Computational Mathematics for Complex Simulation in Science and Engineering

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

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

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

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

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

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

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

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 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 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

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

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

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 semidefinite program. We establish explicit formulas for the optimal convergence rates and extend the proposed synthesis solution to extremum control problems.

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

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

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

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

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 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

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

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

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

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

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

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

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

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

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

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

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

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

"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

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

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

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

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

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

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

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

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

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

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

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.

Abstract

We investigate the influence of uncertain input parameters on the aeroacoustic feedback of cavity flows. The so-called Rossiter feedback requires a direct numerical computation of the acoustic noise, which solves hydrodynamics and acoustics simultaneously, in order to capture the interaction of acoustic waves and the hydrodynamics of the flow. Due to the large bandwidth of spatial and temporal scales, a high order numerical scheme with low dissipation and dispersion error is necessary to preserve important small scale information. Therefore, the open-source CFD solver FLEXI, which is based on a high-order discontinuous Galerkin spectral element method, is used to perform the aforementioned direct simulations of an open cavity configuration with a laminar upstream boundary layer. To analyse the precision of the deterministic cavity simulation with respect to random input parameters we establish a framework for uncertainty quantification. In particular, a non-intrusive spectral projection method with Legendre and Hermite polynomial basis functions is employed in order to treat uniform and normal probability distributions of the uncertain input. The results indicate a strong, non-linear dependency of the acoustic feedback mechanism on the investigated uncertain input parameters. An analysis of the stochastic results offers new insights into the noise generation process of open cavity flows and reveals the strength of the implemented uncertainty quantification framework.

Abstract

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Abstract

In this presentation we address the dynamic concentration reconstruction problem in magnetic particle imaging. The tradeoff between the signal-to-noise ratio in the processed data used for the reconstruction and the concentration dynamics of the tracer requires sophisticated reconstruction techniques to avoid motion artifacts. We incorporate motion priors in the concentration reconstruction and as an extra we can obtain an estimate for a flow field.

Abstract

We present a novel acceleration method for the solution of parametric ODEs by single-step implicit solvers by means of greedy kernel-based surrogate models. In an offline phase, a set of trajectories is precomputed with a high-accuracy ODE solver for a selected set of parameter samples, and used to train a kernel model which predicts the next point in the trajectory as a function of the last one. This model is cheap to evaluate, and it is used in an online phase for new parameter samples to provide a good initialization point for the nonlinear solver of the implicit integrator. The accuracy of the surrogate reflects into a reduction of the number of iterations until convergence of the solver, thus providing an overall speedup of the full simulation. Interestingly, in addition to providing an acceleration, the accuracy of the solution is maintained, since the ODE solver is still used to guarantee the required precision. Although the method can be applied to a large variety of solvers and different ODEs, we will present in details its use with the Implicit Euler method for the solution of the Burgers equation, which results to be a meaningful test case to demonstrate the method's features.

Abstract

We study symmetric Killing 2-tensors on Riemannian manifolds. We show that several additional conditions can be realized only for Sasakian manifolds and Euclidean spheres. In particular we recover a result of S. Gallot on the characterization of spheres by means of functions satisfying a certain differential equation of order three. (C) 2018 Published by Elsevier B.V.

Abstract

This article concerns with incorporating wavelet bases into existing streamline upwind Petrov-Galerkin (SUPG) methods for the numerical solution of nonlinear hyperbolic conservation laws which are known to develop shock solutions. Here, we utilize an SUPG formulation using continuous Galerkin in space and discontinuous Galerkin in time. The main motivation for such a combination is that these methods have good stability properties thanks to adding diffusion in the direction of streamlines. But they are more expensive than explicit semidiscrete methods as they have to use space-time formulations. Using wavelet bases we maintain the stability properties of SUPG methods while we reduce the cost of these methods significantly through natural adaptivity of wavelet expansions. In addition, wavelet bases have a hierarchical structure. We use this property to numerically investigate the hierarchical addition of an artificial diffusion for further stabilization in spirit of spectral diffusion. Furthermore, we add the hierarchical diffusion only in the vicinity of discontinuities using the feature of wavelet bases in detection of location of discontinuities. Also, we again use the last feature of the wavelet bases to perform a postprocessing using a denosing technique based on a minimization formulation to reduce Gibbs oscillations near discontinuities while keeping other regions intact. Finally, we show the performance of the proposed combination through some numerical examples including Burgers�, transport, and wave equations as well as systems of shallow water equations.� 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2062�2089, 2017

Abstract

Error estimates for DGFE solutions are well investigated if one assumes that the exact solution is sufficiently regular. In this article, we consider a Dirichlet and a mixed boundary value problem for a linear elliptic equation in a polygon. It is well known that the first derivatives of the solutions develop singularities near reentrant corner points or points where the boundary conditions change. On the basis of the regularity results formulated in Sobolev--Slobodetskii spaces and weighted spaces of Kondratiev type, we prove error estimates of higher order for DGFE solutions using a suitable graded mesh refinement near boundary singular points. The main tools are as follows: regularity investigation for the exact solution relying on general results for elliptic boundary value problems, error analysis for the interpolation in Sobolev--Slobodetskii spaces, and error estimates for DGFE solutions on special graded refined meshes combined with estimates in weighted Sobolev spaces. Our main result is that there exist a local grading of the mesh and a piecewise interpolation by polynoms of higher degree such that we will get the same order O (ha) of approximation as in the smooth case. � 2011 Wiley Periodicals, Inc. Numer Mehods Partial Differential Eq, 2012

Abstract

This paper continues studies in Lalegname et al. (Int J Fract 152:97--125, 2008 ) on crack propagation in a bounded linear elastic body under the influence of incident waves. In Lalegname et al. ( 2008 ) we have considered shear waves, whereas in this paper we discuss the influence of plane elastic waves to a running crack. Actually, the time dependent problem is formulated in a two-dimensional current cracked configuration by a system of linear elasto-dynamic equations. In order to describe the behaviour of the elastic fields near the straight crack tip, we transform these equations to a reference configuration and derive the dynamic stress singularities. Furthermore, we assume that an energy balance law is valid. Exploiting the knowledge on the singular behaviour of the crack fields, we derive from the energy balance law a dynamic energy release rate. Comparing this energy release rate with an experimentally given fracture toughness we get an ordinary differential equation for the crack tip motion. We present first numerical simulations for a Mode I crack propagation.