Kühn, M.J., Kruse, C. and Ruede, U. (2022) Implicitly Extrapolated Geometric Multigrid on Disk-Like Domains for the Gyrokinetic Poisson Equation from Fusion Plasma Applications, Journal of Scientific Computing, 91 (28) , doi: 10.1007/s10915-022-01802-1

[bibtex]

@ARTICLE{AR-PA-22-23,
author = {Kühn, M.J. and Kruse, C. and Ruede, U. },
title = {Implicitly Extrapolated Geometric Multigrid on Disk-Like Domains for the Gyrokinetic Poisson Equation from Fusion Plasma Applications},
year = {2022},
number = {28},
volume = {91},
doi = {10.1007/s10915-022-01802-1},
journal = {Journal of Scientific Computing},
abstract = {The gyrokinetic Poisson equation arises as a subproblem of Tokamak fusion reactor simulations. It is often posed on disk-like cross sections of the Tokamak that are represented in generalized polar coordinates. On the resulting curvilinear anisotropic meshes, we discretize the differential equation by finite differences or low order finite elements. Using an implicit extrapolation technique similar to multigrid τ-extrapolation, the approximation order can be increased. This technique can be naturally integrated in a matrix-free geometric multigrid algorithm. Special smoothers are developed to deal with the mesh anisotropy arising from the curvilinear coordinate system and mesh grading.}}

Kruse, C., Sosonkina, M., Arioli, M., Tardieu, N. and Ruede, U. (2021) Parallel solution of saddle point systems with nested iterative solvers based on the Golub‐Kahan Bidiagonalization., Concurrency and computation-Practice and Experience, 33 (11) , pp. e5914, doi: 10.1002/cpe.5914

[bibtex]

@ARTICLE{AR-PA-21-217,
author = {Kruse, C. and Sosonkina, M. and Arioli, M. and Tardieu, N. and Ruede, U. },
title = {Parallel solution of saddle point systems with nested iterative solvers based on the Golub‐Kahan Bidiagonalization.},
year = {2021},
number = {11},
volume = {33},
pages = {e5914},
doi = {10.1002/cpe.5914},
journal = {Concurrency and computation-Practice and Experience},
abstract = {The Golub-Kahan bidiagonalization is widely used in the singular value decomposition of rectangular matrices and has been generalized to an iterative solver for symmetric indefinite linear systems with a two-by-two block structure. In this work, we present a scalability study of this generalized solver as implemented in a recent release of the parallel numerical library PETSc (Portable, Extensible Toolkit for Scientific Computation). We present an improved solver performance for the two-dimensional (2D) Stokes equations as compared to previous work. Furthermore, we investigate the performance of different parallel inner solvers in the outer Golub-Kahan iteration for a three-dimensional Stokes problem. The study includes parallel sparse direct solvers and multigrid methods. When increasing the number of cores for a fixed total problem size, the solver exhibits good speedups of up to 50% at the 1024 core count. For the tests in which the total problem size grows while the workload in each core stays constant, the parallel performance of the solver scales almost linearly with the increase in the core counts. In particular, the computation time increases only by about 15% when the number of cores increases from 80 to 1024 for a 2D test case.},
keywords = {Special Issue paper - Special Issue: Intelligent Systems for the Internet of Things (ISIT2018). Special Issue to 13th International Conference on Parallel Processing and Applied Mathematics (PPAM2019). Novel Parallel Processing Techniques for IoT‐based Machine Learning Applications (MPP2019)}}

Kühn, M.J., Kruse, C. and Ruede, U. (2021) Energy-Minimizing, Symmetric Discretizations for Anisotropic Meshes and Energy Functional Extrapolation, SIAM Journal on Scientific Computing, 43 (4) , pp. A2448-A2472, doi: 10.1137/21m1397520

[bibtex]

@ARTICLE{AR-PA-21-219,
author = {Kühn, M.J. and Kruse, C. and Ruede, U. },
title = {Energy-Minimizing, Symmetric Discretizations for Anisotropic Meshes and Energy Functional Extrapolation},
year = {2021},
number = {4},
volume = {43},
pages = {A2448-A2472},
doi = {10.1137/21m1397520},
journal = {SIAM Journal on Scientific Computing},
abstract = {Self-adjoint differential operators often arise from variational calculus on energy
functionals. In this case, a direct discretization of the energy functional induces a discretization of
the differential operator. Following this approach, the discrete equations are naturally symmetric if
the energy functional was self-adjoint, a property that may be lost when using standard difference
formulas on nonuniform meshes or when the differential operator has varying coefficients. Low order
finite difference or finite element systems can be derived by this approach in a systematic way and on
logically structured meshes they become compact difference formulas. Extrapolation formulas used
on the discrete energy can then lead to higher oder approximations of the differential operator. A
rigorous analysis is presented for extrapolation used in combination with nonstandard integration
rules for finite elements. Extrapolation can likewise be applied on matrix-free finite difference stencils.
In our applications, both schemes show up to quartic order of convergence.}}

