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Theisen, Lambert

Lambert Theisen

M.Sc.

PhD student, research assistant
Institute of Applied Analysis and Numerical Simulation
Numerical Mathematics for High Performance Computing

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In April 2024, I started a position at RWTH Aachen.

Subject

Researching PDE eigenvalue problems, asymptotic analysis of expanding domains, directional homogenization, preconditioners for eigenvalue algorithms, preconditioners for linear solvers, spectral coarse spaces for domain decomposition, and Galerkin methods for moment models in rarefied gas modelling.

Publications

Theisen, L., & Stamm, B. (2023). A Scalable Two-Level Domain Decomposition Eigensolver for Periodic Schrödinger Eigenstates in Anisotropically Expanding Domains. https://doi.org/10.48550/arXiv.2311.08757
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.
BibTeX
@preprint{theisen2023scalable, 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.}, archiveprefix = {arXiv}, author = {Theisen, Lambert and Stamm, Benjamin}, doi = {10.48550/arXiv.2311.08757}, eprint = {2311.08757}, primaryclass = {math.NA}, title = {A Scalable Two-Level Domain Decomposition Eigensolver for Periodic Schr\"odinger Eigenstates in Anisotropically Expanding Domains}, url = {https://arxiv.org/abs/2311.08757}, year = 2023 }
Link
https://arxiv.org/abs/2311.08757
DOI
10.48550/arXiv.2311.08757
Stamm, B., & Theisen, L. (2022). A Quasi-Optimal Factorization Preconditioner for Periodic Schrödinger Eigenstates in Anisotropically Expanding Domains. SIAM Journal on Numerical Analysis, 60, Article 5. https://doi.org/10.1137/21m1456005
BibTeX
@article{Stamm_2022, author = {Stamm, Benjamin and Theisen, Lambert}, doi = {10.1137/21m1456005}, journal = {{SIAM} Journal on Numerical Analysis}, month = {09}, number = 5, pages = {2508--2537}, publisher = {Society for Industrial {\&} Applied Mathematics ({SIAM})}, title = {A Quasi-Optimal Factorization Preconditioner for Periodic Schrödinger Eigenstates in Anisotropically Expanding Domains}, url = {https://doi.org/10.1137%2F21m1456005}, volume = 60, year = 2022 }
Link
https://doi.org/10.1137%2F21m1456005
DOI
10.1137/21m1456005
Theisen, L., & Stamm, B. (2022, May). ddEigenLab.jl: Domain-Decomposition Eigenvalue Problem Lab [Zenodo]. https://doi.org/10.5281/zenodo.6576197
BibTeX
@dataset{theisen_lambert_2022_6576197, author = {Theisen, Lambert and Stamm, Benjamin}, doi = {10.5281/zenodo.6576197}, month = {05}, note = {{This work was supported by the German Research Foundation (DFG) under project 411724963.}}, publisher = {Zenodo}, title = {ddEigenLab.jl: Domain-Decomposition Eigenvalue Problem Lab}, url = {https://doi.org/10.5281/zenodo.6576197}, version = {0.2}, year = 2022 }
Link
https://doi.org/10.5281/zenodo.6576197
DOI
10.5281/zenodo.6576197
Theisen, L., & Torrilhon, M. (2021). fenicsR13: A Tensorial Mixed Finite Element Solver for the Linear R13 Equations Using the FEniCS Computing Platform. ACM Trans. Math. Softw., 47, Article 2. https://doi.org/10.1145/3442378
Abstract
We present a mixed finite element solver for the linearized regularized 13-moment equations of non-equilibrium gas dynamics. The Python implementation builds upon the software tools provided by the FEniCS computing platform. We describe a new tensorial approach utilizing the extension capabilities of FEniCS’ Unified Form Language to define required differential operators for tensors above second degree. The presented solver serves as an example for implementing tensorial variational formulations in FEniCS, for which the documentation and literature seem to be very sparse. Using the software abstraction levels provided by the Unified Form Language allows an almost one-to-one correspondence between the underlying mathematics and the resulting source code. Test cases support the correctness of the proposed method using validation with exact solutions. To justify the usage of extended gas flow models, we discuss typical application cases involving rarefaction effects. We provide the documented and validated solver publicly.
BibTeX
@article{Theisen2021, abstract = {We present a mixed finite element solver for the linearized regularized 13-moment equations of non-equilibrium gas dynamics. The Python implementation builds upon the software tools provided by the FEniCS computing platform. We describe a new tensorial approach utilizing the extension capabilities of FEniCS’ Unified Form Language to define required differential operators for tensors above second degree. The presented solver serves as an example for implementing tensorial variational formulations in FEniCS, for which the documentation and literature seem to be very sparse. Using the software abstraction levels provided by the Unified Form Language allows an almost one-to-one correspondence between the underlying mathematics and the resulting source code. Test cases support the correctness of the proposed method using validation with exact solutions. To justify the usage of extended gas flow models, we discuss typical application cases involving rarefaction effects. We provide the documented and validated solver publicly.}, address = {New York, NY, USA}, author = {Theisen, Lambert and Torrilhon, Manuel}, day = 20, doi = {10.1145/3442378}, issn = {0098-3500}, issue_date = {June 2021}, journal = {ACM Trans. Math. Softw.}, month = {04}, number = 2, numpages = {29}, pages = {1–29}, publisher = {Association for Computing Machinery}, title = {fenicsR13: A Tensorial Mixed Finite Element Solver for the Linear R13 Equations Using the FEniCS Computing Platform}, url = {https://doi.org/10.1145/3442378}, volume = 47, year = 2021 }
Link
https://doi.org/10.1145/3442378
DOI
10.1145/3442378
Theisen, L., & Torrilhon, M. (2020, November). fenicsR13: A Tensorial Mixed Finite Element Solver for the Linear R13 Equations Using the FEniCS Computing Platform [Zenodo]. https://doi.org/10.5281/zenodo.4172951
Abstract
fenicsR13: A Tensorial Mixed Finite Element Solver for the Linear R13 Equations Using the FEniCS Computing Platform The project can be accessed at https://git.rwth-aachen.de/lamBOO/fenicsR13
BibTeX
@dataset{theisen_lambert_2020_4172951, abstract = {fenicsR13: A Tensorial Mixed Finite Element Solver for the Linear R13 Equations Using the FEniCS Computing Platform The project can be accessed at https://git.rwth-aachen.de/lamBOO/fenicsR13}, author = {Theisen, Lambert and Torrilhon, Manuel}, doi = {10.5281/zenodo.4172951}, month = {11}, publisher = {Zenodo}, title = {fenicsR13: A Tensorial Mixed Finite Element Solver for the Linear R13 Equations Using the FEniCS Computing Platform}, url = {https://doi.org/10.5281/zenodo.4172951}, version = {1.4}, year = 2020 }
Link
https://doi.org/10.5281/zenodo.4172951
DOI
10.5281/zenodo.4172951

