster convergence. In particular, for the first time we obtain convergence rates for target data adaptive interpolation that are faster than the ones given by uniform points, without the need of any special assumption on the target function. These results are made possible by refining an earlier analysis of greedy algorithms in general Hilbert spaces. The rates are confirmed by a number of numerical examples.

Abstract

We consider Gaussian subordinated Lévy fields (GSLFs) that arise by subordinating Lévy processes with positive transformations of Gaussian random fields on some spatial domain. The resulting random fields are distributionally flexible and have in general discontinuous sample paths. Theoretical investigations of the random fields include pointwise distributions, possible approximations and their covariance function. As an application, a random elliptic PDE is considered, where the constructed random fields occur in the diffusion coefficient. Further, we present various numerical examples to illustrate our theoretical findings.

Abstract

In this letter, we revisit structure exploiting SDP solvers dedicated to the solution of Kalman-Yakubovic-Popov semi-definite programs (KYP-SDPs). These SDPs inherit their name from the KYP lemma and they play a crucial role in, e.g., robustness analysis, robust state feedback synthesis, and robust estimator synthesis for uncertain dynamical systems. Off-the-shelve SDP solvers require O(n^6) arithmetic operations per Newton step to solve this class of problems, where n is the state dimension of the dynamical system under consideration. Specialized solvers reduce this complexity to O(n^3) . However, existing specialized solvers do not include semi-definite constraints on the Lyapunov matrix, which is necessary for controller synthesis. In this letter, we show how to include such constraints in structure exploiting KYP-SDP solvers.

Abstract

This paper introduces a novel representation of convolutional Neural Networks (CNNs) in terms of 2-D dynamical systems. To this end, the usual description of convolutional layers with convolution kernels, i.e., the impulse responses of linear filters, is realized in state space as a linear time-invariant 2-D system. The overall convolutional Neural Network composed of convolutional layers and nonlinear activation functions is then viewed as a 2-D version of a Lur'e system, i.e., a linear dynamical system interconnected with static nonlinear components. One benefit of this 2-D Lur'e system perspective on CNNs is that we can use robust control theory much more efficiently for Lipschitz constant estimation than previously possible.

Abstract

This paper proposes a distributionally robust approach to regret optimal control of discrete-time linear dynam-ical systems with quadratic costs subject to a stochastic additive disturbance on the state process. The underlying probability distribution of the disturbance process is unknown, but assumed to lie in a given ball of distributions defined in terms of the type-2 Wasserstein distance. In this framework, strictly causal linear disturbance feedback controllers are designed to minimize the worst-case expected regret. The regret incurred by a controller is defined as the difference between the cost it incurs in response to a realization of the disturbance process and the cost incurred by the optimal noncausal controller which has perfect knowledge of the disturbance process realization at the outset. Building on a well-established duality theory for optimal transport problems, we derive a reformulation of the minimax regret optimal control problem as a tractable semidefinite program. Using the equivalent dual reformulation, we characterize a worst-case distribution achieving the worst-case expected regret in relation to the distribution at the center of the Wasserstein ball. We compare the minimax regret optimal control design method with the distributionally robust optimal control approach using an illustrative example and numerical experiments.

Abstract

Building on the well-known total variation, this paper develops a general regularization technique based on nonlinear isotropic diffusion (NID) for inverse problems with piecewise smooth solutions. The novelty of our approach is to be adaptive (we speak of A-NID), i.e., the regularization varies during the iterates in order to incorporate prior information on the edges, deal with the evolution of the reconstruction and circumvent the limitations due to the nonconvexity of the proposed functionals. After a detailed analysis of the convergence and well-posedness of the method, the latter is validated by simulations performed on synthetic and real data on computerized tomography.

Abstract

We present a novel model for fluid-driven fracture propagation in poro-elastic media. Our approach combines ideas from dimensionally reduced discrete fracture models with diffuse phase-field models. The main advantage of this combined approach is that the fracture geometry is always represented explicitly, while the propagation remains geometrically flexible. We prove that our model is thermodynamically consistent. In order to solve our model numerically, we propose a mixed-dimensional discontinuous Galerkin scheme with a computational grid fully conforming to the fractures. As the fracture propagates, the diffuse phase-field acts as indicator to identify new fracture facets to be added to the discrete fracture network. Numerical experiments demonstrate that our approach reproduces classical scenarios for fracturing porous media

Abstract

Abstract. We study the long-time behavior of solutions to quasilinear doubly degenerate parabolic problems of fourth order. The equations model, for instance, the dynamic behavior of a non-Newtonian thin-film flow on a flat impermeable bottom and with zero contact angle. We consider a shear-rate dependent fluid the rheology of which is described by a constitutive power law or Ellis law for the fluid viscosity. In all three cases, positive constants (i.e., positive flat films) are the only positive steady-state solutions. Moreover, we can give a detailed picture of the long-time behavior of solutions with respect to the \(H^1(Ømega )\) -norm. In the case of shear-thickening power-law fluids, one observes that solutions which are initially close to a steady state converge to equilibrium in finite time. In the shear-thinning power-law case, we find that steady states are polynomially stable in the sense that, as time tends to infinity, solutions which are initially close to a steady state converge to equilibrium at rate \(1/t^1/\) for some \(0\) . Finally, in the case of an Ellis fluid, steady states are exponentially stable in \(H^1(Ømega )\) .

Abstract

We consider a phase-field model for the incompressible flow of two immiscible fluids. This model extends widespread models for two fluid phases by including a third, solid phase, which can evolve due to e.g. precipitation and dissolution. We consider a simple, two-dimensional geometry of a thin strip, which can still be seen as the representation of a single pore throat in a porous medium. Under moderate assumptions on the Péclet number and the capillary number, we investigate the limit case when the ratio between the width and the length of the strip goes to zero. In this way, and employing transversal averaging, we derive an upscaled model. The result is a multi-scale model consisting of the upscaled equations for the total flux and the ion transport, while the phase-field equation has to be solved in cell problems at the pore scale to determine the position of interfaces. We also investigate the sharp-interface limit of the multi-scale model, in which the phase-field parameter approaches 0. The resulting sharp-interface model consists only of Darcy-scale equations, as the cell problems can be solved explicitly. Notably, we find asymptotic consistency, that is, the upscaling process and the sharp-interface limit commute. We use numerical results to investigate the validity of the upscaling when discontinuities are formed in the upscaled model.

Abstract

This paper considers the problem of designing accelerated gradient-based algorithms for optimization and saddle-point problems. The class of objective functions is defined by a generalized sector condition. This class of functions contains strongly convex functions with Lipschitz gradients but also non-convex functions, which allows not only to address optimization problems but also saddle-point problems. The proposed design procedure relies on a suitable class of Lyapunov functions and on convex semi-definite programming. The proposed synthesis allows the design of algorithms that reach the performance of state-of-the-art accelerated gradient methods and beyond.

Abstract

We propose novel quadratic performance tests for linear discrete-time impulsive systems based on viewing these systems as feedback interconnections of some non-impulsive linear system with an impulsive operator. In order to systematically analyze such interconnections, we employ the framework of integral quadratic constraints and propose novel constraints of this kind for capturing the behavior of the involved impulsive operator. As a major benefit, the modularity of this framework permits seamless extensions to interconnections affected by heterogeneous uncertainties in a straightforward fashion. This contrasts with alternative approaches which are based on capturing the system's impulsive behavior by means of a clock. Building upon the developed analysis criteria, we characterize the existence of non-impulsive estimators for such impulsive interconnections in a lossless fashion and in terms of linear matrix inequalities. Finally, our approach is illustrated by means of several numerical examples.

Abstract

This paper provides a brief overview of uncertainty visualization along with some fundamental considerations on uncertainty propagation and modeling. Starting from the visualization pipeline, we discuss how the different stages along this pipeline can be affected by uncertainty and how they can deal with this and propagate uncertainty information to subsequent processing steps. We illustrate recent advances in the field with a number of examples from a wide range of applications: uncertainty visualization of hierarchical data, multivariate time series, stochastic partial differential equations, and data from linguistic annotation.

Abstract

Keywords: Transient wave scattering · Elastodynamics · Time-dependent boundary integral equations · Convolution quadrature

Abstract

Careful donor selection is important for kidney transplantations (KT) from suboptimal donors aged ≥65 years. Several tools such as histopathological assessment of preimplant biopsies have been shown to predict allograft survival and can be applied for selection. Whether the explanting surgeon's appraisal is associated with outcomes of KTs from suboptimal donors is unknown.