Kruse, C., Sosonkina, M., Arioli, M., Tardieu, N. and Ruede, U. (2020) Parallel solution of saddle point systems with nested iterative solvers based on the Golub-Kahan Bidiagonalization, Concurrency and computation-Practice and Experience, 33 (11) , pp. e5914, ISSN 1532-0634, doi: 10.1002/cpe.5914

[bibtex] [pdf]

@ARTICLE{AR-PA-20-17,
author = {Kruse, C. and Sosonkina, M. and Arioli, M. and Tardieu, N. and Ruede, U. },
title = {Parallel solution of saddle point systems with nested iterative solvers based on the Golub-Kahan Bidiagonalization},
year = {2020},
number = {11},
volume = {33},
pages = {e5914},
issn = {1532-0634},
doi = {10.1002/cpe.5914},
journal = {Concurrency and computation-Practice and Experience},
abstract = {The Golub‐Kahan bidiagonalization is widely used in the singular value decomposition of rectangular matrices and has been generalized to an iterative solver for symmetric indefinite linear systems with a two‐by‐two block structure. In this work, we present a scalability study of this generalized solver as implemented in a recent release of the parallel numerical library PETSc (Portable, Extensible Toolkit for Scientific Computation). We present an improved solver performance for the two‐dimensional (2D) Stokes equations as compared to previous work. Furthermore, we investigate the performance of different parallel inner solvers in the outer Golub‐Kahan iteration for a three‐dimensional Stokes problem. The study includes parallel sparse direct solvers and multigrid methods. When increasing the number of cores for a fixed total problem size, the solver exhibits good speedups of up to 50% at the 1024 core count. For the tests in which the total problem size grows while the workload in each core stays constant, the parallel performance of the solver scales almost linearly with the increase in the core counts. In particular, the computation time increases only by about 15% when the number of cores increases from 80 to 1024 for a 2D test case.},
keywords = {Golub‐Kahan bidiagonalization, iterative solver, parallel performance, PETSc},
pdf = { https://doi.org/10.1002/cpe.5914}}

Kühn, M.J., Kruse, C. and Ruede, U. (2020) Energy-minimizing, symmetric finite differences for anisotropic meshes and energy functional extrapolation

[bibtex]

@ARTICLE{AR-PA-20-137,
author = {Kühn, M.J. and Kruse, C. and Ruede, U. },
title = {Energy-minimizing, symmetric finite differences for anisotropic meshes and energy functional extrapolation},
year = {2020},
abstract = { Self-adjoint differential operators often arise from variational calculus on energy functionals. In this case, a direct discretization of the energy functional induces a discretization of the differential operator. Following this approach, the discrete equations are naturally symmetric if the energy functional was self-adjoint, a property that may be lost when using standard difference formulas on nonuniform meshes or when the differential operator has varying coefficients. Low order finite difference or finite element systems can be derived by this approach in a systematic way and on logically structured meshes they become compact difference formulas. Extrapolation formulas used on the discrete energy can then lead to higher oder approximations of the differential operator. A rigorous analysis is presented for extrapolation used in combination with nonstandard integration rules for finite elements. Extrapolation can likewise be applied on matrix-free finite difference stencils. In our applications, both schemes show up to quartic order of convergence.}}

Kruse, C., Darrigrand, V., Tardieu, N., Arioli, M. and Ruede, U. (2020) Application of an iterative Golub-Kahan algorithm to structural mechanics problems with multi-point constraints, Advanced Modeling and Simulation in Engineering Sciences, 7 (1) , pp. 1-20, doi: 10.1186/s40323-020-00181-2

[bibtex] [pdf]

@ARTICLE{AR-PA-20-159,
author = {Kruse, C. and Darrigrand, V. and Tardieu, N. and Arioli, M. and Ruede, U. },
title = {Application of an iterative Golub-Kahan algorithm to structural mechanics problems with multi-point constraints},
year = {2020},
number = {1},
volume = {7},
pages = {1-20},
doi = {10.1186/s40323-020-00181-2},
journal = {Advanced Modeling and Simulation in Engineering Sciences},
pdf = {https://doi.org/10.1186/s40323-020-00181-2}}

Barucq, H., M'Barek, F., Kruse, C. and Tordeux, S. (2019) Sparsified discrete wave problem involving a radiation condition on a prolate spheroidal surface, IMA Journal of Numerical Analysis, doi: 10.1093/imanum/drz051

[bibtex]
[url] [pdf]

@ARTICLE{AR-PA-19-226,
author = {Barucq, H. and M'Barek, F. and Kruse, C. and Tordeux, S. },
title = {Sparsified discrete wave problem involving a radiation condition on a prolate spheroidal surface},
year = {2019},
doi = {10.1093/imanum/drz051},
journal = {IMA Journal of Numerical Analysis},
abstract = {We develop and analyse a high-order outgoing radiation boundary condition for solving three-dimensional scattering problems by elongated obstacles. This Dirichlet-to-Neumann condition is constructed using the classical method of separation of variables that allows one to define the scattered field in a truncated domain. It reads as an infinite series that is truncated for numerical purposes. The radiation condition is implemented in a finite element framework represented by a large dense matrix. Fortunately, the dense matrix can be decomposed into a full block matrix that involves the degrees of freedom on the exterior boundary and a sparse finite element matrix. The inversion of the full block is avoided by using a Sherman–Morrison algorithm that reduces the memory usage drastically. Despite being of high order, this method has only a low memory cost.},
pdf = {https://doi.org/10.1093/imanum/drz051},
url = {https://academic.oup.com/imajna/advance-article-abstract/doi/10.1093/imanum/drz051/5627734}}