Teaching

WS22: Assistant for Höhere Mathematik 1 für Ingenieure (higher mathematics 1 for engineers) @ LExMath - Lehrexportzentrum Mathematik, University of Stuttgart, with Prof. Stroppel
SS22: Assistant for Mathematical aspects in computational chemistry @ ACoM, RWTH Aachen, with Prof. Stamm
WS21: Assistant for Mathematische Grundlagen III (CES) @ MathCCES, RWTH Aachen, with Prof. Torrilhon & Prof. Krumscheid
WS20: Assistant for Mathematische Grundlagen III (CES) @ MathCCES, RWTH Aachen, with Prof. Stamm & Prof. Hakon Hoel
SS20: Assistant for Mathematische Grundlagen IV (CES) @ MathCCES, RWTH Aachen, with Prof. Stamm & Prof. Krumscheid
WS19/20: Assistant for Mathematische Grundlagen I (CES) @ MathCCES, RWTH Aachen, with Prof. Stamm & Prof. Krumscheid
SS19: Tutor for Mathematische Grundlagen IV (CES) @ MathCCES, RWTH Aachen
WS18/19: Tutor for Partielle Differentialgleichungen (CES) @ MathCCES, RWTH Aachen

Vita

Education

10/19 – now: PhD Student @ RWTH Aachen University, Aachen, Germany
10/14 – 04/18: Bachelor of Science in Computational Engineering Science (CES) @ RWTH Aachen University, Aachen, Germany
04/18 – 09/19: Master of Science in Computational Engineering Science (CES) @ RWTH Aachen University, Aachen, Germany

Professional

10/22 – now: Research Assistant @ Chair of Numerical Mathematics for High Performance Computing (NMH/IANS), University of Stuttgart
10/19 – 09/22: Research Assistant @ MathCCES / Applied and Computational Mathematics, RWTH Aachen
10/17 – 03/18: Internship about 3D CFD Simulation and Hexahedral Meshing @ ABB Corporate Research Center Baden-Dättwil, Switzerland
08/15 – 10/15: Working Student in IT and EDV Systems @ Sanofi, Frankfurt am Main, Germany
07/14 – 10/14: Working Student in Export & Dispatch @ Sanofi, Frankfurt am Main, Germany

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