Abstract

In this paper we analyze a greedy procedure to approximate a linear functional defined in a reproducing kernel Hilbert space by nodal values. This procedure computes a quadrature rule which can be applied to general functionals. For a large class of functionals, that includes integration functionals and other interesting cases, but does not include differentiation, we prove convergence results for the approximation by means of quasi-uniform and greedy points which generalize in various ways several known results. A perturbation analysis of the weights and node computation is also discussed. Beyond the theoretical investigations, we demonstrate numerically that our algorithm is effective in treating various integration densities, and that it is even very competitive when compared to existing methods for Uncertainty Quantification.

Abstract

We present an integrated approach for the use of simulated data from full order discretization as well as projection-based Reduced Basis reduced order models for the training of machine learning approaches, in particular Kernel Methods, in order to achieve fast, reliable predictive models for the chemical conversion rate in reactive flows with varying transport regimes.

Abstract

We present efficient reduced basis (RB) methods for the simulation of a coupled problem consisting of a rigid robot hand interacting with soft tissue material. The soft tissue is modeled by the linear elasticity equation and discretized with the Finite Element Method. We look at two different scenarios: (i) the forward simulation and (ii) a feedback control formulation of the model. In both cases, large-scale systems of equations appear, which need to be solved in real-time. This is essential in practice for the implementation in a real robot. For the feedback-scenario, we encounter a high-dimensional Algebraic Riccati Equation (ARE) in the context of the linear quadratic regulator. To overcome the real-time constraint by significantly reducing the computational complexity, we use several structure-preserving and non-structure-preserving reduction methods. These include reduced basis techniques based on the Proper Orthogonal Decomposition. For the ARE, we compute a low-rank-factor and hence solve a low-dimensional ARE instead of solving a full dimensional problem. Numerical examples for both cases (i) and (ii) are provided. These illustrate the approximation quality of the reduced solution and speedup factors of the different reduction approaches.

Abstract

The modelling of liquid--vapour flow with phase transition poses many challenges, both on the theoretical level, as well as on the level of discretisation methods. Therefore, accurate mathematical models and efficient numerical methods are required. In that, we focus on two modelling approaches: the sharp-interface (SI) approach and the diffuse-interface (DI) approach. For the SI-approach, representing the phase boundary as a co-dimension-1 manifold, we develop and validate analytical Riemann solvers for basic isothermal two-phase flow scenarios. This ansatz becomes cumbersome for increasingly complex thermodynamical settings. A more versatile multiscale interface solver, that is based on molecular dynamics simulations, is able to accurately describe the evolution of phase boundaries in the temperature-dependent case. It is shown to be even applicable to two-phase flow of multiple components. Despite the successful developments for the SI approach, these models fail if the interface undergoes topological changes. To understand merging and splitting phenomena for droplet ensembles, we consider DI models of second gradient type. For these Navier--Stokes--Korteweg systems, that can be seen as a third order extension of the Navier--Stokes equations, we propose variants that are more accessible to standard numerical schemes. More precisely, we reformulate the capillarity operator to restore the hyperbolicity of the Euler operator in the full system.

Abstract

The accuracy of the training data limits the accuracy of bulk properties from machine-learned potentials. For example, hybrid functionals or wave-function-based quantum chemical methods are readily available for cluster data but effectively out of scope for periodic structures. We show that local, atom-centered descriptors for machine-learned potentials enable the prediction of bulk properties from cluster model training data, agreeing reasonably well with predictions from bulk training data. We demonstrate such transferability by studying structural and dynamical properties of bulk liquid water with density functional theory and have found an excellent agreement with experimental and theoretical counterparts.

Abstract

In this paper, we consider a non-linear fourth-order evolution equation of Cahn–Hilliard-type on evolving surfaces with prescribed velocity, where the non-linear terms are only assumed to have locally Lipschitz derivatives. High-order evolving surface finite elements are used to discretise the weak equation system in space, and a modified matrix–vector formulation for the semi-discrete problem is derived. The anti-symmetric structure of the equation system is preserved by the spatial discretisation. A new stability proof, based on this structure, combined with consistency bounds proves optimal-order and uniform-in-time error estimates. The paper is concluded by a variety of numerical experiments.

Abstract

This work presents a framework to synthesize structured gain-scheduled controllers for structured plants that are affected by time-varying parametric scheduling blocks. Using a so-called lifting approach, we are able to handle several structured gain-scheduling problems arising from a nested inner and outer loop configuration with partial or full dependence on the scheduling block. Our resulting design conditions are formulated in terms of convex linear matrix inequalities and permit to handle multiple performance objectives.

Abstract

We establish a framework for systematically analyzing and designing output-feedback controllers for linear impulsive and related hybrid systems that might even be affected by various types of uncertainties. In particular, the framework encompasses uncertain switched and sampled-data systems as well as networked systems with switching communication topologies. The framework is based on recently developed convex criteria involving a so-called clock for analyzing impulsive systems under dwell-time constraints. We elaborate on the extension of those criteria for dynamic output-feedback controller synthesis by means of convex optimization and generalize the so-called dual iteration to impulsive systems. The latter originally and still constitutes a promising heuristic procedure for the challenging and non-convex design of static output-feedback controllers for standard linear time-invariant systems. Moreover, for uncertain impulsive systems as modeled in terms of linear fractional representations, we generalize the nominal analysis criteria by providing novel robust analysis conditions based on a novel time-domain and clock-dependent formulation of integral quadratic constraints. Finally, by combining the insights on nominal synthesis and robust analysis, we are able to tackle challenging output-feedback designs of practical relevance, such as the design of gain-scheduled, robust or robust gain-scheduled controllers for impulsive systems. Most of the obtained analysis and synthesis conditions involve infinite-dimensional (differential) linear matrix inequalities which can be numerically solved by using relaxation methods based on, e.g., linear splines, B-splines or matrix sum-of-squares that we discuss as well.

Abstract

Process metadata constitute a relevant part of the documentation of research processes and the creation and use of research data. As an addition to publications on the research question, they capture details needed for reusability and comparability and are thus important for a sustainable handling of research data. In the DH project SDC4Lit we want to capture process metadata when researchers work with literary material, conducting manual and automatic processing steps, which need to be treated with equal emphasis. We present a content-related mapping between two process metadata schemas from the area of Digital Humanities (GRAIN, RePlay-DH) and one from Computational Engineering (EngMeta) and find that there are no basic obstacles preventing the use of any of them for our purposes. Actually a basic difference rather exists between GRAIN on the one hand and RePlay-DH and EngMeta on the other, regarding the treatment of tools and actors in a workflow step.

Abstract

The analysis of eigenvalues of Laplace and Schrödinger operators is an important and classical topic in mathematical physics with many applications. This book presents a thorough introduction to the area, suitable for masters and graduate students, and includes an ample amount of background material on the spectral theory of linear operators in Hilbert spaces and on Sobolev space theory. Of particular interest is a family of inequalities by Lieb and Thirring on eigenvalues of Schrödinger operators, which they used in their proof of stability of matter. The final part of this book is devoted to the active research on sharp constants in these inequalities and contains state-of-the-art results, serving as a reference for experts and as a starting point for further research.

Abstract

Summary: "We study the dynamic behaviour of two viscous fluid films confined between two concentric cylinders rotating at a small relative velocity. It is assumed that the fluids are immiscible and that the volume of the outer fluid film is large compared to the volume of the inner one. Moreover, while the outer fluid is considered to have constant viscosity, the rheological behaviour of the inner thin film is determined by a strain-dependent power-law. Starting from a Navier-Stokes system, we formally derive evolution equations for the interface separating the two fluids. Two competing effects drive the dynamics of the interface, namely the surface tension and the shear stresses induced by the rotation of the cylinders. When the two effects are comparable, the solutions behave, for large times, as in the Newtonian regime. We also study the regime in which the surface tension effects dominate the stresses induced by the rotation of the cylinders. In this case, we prove local existence of positive weak solutions both for shear-thinning and shear-thickening fluids. In the latter case, we show that interfaces which are initially close to a circle converge to a circle in finite time and keep that shape for later times.''