Banks, H.T., Doumic, M. and Kruse, C. (2017) A numerical scheme for the early steps of nucleation-aggregation models, Journal of Mathematical Biology, 74 (1-2) , pp. 259–287, ISSN 0303-6812, doi: 10.1007/s00285-016-1026-0

[bibtex] [pdf]

@ARTICLE{AR-PA-17-326,
author = {Banks, H.T. and Doumic, M. and Kruse, C. },
title = {A numerical scheme for the early steps of nucleation-aggregation models},
year = {2017},
number = {1-2},
volume = {74},
pages = {259–287},
issn = {0303-6812},
doi = {10.1007/s00285-016-1026-0},
journal = {Journal of Mathematical Biology},
keywords = {Polymerization, Aggregation-fragmentation models, Finite volume schemes, Adaptive grid },
pdf = {https://doi.org/10.1007/s00285-016-1026-0}}

Kruse, C., Sosonkina, M., Arioli, M., Tardieu, N. and Ruede, U. (2020) Parallel Performance of an Iterative Solver Based on the Golub-Kahan Bidiagonalization, Parallel Processing and Applied Mathematics. PPAM 2019 - 13th INTERNATIONAL CONFERENCE ON PARALLEL PROCESSING AND APPLIED MATHEMATICS, Bialystok, Poland, 11 2020, Roman WyrzykowskiEwa DeelmanJack DongarraKonrad Karczewski, doi: 10.1007/978-3-030-43229-4

[bibtex]
[url]

@CONFERENCE{PR-PA-20-11,
author = {Kruse, C. and Sosonkina, M. and Arioli, M. and Tardieu, N. and Ruede, U. },
title = {Parallel Performance of an Iterative Solver Based on the Golub-Kahan Bidiagonalization},
year = {2020},
month = {11},
booktitle = {Parallel Processing and Applied Mathematics},
editor = {Roman WyrzykowskiEwa DeelmanJack DongarraKonrad Karczewski},
publisher = {Springer},
volume = {12043},
pages = {104-116},
isbn = {978-3-030-43228-7},
organization = {PPAM 2019 - 13th INTERNATIONAL CONFERENCE ON PARALLEL PROCESSING AND APPLIED MATHEMATICS},
address = {Bialystok, Poland},
doi = {10.1007/978-3-030-43229-4},
abstract = {We present an iterative method based on a generalization of the Golub-Kahan bidiagonalization for solving indenite matrices with a 2x2 block structure. We focus in particular on our recent implementation of the algorithm using the parallel numerical library PETSc. Since the algorithm is a nested solver, we investigate dierent choices for parallel inner solvers and show its strong scalability for two Stokes test problems. The algorithm is found to be scalable for large sparse problems.},
keywords = {Golub-Kahan bidiagonalization, Iterative solver, PETSc, Parallel performance },
url = {https://link.springer.com/chapter/10.1007/978-3-030-43229-4_10}}

Arioli, M., Kruse, C., Ruede, U. and Tardieu, N. (2018) An iterative generalized Golub-Kahan algorithm for problems in structural mechanics, Cerfacs, Toulouse, France

[bibtex]

@TECHREPORT{TR-PA-18-159,
author = {Arioli, M. and Kruse, C. and Ruede, U. and Tardieu, N. },
title = {An iterative generalized Golub-Kahan algorithm for problems in structural mechanics},
year = {2018},
institution = {Cerfacs, Toulouse, France},
month = {8},
abstract = {This paper studies the Craig variant of the Golub-Kahan bidiagonalization algorithm as an iterative solver for linear systems with saddle point structure. Such symmetric indefinite systems in 2x2 block form arise in many applications, but standard iterative solvers are often found to perform poorly on them and robust preconditioners may not be available. Specifically, such systems arise in structural mechanics, when a semidefinite finite element stiffness matrix is augmented with linear multi-point constraints via Lagrange multipliers. Engineers often use such multi-point constraints to introduce boundary or coupling conditions into complex finite element models. The article will present a systematic convergence study of the Golub-Kahan algorithm for a sequence of test problems of increasing complexity, including concrete structures enforced with pretension cables and the coupled finite element model of a reactor containment building. When the systems are suitably transformed using augmented Lagrangians on the semidefinite block and when the constraint equations are properly scaled, the Golub-Kahan algorithm is found to exhibit excellent convergence that depends only weakly on the size of the model. The new algorithm is found to be robust in practical cases that are otherwise considered to be difficult for iterative solvers.}}