Abstract

Results of our proposed optical flow method on the Sintel and KITTI datasets. We provide the benchmark results before and after applying our refinement approach.Additionally, we provide a supplementary material to our paper with more details on the minimisation, the numerical solution and additional results.The data is structured as follows:The file supplement.pdf contains the supplementary material of the paper.Additionally, there are 10 data folders according to the test results in Table 3 of our paper. They are named using the scheme input/output_datasetName_methodName:input/output: whether the files are the input for our method or the output of our methoddatasetName: one of kitti, sintelClean or sintelFinal relating to either the KITTI dataset Menze, CVPR 2015 or the Sintel dataset Butler, ECCV 2012methodName: one of raft or raftWarm Teed, ECCV 2020 or the refined variants from our method.The data can be used to build on the output of our method or to compare a novel approach against our method by using the same input data.The dataformat is pdf for the supplementary material and png and flo for the optical flow files as used in the respective benchmarks.

Abstract

In this note it is shown that the famous multiplier absolute stability test of R. O'Shea, G. Zames and P. Falb is necessary and sufficient if the set of Lur'e interconnections is lifted to a Kronecker structure and an explicit method to construct the destabilizing static nonlinearity is presented.

Abstract

As an extension to the well-established stationary elliptic partial differential equation (PDE) with random continuous coefficients we study a time-dependent advection-diffusion problem, where the coefficients may have random spatial discontinuities. In a subsurface flow model, the randomness in a parabolic equation may account for insufficient measurements or uncertain material procurement, while the discontinuities could represent transitions in heterogeneous media. Specifically, a scenario with coupled advection and diffusion coefficients that are modeled as sums of continuous random fields and discontinuous jump components are considered. The respective coefficient functions allow a very flexible modeling, however, they also complicate the analysis and numerical approximation of the corresponding random parabolic PDE. We show that the model problem is indeed well-posed under mild assumptions and show measurability of the pathwise solution. For the numerical approximation we employ a sample-adapted, pathwise discretization scheme based on a finite element approach. This semi-discrete method accounts for the discontinuities in each sample, but leads to stochastic, finite-dimensional approximation spaces. We ensure measurability of the semi-discrete solution, which in turn enables us to derive moments bounds on the mean-squared approximation error. By coupling this semi-discrete approach with suitable coefficient approximation and a stable time stepping, we obtain a fully discrete algorithm to solve the random parabolic PDE. We provide an overall error bound for this scheme and illustrate our results with several numerical experiments.

Abstract

Standard kernel methods for machine learning usually struggle when dealing with large datasets. We review a recently introduced Structured Deep Kernel Network (SDKN) approach that is capable of dealing with high-dimensional and huge datasets - and enjoys typical standard machine learning approximation properties.

Abstract

Multiphase flow in fractured porous media can be described by discrete fracture matrix models that represent the fractures as dimensionally reduced manifolds embedded in the bulk porous medium. Generalizing earlier work on this approach we focus on immiscible two-phase flow in time-dependent fracture geometries, i.e., the fracture itself and the aperture of the fractures might evolve in time. For dynamic fracture geometries of that kind, neglecting capillary forces, we deduce by transversal averaging of a full dimensional description a dimensionally reduced model that governs the geometric evolution and the flow dynamics. The core computational contribution is a mixed-dimensional finite-volume discretization based on a conforming moving-mesh ansatz. This finite-volume moving-mesh (FVMM) algorithm is tracking the fractures' motions as a family of unions of facets of the mesh. Notably, the method permits arbitrary movement of facets of the triangulation while keeping the mass conservation constraint. In a series of numerical examples we investigate the modeling error of the reduced model as it compares to the original full dimensional model. Moreover, we show the performance of the finite-volume moving-mesh algorithm for the complex wave pattern that is induced by the interaction of saturation fronts and evolving fractures.

Abstract

We prove that two-layer (Leaky)ReLU networks initialized by e.g. the widely used method proposed by He et al. (2015) and trained using gradient descent on a least-squares loss are not universally consistent. Specifically, we describe a large class of one-dimensional data-generating distributions for which, with high probability, gradient descent only finds a bad local minimum of the optimization landscape, since it is unable to move the biases far away from their initialization at zero. It turns out that in these cases, the found network essentially performs linear regression even if the target function is non-linear. We further provide numerical evidence that this happens in practical situations, for some multi-dimensional distributions and that stochastic gradient descent exhibits similar behavior. We also provide empirical results on how the choice of initialization and optimizer can influence this behavior.

Abstract

Data-dependent greedy algorithms in kernel spaces are known to provide fast converging interpolants, while being extremely easy to implement and efficient to run. Despite this experimental evidence, no detailed theory has yet been presented. This situation is unsatisfactory, especially when compared to the case of the data-independent P-greedy algorithm, for which optimal convergence rates are available, despite its performances being usually inferior to the ones of target data-dependent algorithms. In this work, we fill this gap by first defining a new scale of greedy algorithms for interpolation that comprises all the existing ones in a unique analysis, where the degree of dependency of the selection criterion on the functional data is quantified by a real parameter. We then prove new convergence rates where this degree is taken into account, and we show that, possibly up to a logarithmic factor, target data-dependent selection strategies provide faster convergence. In particular, for the first time we obtain convergence rates for target data adaptive interpolation that are faster than the ones given by uniform points, without the need of any special assumption on the target function. These results are made possible by refining an earlier analysis of greedy algorithms in general Hilbert spaces. The rates are confirmed by a number of numerical examples.

Abstract

Starting from a linear fractional representation of a linear system affected by constant parametric uncertainties, we demonstrate how to enhance standard robust analysis tests by taking available (noisy) input-output data of the uncertain system into account. Our approach relies on a lifting of the system and on the construction of data-dependent multipliers. It leads to a test in terms of linear matrix inequalities which guarantees stability and performance for all systems compatible with the observed data if it is in the affirmative. In contrast to many other data-based approaches, prior physical knowledge is included at the outset due to the underlying linear fractional representation.

Abstract

In this work we present a novel exact Riemann solver for an artificial Equation of State (EoS) based modification of the incompressible Euler equations. Differently from the well known artificial compressibility method, this new approach overcomes the lack of the pressure-velocity coupling by using a suitably designed artificial EoS. The modified set of equations fits into the framework of first order hyperbolic conservation laws. Accordingly, an exact Riemann solver is derived. This new solver has the advantage to avoid a wave pattern violation issue which can affect the exact Riemann problem solution obtained by the standard artificial compressibility approach. The new artificial EoS based Riemann solver can be used as a tool for the definition of the advective Godunov fluxes in a Finite Volume or a discontinuous Galerkin discretization of the incompressible Navier–Stokes (INS) equations. We assess and analyse the new exact Riemann solver on 1D Riemann problems. Its capability and effectiveness are then shown in the context of a high-order accurate discontinuous Galerkin discretization of the INS equations. In particular, we verify the convergence properties, both in space and time, considering two test cases in 2D, namely, the Kovasznay test case and a damped travelling waves test case. Finally, we perform an implicit large eddy simulation of the incompressible turbulent flow over periodic hills where the classic forcing term formulation is modified to deal with variable time steps. Solution comparisons with respect to numerical and experimental results from the literature are given.

Abstract

Python Code to generate the meshes and FEM solutions to Section 5 of the paper Uncertainty visualization: Fundamentals and recent developments.Commentaries are in the code to explain it.Paraview is used for the visualization.

Abstract

Quantum systems composed of N distinct particles in \$\$\backslashmathbb \R\^2\$\$with two-body contact interactions of TMS type are shown to arise as limits---in the norm resolvent sense---of Schrödinger operators with suitably rescaled pair potentials.

Abstract

Summary: "We study the flow of two immiscible fluids located on a solid bottom, where the lower fluid is Newtonian and the upper fluid is a non-Newtonian Ellis fluid. Neglecting gravitational effects, we consider the formal asymptotic limit of small film heights in the twop-hase Navier-Stokes system. This leads to a strongly coupled system of two parabolic equations of fourth order with merely Hölder-continuous dependence on the coefficients. For the case of strictly positive initial film heights we prove local existence of strong solutions by abstract semigroup theory. Uniqueness is proved by energy methods. Under additional regularity assumptions, we investigate asymptotic stability of the unique equilibrium solution, which is given by constant film heights.''

Abstract

Python Code to generate the meshes and FEM solutions to Section 5 of the paper Uncertainty visualization: Fundamentals and recent developments.Commentaries are in the code to explain it.Paraview is used for the visualization.

Abstract

The dynamics of compressible liquid–vapor flow depends sensitively on the microscale behavior at the phase boundary. We consider a sharp-interface approach, and propose a multiscale model to describe liquid–vapor flow accurately, without imposing ad-hoc closure relations on the continuum scale. The multiscale model combines the Euler equations on the continuum scale with molecular-scale particle simulations that govern the interface motion. We rely on an interface-preserving moving mesh finite volume method to discretize the continuum-scale sharp-interface flow in a conservative manner. Computational efficiency, while preserving physical properties, is achieved by a surrogate solver for the interface dynamics based on constraint-aware neural networks. The multiscale model is presented in its general form, and applied to regimes of temperature-dependent liquid–vapor flow which have not been accessible before.

Abstract

We show how to compose robust stability tests for uncertain systems modeled as linear fractional representations and affected by various types of dynamic uncertainties. Our results are formulated in terms of linear matrix inequalities and rest on the recently established notion of finite-horizon integral quadratic constraints with a terminal cost. The construction of such constraints is motivated by an unconventional application of the S-procedure in the frequency domain with dynamic Lagrange multipliers. Our technical contribution reveals how this construction can be performed by dissipativity arguments in the time-domain and in a lossless fashion. This opens the way for generalizations to time-varying or hybrid systems.

Abstract

Naturally reductive spaces, in general, can be seen as an adequate generalization of Riemannian symmetric spaces. Nevertheless, there are some that are closer to symmetric spaces than others. On the one hand, there is the series of Hopf fibrations over complex space forms, including the Heisenberg groups with their metrics of type H. On the other hand, there exist certain naturally reductive spaces in dimensions 6 and 7 whose torsion forms have a distinguished algebraic property. All these spaces generalize geometric or algebraic properties of three-dimensional naturally reductive spaces and have the following point in common: along every geodesic, the Jacobi operator satisfies an ordinary differential equation with constant coefficients which can be chosen independently of the given geodesic.

Abstract

The Lorenz attractor is considered which is a well known attractor of a dynamical system. As the Lorenz attractor shows some fractal structure, its box dimension is investigated. A general algorithmic approach is developed which enables us to calculate its box dimension, and special attention will be paid to numerical aspects of the algorithm. The algorithm gets tested with the help of some well known fractals - self-similar fractals and fractal graphs of deterministic as well as random functions. For these examples, the algorithm yields values which are very close to the theoretical values. However, in the case of the Lorenz attractor the algorithm seems to fail. The reason for this phenomenon is discussed, the notion of local box dimension is introduced, and the algorithm is used in order to reveal a hidden multifractal structure of the Lorenz attractor. (C) 2020 Elsevier B.V. All rights reserved.

Abstract

We consider the incompressible flow of two immiscible fluids in the presence of a solid phase that undergoes changes in time due to precipitation and dissolution effects. Based on a seminal sharp interface model a phase-field approach is suggested that couples the Navier–Stokes equations and the solid’s ion concentration transport equation with the Cahn–Hilliard evolution for the phase fields. The model is shown to preserve the fundamental conservation constraints and to obey the second law of thermodynamics for a novel free energy formulation. An extended analysis for vanishing interfacial width reveals that in this limit the sharp interface model is recovered, including all relevant transmission conditions. Notably, the new phase-field model is able to realize Navier-slip conditions for solid–fluid interfaces in the limit.

Abstract

We present a convex solution for the design of generalized accelerated gradient algorithms for strongly convex objective functions with Lipschitz continuous gradients. We utilize integral quadratic constraints and the Youla parameterization from robust control theory to formulate a solution of the algorithm design problem as a convex semi-definite program. We establish explicit formulas for the optimal convergence rates and extend the proposed synthesis solution to extremum control problems.

Abstract

Genetic risk analysis is increasingly in demand by participants. Hybrid genetic testing has the advantage over direct to consumer testing by involving a physician who guides the process and offers counseling after receiving the results. The objective of this study was to determine whether a structured physician moderated primary preventive, hybrid genetic risk assessment enhanced counseling program leads to improvement in lifestyle and does not impair quality of life. Risk genes for malignant, cardiovascular, coagulation, storage diseases and pharmacogenetics (>þinspace100 genes) were tested. Screening, consultation and genetic counseling embedded in a primary/secondary prevention check-up program for executives of surrounding companies took place in a single center in Germany. Follow-up included established questionnaires for quality of life, nutrition and physical activity. Analysis included nþinspace=þinspace244 participants. Median age at baseline was 49 years (interquartile range: 44--55), 93\% were male, 3\% (nþinspace=þinspace7 of 136 responses) were smoker. Mean body mass index was 25.2 kg/m2. Follow-up response rate was 74\% (nþinspace=þinspace180), mean follow-up time was 6.8 months (standard deviationþinspace=þinspace2.1). In 91 participants (37.8\%, 91/241) at least one pathogenic variant was found, 60 thereof were clinically relevant (24.9\%, 60/241). 238 participants (98\%, 238/241) hadþinspace>þinspace1 pharmacogenetic variant, only 2 (0.8\%, 2/241) took a correspondingly affected drug (56 participants tookþinspace≥þinspace1 drug/day). The energy expenditure significantly increased by ≈ 35\% median multiple of energy expenditure of 1.34 (confidence intervalþinspace=þinspace1.15--1.57, pþinspace<þinspace0.001) metabolic equivalents of task (MET)-min/week; participants spent on average 41 min (pþinspace<þinspace0.001) less in sedentary activities per day and spent more time for lunch (≈ 2 additional minutes/day; pþinspace=þinspace0.031). Indicators of the consumption of red meat and sweet pastries significantly decreased (both adjusted pþinspace=þinspace0.049). Neither quality of life in general nor subgroup analysis of participants with at least one conspicuous genetic risk differed significantly over follow-up. Hybrid genetic testing and counseling exerted positive effects on health-related behavior and was not associated with major psychological adverse effects in the short-term follow-up. The approach seems to be feasible for use in preventive health care.

Abstract

We present a Stochastic Galerkin (SG) scheme for Uncertainty Quantification (UQ) of the compressible Navier-Stokes and Euler equations. For spatial discretization, we rely on the high-order Discontinuous Galerkin Spectral Element Method (DGSEM) in combination with an explicit time-stepping scheme. We simulate complex flow problems in two- and three-dimensional domains using curved unstructured hexahedral meshes. The dimension of the stochastic space can be arbitrarily large. In order to treat discontinuities and to ensure hyperbolicity, we employ a multi-element approach in combination with a hyperbolicity-preserving limiter in the stochastic domain and a Finite Volume subcell shock capturing scheme in physical space. Special emphasis is put on code performance and massive parallel scalability. We demonstrate the versatility and broad applicability of our code with various numerical experiments including the supersonic flow around a spacecraft geometry. By this, we wish to contribute to the path of the Stochastic Galerkin method towards practical applicability in research and industrial engineering.

Abstract

A proper isometric Lie group action on a Riemannian manifold is called polar if there exists a closed connected submanifold which meets all orbits orthogonally. In this article we study polar actions on Damek-Ricci spaces. We prove criteria for isometric actions on Damek-Ricci spaces to be polar, find examples and give some partial classifications of polar actions on Damek-Ricci spaces. In particular, we show that non-trivial polar actions exist on all Damek-Ricci spaces.

Abstract

In this paper we revisit the classical problem of nonparametric regression, but impose local differential privacy constraints. Under such constraints, the raw data (X-1, Y-1), ..., (X-n, Y-n), taking values in R-d x R, cannot be directly observed, and all estimators are functions of the randomised output from a suitable privacy mechanism. The statistician is free to choose the form of the privacy mechanism, and here we add Laplace distributed noise to a discretisation of the location of a feature vector X-i and to the value of its response variable Y-i. Based on this randomised data, we design a novel estimator of the regression function, which can be viewed as a privatised version of the well-studied partitioning regression estimator. The main result is that the estimator is strongly universally consistent, and we further establish an upper bound on the rate of convergence. Our methods and analysis also give rise to a strongly universally consistent binary classification rule for locally differentially private data.

Abstract

The dual iteration was introduced in a conference paper in 1997 by Iwasaki as an iterative and heuristic procedure for the challenging and non-convex design of static output-feedback controllers. We recall in detail its essential ingredients and go beyond the work of Iwasaki by demonstrating that the framework of linear fractional representations allows for a seamless extension the dual iteration to output-feedback designs of tremendous practical relevance such as the design of robust or robust gain-scheduled controllers. In the paper of Iwasaki the dual iteration is solely based on, and motivated by algebraic manipulations resulting from the elimination lemma. We provide a novel control theoretic interpretation of the individual steps, which paves the way for further generalizations of the powerful scheme to situations where the elimination lemma is not applicable. Exemplary, we extend the dual iteration to the multi-objective design of static output-feedback H∞-controllers, which guarantee that the closed-loop poles are contained in an a priori specified generalized stability region. We demonstrate the approach with numerous numerical examples inspired from the literature.

Abstract

We provide a convex solution to the robust gain-scheduled estimation problem based on integral quadratic constraints with dynamic D-scalings for both the uncertain and the scheduled component. This closes an important gap since so far merely static scalings or multipliers could be used for the scheduled component in estimation problems. To this end, we provide novel synthesis criteria in terms of linear matrix inequalities for the design of nominal gain-scheduled estimators which allow for a direct combination with available results on robust estimation. We illustrate the benefit of our design approach by means of a numerical example.

Abstract

I Mathematische Grundlagen. Elementare Logik -- Mengen und Abbildungen -- Kombinatorik und Wahrscheinlichkeit -- Komplexe Zahlen -- II Vektorrechnung. Vektoren -- Längen, Winkel und Skalarprodukt -- Vektor- und Spatprodukt -- Geraden und Ebenen -- III Differentialrechnung. Polynome und rationale Funktionen -- Exponentialfunktion, Logarithmus und trigonometrische Funktionen -- Grenzwerte, Reihen und Stetigkeit -- Differentiationsregeln und Anwendungen -- Taylor-Entwicklung -- Extremwerte und Funktionsuntersuchung -- IV Integralrechnung. Integral und Stammfunktion -- Partielle Integration, Substitution und spezielle Integranden -- Uneigentliche Integrale -- V Anwendung mathematischer Software. MATLAB® -- Maple™. VI. Formelsammlung. Mathematische Grundlagen -- Vektorrechnung -- Differentialrechnung -- Integralrechnung. Literaturverzeichnis.

Abstract

The combination of machine learning with control offers many opportunities, in particular for robust control. However, due to strong safety and reliability requirements in many real-world applications, providing rigorous statistical and control-theoretic guarantees is of utmost importance, yet difficult to achieve for learning-based control schemes. We present a general framework for learning-enhanced robust control that allows for systematic integration of prior engineering knowledge, is fully compatible with modern robust control and still comes with rigorous and practically meaningful guarantees. Building on the established Linear Fractional Representation and Integral Quadratic Constraints framework, we integrate Gaussian Process Regression as a learning component and state-of-the-art robust controller synthesis. In a concrete robust control example, our approach is demonstrated to yield improved performance with more data, while guarantees are maintained throughout.

Abstract

Approaches based on order-adaptive regularisation belong to the most accurate variational methods for computing the optical flow. By locally deciding between first- and second-order regularisation, they are applicable to scenes with both fronto-parallel and ego-motion. So far, however, existing order-adaptive methods have a decisive drawback. While the involved first- and second-order smoothness terms already make use of anisotropic concepts, the underlying selection process itself is still isotropic in that sense that it locally chooses the same regularisation order for all directions. In our paper, we address this shortcoming. We propose a generalised order-adaptive approach that allows to select the local regularisation order for each direction individually. To this end, we split the order-adaptive regularisation across and along the locally dominant direction and perform an energy competition for each direction separately. This in turn offers another advantage. Since the parameters can be chosen differently for both directions, the approach allows for a better adaption to the underlying scene. Experiments for MPI Sintel and KITTI 2015 demonstrate the usefulness of our approach. They not only show improvements compared to an isotropic selection scheme. They also make explicit that our approach is able to improve the results from state-of-the-art learning-based approaches, if applied as a final refinement step – thereby achieving top results in both benchmarks.

Abstract

For a partially unknown linear systems, we present a systematic control design approach based on generated data from measurements of closed-loop experiments with suitable test controllers. These experiments are used to improve the achieved performance and to reduce the uncertainty about the unknown parts of the system. This is achieved through a parametrization of auspicious controllers with convex relaxation techniques from robust control, which guarantees that their implementation on the unknown plant is safe. This approach permits to systematically incorporate available prior knowledge about the system by employing the framework of linear fractional representations.

Abstract

This volume is dedicated to Ari Laptev on the occasion of his 70th birthday. It collects contributions by his numerous colleagues sharing with him research interests in analysis and spectral theory. In brief, the topics covered include Friedrichs, Hardy, and Lieb–Thirring inequalities, eigenvalue bounds and asymptotics, Feshbach–Schur maps and perturbation theory, scattering theory and orthogonal polynomials, stability of matter, electron density estimates, Bose–Einstein condensation, Wehrl-type entropy inequalities, Bogoliubov theory, wave packet evolution, heat kernel estimates, homogenization, d-bar problems, Brezis–Nirenberg problems, the nonlinear Schrödinger equation in magnetic fields, classical discriminants, and the two-dimensional Euler–Bardina equations. In addition, Ari’s multifaceted service to the mathematical community is also touched upon. Altogether the volume presents a collection of research articles which will be of interest to any active scientist working in one of the above mentioned fields.

Abstract

Let G be a connected reductive algebraic group defined over a finite field with q elements. In the 1980's, Kawanaka introduced generalised Gelfand--Graev representations of the finite group \$\$ G\backslashleft(\\backslashmathbbm\F\\\_q\backslashright) \$\$, assuming that q is a power of a good prime for G. These representations have turned out to be extremely useful in various contexts. Here we investigate to what extent Kawanaka's construction can be carried out when we drop the assumptions on q. As a curious by-product, we obtain a new, conjectural characterisation of Lusztig's concept of special unipotent classes of G in terms of weighted Dynkin diagrams.

Abstract

In this report we present advances in our research on direct aeroacoustics and uncertainty quantification, based on the high-order Discontinuous Galerkin solver FLEXI. Oscillation phenomena triggered by flow over cavities can lead to an unpleasant tonal (whistling) noise, which provides motivation for industry and academia to better understand the underlying mechanisms. We present a numerical setup capable of capturing these phenomena with high efficiency, as we show by comparison to experimental data and results from industry. Some of these phenomena are highly sensitive towards flow conditions, which makes an integrated approach regarding these conditions necessary. This is the goal of uncertainty quantification. We present software for both intrusive and non-intrusive uncertainty quantification methods. We investigate convergence and computational performance. The development of both codes was in parts carried out in cooperation with HLRS. Apart from validation results, we show a non-intrusive simulation of 3D turbulent cavity noise.

Abstract

Sparse grids enable accurate function approximation in higher-dimensional settings with a moderate number of grid points. In this paper, we introduce generalized versions of various efficient sparse grid algorithms in a unified notation: For local basis functions, transformations between function values at grid points and interpolation coefficients can be efficiently realized using the well-known unidirectional principle (Balder and Zenger, SIAM Journal on Scientific Computing, 17(3):631--646, May 1996; Bungartz, Finite Elements of Higher Order on Sparse Grids, Universität München, Aachen, November 1998). For general basis functions without spatial adaptivity, Avila and Carrington show in (The Journal of Chemical Physics, 143(21):214108, 2015) and follow-up work that the unidirectional principle can instead be applied to an equivalent incremental basis, and we show that a second application of the unidirectional principle can be used to compute interpolation coefficients for the original, non-incremental basis. Our Open Source implementation for the latter case is able to compute interpolation coefficients for up to 20 million sparse grid points in one second on a single CPU core. We also discuss the efficient application of linear operators occurring in derivative evaluations of interpolants and in the solution of partial differential equations on sparse grids with finite elements, e.g., using the UpDown scheme (Bungartz, Dünne Gitter und deren Anwendung bei der adaptiven Lösung der dreidimensionalen Poisson-Gleichung, PhD thesis, Technische Universität München, 1992; Pflüger, Spatially Adaptive Sparse Grids for High-Dimensional Problems, Verlag Dr. Hut, München, February 2010; Zeiser, Journal of Scientific Computing, 47(3):328--346, 2011; Bungartz et al., Journal of Computational and Applied Mathematics, 236(15):3741--3750, 2012).

Abstract

A novel convex state-space solution to the robust output-feedback synthesis problem based on dynamic integral quadratic constraints for a particular class of systems is presented. Convexification rests upon the assumption that the control-to-uncertainty block in the generalized plant commutes with the off-diagonal block of the multiplier. We demonstrate that this commutation property is valid for several by themselves interesting concrete scenarios, such as in extremum control, a generalization of the convex design of first-order optimization algorithms involving filtered gradient evaluations.

Abstract

We study a nonlinear fourth-order extension of Richards' equation that describes infiltration processes in unsaturated soils. We prove the well-posedness of the fourth-order equation by first applying Kirchhoff's transformation to linearize the higher-order terms. The transformed equation is then discretized in time and space and a set of a priori estimates is established. These allow, by means of compactness theorems, extracting a unique weak solution. Finally, we use the inverse of Kirchhoff's transformation to prove the well-posedness of the original equation.

Abstract

We derive novel criteria for designing stabilizing dynamic output-feedback controllers for a class of aperiodic impulsive systems subject to a range dwell-time condition. Our synthesis conditions are formulated as clock-dependent linear matrix inequalities (LMIs) which can be solved numerically, e.g., by using matrix sum-of-squares relaxation methods. We show that our results allow us to design dynamic output-feedback controllers for aperiodic sample-data systems and illustrate the proposed approach by means of a numerical example.

Abstract

In many natural and technical applications in porous media fluid's flow behavior is highly affected by fractures. Many approaches employ mixed-dimensional models that model thin features as dimension-reduced manifolds. Following this idea, we consider porous media where dominant heterogeneities are geometrically represented by sharp interfaces. We model incompressible two-phase flow in porous media both in the bulk porous medium and within the fractures. We present a reliable and geometrically flexible implementation of a fully conforming finite volume approach within the DUNE framework for two and three spatial dimensions. The implementation is based on the new dune-mmesh grid implementation that manages bulk and surface triangulation simultaneously. The model and the implementation are extended to handle fracture junctions. We apply our scheme to benchmark cases with complex fracture networks to show the reliability of the approach.

Abstract

The classic regularization theory for solving inverse problems is built on the assumption that the forward operator perfectly represents the underlying physical model of the data acquisition. However, in many applications, for instance in microscopy or magnetic particle imaging, this is not the case. Another important example represent dynamic inverse problems, where changes of the searched-for quantity during data collection can be interpreted as model uncertainties. In this article, we propose a regularization strategy for linear inverse problems with inexact forward operator based on sequential subspace optimization methods (SESOP). In order to account for local modelling errors, we suggest to combine SESOP with the Kaczmarz’ method. We study convergence and regularization properties of the proposed method and discuss several practical realizations. Relevance and performance of our approach are evaluated at simulated data from dynamic computerized tomography with various dynamic scenarios.

Abstract

We show that simply connected Riemannian homogeneous spaces of compact semisimple Lie groups with polar isotropy actions are symmetric, generalizing results of Fabio Podestà and the third named author. Without assuming compactness, we give a classification of Riemannian homogeneous spaces of semisimple Lie groups whose linear isotropy representations are polar. We show for various such spaces that they do not have polar isotropy actions. Moreover, we prove that Heisenberg groups and non-symmetric Damek--Ricci spaces have non-polar isotropy actions.

Abstract

The problem of dynamic 2D vector tomography is considered. Object motion is a combination of rotation and shifting. Properties of the dynamic ray transform operators are investigated. Singular value decomposition of the operators is constructed with usage of classic orthogonal polynomials.

Abstract

The N-wave interaction (NWI) system appears as an amplitude system in the description of N nonlinearly interacting and linearly transported wave packets in dispersive wave systems such as the water wave problem or problems in nonlinear optics. The purpose of this paper is twofold. First we give a new simplified proof for the failure of the NWI approximation in case of resonances which are located at integer multiples of a basic wave number k(0) in the original 2 pi/k0-spatially periodic dispersive wave system. Secondly, we give a first rigorous proof that an amplitude system fails in the description of an original system, without imposing periodic boundary conditions on the original system.

Abstract

We present an a posteriori error analysis for one-dimensional ran-dom hyperbolic systems of conservation laws. For the discretization of therandom space we consider the Non-Intrusive Spectral Projection method, thespatio-temporal discretization uses the Runge–Kutta Discontinuous Galerkinmethod. We derive an a posteriori error estimator using smooth reconstructionsof the numerical solution, which combined with the relative entropy stabilityframework yields computable error bounds for the space-stochastic discretiza-tion error. Moreover, we show that the estimator admits a splitting into astochastic and deterministic part.

Abstract

Bereitstellung einer Sammlung von Android-Apps zum Einsatz in der universitäten Mathematik-Ausbildung, gleichermaßen als Demonstratoren und zum Selbststudium.

Abstract

In this article we consider one-dimensional random systems of hyperbolic conservation laws. We first establish existence and uniqueness of random entropy admissible solutions for initial value problems of conservation laws which involve random initial data and random flux functions. Based on these results we present an a posteriori error analysis for a numerical approximation of the random entropy solution. For the stochastic discretization, we consider a non-intrusive approach, the Stochastic Collocation method. The spatial-temporal discretization relies on the Runge--Kutta Discontinuous Galerkin method. We derive the a posteriori estimator using smooth reconstructions of the discrete solution. Combined with the relative entropy stability framework this yields computable error bounds for the entire space-stochastic discretization error. The estimator admits a splitting into a stochastic and a deterministic (space-time) part, allowing for a novel residual-based space-stochastic adaptive mesh refinement algorithm. We conclude with various numerical examples investigating the scaling properties of the residuals and illustrating the efficiency of the proposed adaptive algorithms.

Abstract

We obtain well-posedness results in Lp-based weighted Sobolev spaces for a transmission problem for anisotropic Stokes and Navier–Stokes systems with L∞ strongly elliptic coefficient tensor, in complementary Lipschitz domains of Rn, n≥3. The strong ellipticity allows to explore the associated pseudostress setting. First, we use a variational approach that reduces the anisotropic Stokes system in the whole Rn to an equivalent mixed variational formulation with data in Lp-based weighted Sobolev spaces. We show that such mixed variational formulation is well-posed in the space Hp1(Rn)n×Lp(Rn), n≥ 3, for any p in an open interval containing 2. Then similar well-posedness results are obtained for two linear transmission problems. These results are used to define the Newtonian and layer potential operators for the considered anisotropic Stokes system and to obtain mapping properties of these operators. The potentials are employed to show the well-posedness of some linear transmission problems, which then is combined with a fixed point theorem in order to show the well-posedness of a nonlinear transmission problem for the anisotropic Stokes and Navier–Stokes systems in Lp-based weighted Sobolev spaces, whenever the given data are small enough.

Abstract

Neural networks are increasingly used in complex (data-driven) simulations as surrogates or for accelerating the computation of classical surrogates. In many applications physical constraints, such as mass or energy conservation, must be satisfied to obtain reliable results. However, standard machine learning algorithms are generally not tailored to respect such constraints. We propose two different strategies to generate constraint-aware neural networks. We test their performance in the context of front-capturing schemes for strongly nonlinear wave motion in compressible fluid flow. Precisely in this context so-called Riemann problems have to be solved as surrogates. Their solution describes the local dynamics of the captured wave front in numerical simulations. Three model problems are considered: a cubic flux model problem, an isothermal two-phase flow model, and the Euler equations. We observe, that constraint-aware neural networks do not only account for the constraint but lead to an improvement of the numerical accuracy of the overall fluid simulation.

Abstract

Learning rates for least-squares regression are typically expressed in terms of $L_2$-norms. In this paper we extend these rates to norms stronger than the $L_2$-norm without requiring the regression function to be contained in the hypothesis space. In the special case of Sobolev reproducing kernel Hilbert spaces used as hypotheses spaces, these stronger norms coincide with fractional Sobolev norms between the used Sobolev space and $L_2$. As a consequence, not only the target function but also some of its derivatives can be estimated without changing the algorithm. From a technical point of view, we combine the well-known integral operator techniques with an embedding property, which so far has only been used in combination with empirical process arguments. This combination results in new finite sample bounds with respect to the stronger norms. From these finite sample bounds our rates easily follow. Finally, we prove the asymptotic optimality of our results in many cases.

Abstract

Gaussian process emulators (GPE) are a machine learning approach that replicates computational demanding models using training runs of that model. Constructing such a surrogate is very challenging and, in the context of Bayesian inference, the training runs should be well invested. The current paper offers a fully Bayesian view on GPEs for Bayesian inference accompanied by Bayesian active learning (BAL). We introduce three BAL strategies that adaptively identify training sets for the GPE using information-theoretic arguments. The first strategy relies on Bayesian model evidence that indicates the GPE's quality of matching the measurement data, the second strategy is based on relative entropy that indicates the relative information gain for the GPE, and the third is founded on information entropy that indicates the missing information in the GPE. We illustrate the performance of our three strategies using analytical- and carbon-dioxide benchmarks. The paper shows evidence of convergence against a reference solution and demonstrates quantification of post-calibration uncertainty by comparing the introduced three strategies. We conclude that Bayesian model evidence-based and relative entropy-based strategies outperform the entropy-based strategy because the latter can be misleading during the BAL. The relative entropy-based strategy demonstrates superior performance to the Bayesian model evidence-based strategy.

Abstract

We study the limiting behavior of the Dirichlet and Neumann eigenvalue counting function of generalized second-order differential operators d/d mu d/dx, where mu is a finite atomless Borel measure on some compact interval a, b. Therefore, we firstly recall the results of the spectral asymptotics for these operators received so far. Afterwards, we make a proposition about the convergence behavior for so-called random V-variable Cantor measures.

Abstract

We are interested in the description of small modulations in time and space of wave-train solutions to the complex Ginzburg-Landau equation partial derivative(T)Psi = (1 + i alpha)partial derivative(2)(X)Psi + Psi - (1+i beta)Psi vertical bar Psi vertical bar(2) near the Eckhaus boundary, that is, when the wave train is near the threshold of its first instability. Depending on the parameters of, alpha, beta, a number of modulation equations can he derived, such as the KdV equation, the Cahn-Hilliard equation, and a family of Ginzburg-Landau based amplitude equations. Here we establish error estimates showing that the Korteweg-de Vries (KdV) approximation makes correct predictions in a certain parameter regime. Our proof is based on energy estimates and exploits the conservation law structure of the critical mode. In order to improve linear damping, we work in spaces of analytic functions.

Abstract

The Richards' equation is a model for flow of water in unsaturated soils. The coefficients of this (nonlinear) partial differential equation describe the permeability of the medium. Insufficient or uncertain measurements are commonly modeled by random coefficients. For flows in heterogeneous\fracturedmedia, the coefficients are modeled as discontinuous random fields, where the interfaces along the stochastic discontinuities represent transitions in the media. More precisely, the random coefficient is given by the sum of a (continuous) Gaussian random field and a (discontinuous) jump part. In this work moments of the solution to the random partial differential equation are calculated using a path-wise numerical approximation combined with multilevel Monte Carlo sampling. The discontinuities dictate the spatial discretization, which leads to a stochastic grid. Hence, the refinement parameter and problem-dependent constants in the error analysis are random variables and we derive (optimal) a-priori convergence rates in a mean-square sense.

Abstract

In this paper, the authors study a model describing the evolution of the height of a non-Newtonian incompressible thin liquid film that spreads on a solid surface by the Ellis constitutive law. Existence of strong positive solutions in Sobolev-Slobodetskiĭ spaces is shown by using Amann's abstract existence theory. In addition, the authors prove uniqueness of the solutions by the energy approach.

Abstract

We propose a novel $hp$-Multilevel Monte Carlo method for the quantification of uncertainties in compressible flows using the Discontinuous Galerkin method as deterministic solver. The multilevel approach exploits hierarchies of uniformly refined meshes jointly with an increasing polynomial degree for the ansatz space. It allows for a very large range of resolutions in the physical space and thus an efficient decrease of the statistical error. We prove that the overall complexity of the $hp$-Multilevel Monte Carlo method to compute the mean field with prescribed accuracy is of quadratic order with respect to the accuracy. We also propose a novel and simple approach to estimate a lower confidence bound for the optimal number of samples per level, which helps to prevent overshooting the optimal number of samples. The method is in particular adapted to the needs of high-performance computing. Our theoretical results are verified by numerical experiments for the two dimensional compressible Navier-Stokes equations. In particular we consider a cavity flow problem from computational acoustics, demonstrating that the method is suitable to handle complex engineering problems.

Abstract

Entropy numbers are an important tool for quantifying the compactness of operators. Besides establishing new upper bounds on the entropy numbers of diagonal operators $D_\sigma$ from $\ell_p$ to $\ell_q$, where $p\not=q$, we investigate the optimality of these bounds. In case of $p<q$ optimality is proven for fast decaying diagonal sequences, which include exponentially decreasing sequences. In case of $p>q$ we show optimality under weaker assumption than previously used in the literature. In addition, we illustrate the benefit of our results with examples not covered in the literature so far.

Abstract

Using octonions and the triality property of Spin(8), we find explicit formulae for the Lie brackets of the exceptional simple real Lie algebras f4 and f∗4, i.e. the Lie algebras of the isometry groups of the Cayley projective plane and the Cayley hyperbolic plane. As an application, we determine all polar actions on the Cayley hyperbolic plane which leave a totally geodesic subspace invariant.

Abstract

The isothermal Navier-Stokes-Korteweg system is a classical diffuse interface model for compressible two-phase flow which grounds in Van Der Waals' theory of capillarity. However, the numerical solution faces two major challenges: due to a third-order dispersion contribution in the momentum equations, extended numerical stencils are required for the flux calculation. Furthermore, the equation of state given by a Van-der-Waals law, exhibits non-monotone behaviour in the two-phase state space leading to imaginary eigenvalues of the Jacobian of the first-order fluxes. In this work a lower-order parabolic relaxation model is used to approximate solutions of the classical NSK equations. Whereas an additional parabolic evolution equation for the relaxation variable has to be solved, the system involves no differential operator of higher as second order. The use of a modified pressure function guarantees that the first-order fluxes remain hyperbolic. Altogether, the relaxation system is directly accessible for standard compressible flow solvers. We use the higher-order Discontinuous Galerkin spectral element method as realized in the open source code FLEXI. The relaxation model is validated against solutions of the original NSK model for a variety of 1D and 2D test cases. Three-dimensional simulations of head-on droplet collisions for a range of different collision Weber numbers underline the capability of the approach.

Abstract

"The authors study eigenvalues of the Laplacian (the negative second derivative operator) of a compact metric graph equipped with complex δ-vertex conditions. More precisely: (a) a continuity condition is imposed at each vertex; (b) on a selected set of vertices R, the vertex conditions ∑e∼vj∂∂νf|e(vj)+αjf(vj)=0 are imposed, where the interaction strengths αj are complex parameters and the derivative is taken in the direction towards the vertex; and (c) on the vertices in ∖R, Kirchhoff vertex conditions are imposed. The main focus in this article is on the asymptotic behavior of the (purely discrete) spectrum as certain coefficients αj tend to ∞ in the complex plane. If m of these coefficients tend to infinity within some sector in the open left half-plane while the remaining coefficients tend to infinity in a way such that Reαj remains bounded from below, then the authors show that exactly m eigenvalues diverge away from the positive real semi-axis. Moreover, they provide the asymptotics of these eigenvalues; the leading term is quadratic in the αj, with a coefficient depending on the vertex degrees. As variational principles are not available for the study of such non-self-adjoint problems, the authors use a Birman-Schwinger type characterization of eigenvalues in terms of a parameter-dependent Dirichlet-to-Neumann matrix (or Titchmarsh-Weyl function). In addition, they also obtain estimates for the numerical range and the eigenvalues."

Abstract

Hamiltonian systems are central in the formulation of non-dissipative physical systems. They are characterized by a phase-space, a symplectic form and a Hamiltonian function. In numerical simulations of Hamiltonian systems, algorithms show improved accuracy when the symplectic structure is preserved 10. For structure-preserving model order reduction (MOR) of Hamiltonian systems, symplectic MOR 17, 14, 11, 3, 16 can be used. It is based on a reduced-order basis that is symplectic, which requires symplectic basis generation techniques. In our work, we discuss greedy algorithms for symplectic basis generation. We complement the procedure presented in 14 with ideas of the POD-greedy 9, which results in a new greedy symplectic basis generation technique, the PSD-greedy. Inspired by POD-greedy, we use compression techniques in the greedy iterations to enrich the basis iteratively. We prove that this algorithm computes a symplectic basis when symplectic techniques are used for compression. In the numerical experiments, we compare the discussed methods for a linear elasticity problem. The results show that improvements of up to one order of magnitude in the relative reduction error are achievable with the new basis generation technique compared to the existing greedy approach from 14.

Abstract

In this contribution we present a local discontinuous Galerkin (LDG) pressure-correction scheme for the incompressible Navier--Stokes equations. The scheme does not need penalty parameters and satisfies the discrete continuity equation exactly. The scheme is especially suitable for two-phase flow when used with a piecewise-linear interface construction (PLIC) volume-of-fluid (VoF) method and cut-cell quadratures.

Abstract

In this article we present an a posteriori error estimator for the spatial-stochastic error of a Galerkin-type discretisation of an initial value problem for a random hyperbolic conservation law. For the stochastic discretisation we use the Stochastic Galerkin method and for the spatial-temporal discretisation of the Stochastic Galerkin system a Runge-Kutta Discontinuous Galerkin method. The estimator is obtained using smooth reconstructions of the discrete solution. Combined with the relative entropy stability framework of Dafermos dafermos2005hyperbolic, this leads to computable error bounds for the space-stochastic discretisation error. \\ Moreover, it turns out that the error estimator admits a splitting into one part representing the spatial error, and a remaining term, which can be interpreted as the stochastic error. This decomposition allows us to balance the errors arising from spatial and stochastic discretisation. We conclude with some numerical examples confirming the theoretical findings.

Abstract

We present version 3 of the open-source simulator for flow and transport processes in porous media DuMux. DuMux is based on the modular C++ framework Dune (Distributed and Unified Numerics Environment) and is developed as a research code with a focus on modularity and reusability. We describe recent efforts in improving the transparency and efficiency of the development process and community-building, as well as efforts towards quality assurance and reproducible research. In addition to a major redesign of many simulation components in order to facilitate setting up complex simulations in DuMux, version 3 introduces a more consistent abstraction of finite volume schemes. Finally, the new framework for multi-domain simulations is described, and three numerical examples demonstrate its flexibility.

Abstract

The purpose of this paper is to construct small-amplitude breather solutions for a nonlinear Klein-Gordon equation posed on a periodic metric graph via spatial dynamics and center manifold reduction. The major difficulty occurs from the irregularity of the solutions. The persistence of the approximately constructed pulse solutions under higher order perturbations is obtained by symmetry and reversibility arguments. (C) 2019 Elsevier Inc. All rights reserved.

Abstract

We show the existence of breather solutions in a nonlinear Klein-Gordon system on a discrete graph with periodic junctions. The proof is based on the Theorem of Crandall-Rabinowitz.

Abstract

We introduce a Darcy‐scale model to describe compressible multicomponent flow in a fully saturated porous medium. In order to capture cross‐diffusive effects between the different species correctly, we make use of the Maxwell–Stefan theory in a thermodynamically consistent way. For inviscid flow, the model turns out to be a nonlinear system of hyperbolic balance laws. We show that the dissipative structure of the Maxwell‐Stefan operator permits to guarantee the existence of global classical solutions for initial data close to equilibria. Furthermore, it is proven by relative entropy techniques that solutions of the Darcy‐scale model tend in a certain long‐time regime to solutions of a parabolic limit system.

Abstract

We consider the impingement of a droplet onto a wall with high im-pact speed. For this purpose an isothermal Navier–Stokes–Allen–Cahn model5 is used. Properties of the model are discussed. In order to solve the systemnumerically we introduce an energy consistent discontinuous Galerkin schemeand show a numerical example of droplet impact.

Abstract

We consider the impingement of a droplet onto a wall with high impact speed. To model this process we favour a diffuse-interface concept. Precisely, we suggest a compressible Navier--Stokes--Allen--Cahn model following 5. Basic properties of the model are discussed. To cope with the fluid-wall interaction, we derive thermodynamically consistent boundary conditions that account for dynamic contact angles. We briefly discuss a discontinuous Galerkin scheme which approximates the energy dissipation of the system exactly and illustrate the results with a series of numerical simulations. Currently, these simulations are restricted to static contact angle boundary conditions.

Abstract

In every point of a Kahler manifold there exist special holomorphic coordinates well adapted to the underlying geometry. Comparing these Kahler normal coordinates with the Riemannian normal coordinates defined via the exponential map we prove that their difference is a universal power series in the curvature tensor and its iterated covariant derivatives and devise an algorithm to calculate this power series to arbitrary order. As a byproduct we generalize Kahler normal coordinates to the class of complex affine manifolds with (1, 1)-curvature tensor. Moreover we describe the Spencer connection on the infinite order Taylor series of the Kahler normal potential and obtain explicit formulas for the Taylor series of all relevant geometric objects on symmetric spaces.

Abstract

This article is a follow up of our submitted paper (D. Seus et al, Comput Methods Appl Mech Eng 333:331--355, 2018) in which a decomposition of the Richards equation along two soil layers was discussed. A decomposed problem was formulated and a decoupling and linearisation technique was presented to solve the problem in each time step in a fixed point type iteration. This article extends these ideas to the case of two-phase in porous media and the convergence of the proposed domain decomposition method is rigorously shown."

Abstract

A mixed variational formulation of some problems in L2-based Sobolev spaces is used to define the Newtonian and layer potentials for the Stokes system with L∞ coefficients on Lipschitz domains in ℝ3\$\$\\backslashmathbb R\^3\$\$. Then the solution of the exterior Dirichlet problem for the Stokes system with L∞ coefficients is presented in terms of these potentials and the inverse of the corresponding single layer operator.

Abstract

We propose a novel approach for output-feedback controller design for linear systems affected by arbitrary fast time-varying uncertainties. Our approach is based on consecutively solving two related robust synthesis problems which by themselves admit convex solutions. The first one is robust full information controller design, while the subsequent one is similar to the robust estimator design problem. The latter includes a homotopy parameter that serves to construct a solution of the output-feedback synthesis problem from a given solution of the full information problem. In contrast to alternative approaches, ours requires to solve just two linear matrix inequalities and generates guaranteed bounds for the achieved performance. We illustrate our results with several numerical examples taken from the literature.

Abstract

This article deals with the problem of recovering an image from tomographic data corrupted by the motion of an object. Using generalized Radon transforms as modeling for the dynamic imaging operators, we present a novel approach for the reconstruction process. The proposed method, motivated by results on microlocal analysis, approximates the solution via a filtered backprojection-type algorithm which has the advantage of being quick to implement. The numerical results applied on dynamic photoacoustic tomography (PAT) data corroborate the efficiency and relevance of the proposed method.

Abstract

A variety of methods is available to quantify uncertainties arising within the modeling of flow and transport in carbon dioxide storage, but there is a lack of thorough comparisons. Usually, raw data from such storage sites can hardly be described by theoretical statistical distributions since only very limited data is available. Hence, exact information on distribution shapes for all uncertain parameters is very rare in realistic applications. We discuss and compare four different methods tested for data-driven uncertainty quantification based on a benchmark scenario of carbon dioxide storage. In the benchmark, for which we provide data and code, carbon dioxide is injected into a saline aquifer modeled by the nonlinear capillarity-free fractional flow formulation for two incompressible fluid phases, namely carbon dioxide and brine. To cover different aspects of uncertainty quantification, we incorporate various sources of uncertainty such as uncertainty of boundary conditions, of conceptual model definitions and of material properties. We consider recent versions of the following non-intrusive and intrusive uncertainty quantification methods: arbitary polynomial chaos, spatially adaptive sparse grids, kernel-based greedy interpolation and hybrid stochastic Galerkin. The performance of each approach is demonstrated assessing expectation value and standard deviation of the carbon dioxide saturation against a reference statistic based on Monte Carlo sampling. We compare the convergence of all methods reporting on accuracy with respect to the number of model runs and resolution. Finally we offer suggestions about the methods� advantages and disadvantages that can guide the modeler for uncertainty quantification in carbon dioxide storage and beyond.