Article
Mathematics, Applied
Jed Brown, Yunhui He, Scott MacLachlan, Matt Menickelly, Stefan M. Wild
Summary: Local Fourier analysis is a useful tool for predicting and analyzing the performance of efficient algorithms for solving discretized PDEs. It involves minimizing an estimate of the spectral radius or the condition number by optimizing solver parameters, which often requires maximizing over Fourier frequencies. While analytical solutions to minimax problems are rare, optimization algorithms can be used effectively to obtain efficient algorithms.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Jeremy L. Thompson, Jed Brown, Yunhui He
Summary: This paper introduces LFAToolkit.jl, a Julia package for local Fourier analysis of high-order finite element methods. It can analyze preconditioning techniques for arbitrary systems of second order PDEs and supports mixed finite element methods. With this toolkit, analysis of h-multigrid for finite element discretizations or finite difference discretizations is also possible.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Yunhui He
Summary: Local Fourier analysis (LFA) plays an important role in predicting the convergence factor of multigrid methods for discretizations of PDEs. The study demonstrates that the LFA representation for d-dimensional PDEs is independent of the placement of degrees of freedom (DoFs), providing a simple and unified way to compute symbols of discrete operators. This simple representation can aid in generalizing the implementation of LFA for different discretizations and higher order methods.
NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Yunhui He, Sander Rhebergen, Hans De Sterck
Summary: This paper introduces a geometric multigrid method for hybridized and embedded discontinuous Galerkin discretizations of Laplacian, and shows through local Fourier analysis the effectiveness of this method. Numerical examples demonstrate that applying multigrid to an embedded discontinuous Galerkin discretization performs better than applying it to a hybridized discontinuous Galerkin discretization.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Patrick E. Farrell, Yunhui He, Scott P. MacLachlan
Summary: Multigrid methods are popular solution algorithms for discretized PDEs due to their high efficiency, with the choice and optimization of components being crucial for algorithm design. This article presents a local Fourier analysis of monolithic multigrid methods based on additive Vanka relaxation schemes, offering insight into patch selection and proposing parameters for minimizing the two-grid convergence factor. Numerical experiments validate the LFA predictions.
NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Yunhui He
Summary: This work introduces an extension of the classical Local Fourier Analysis (LFA) by extending scalar stencils to matrix-stencils, proving that any scalar stencil operator can be described by a matrix-stencil operator, and showing that symbols based on scalar stencils and matrix-stencils of a given discrete operator are unitarily similar. This leads to a simple and unified framework of two-grid LFA based on matrix-stencils, which fits well with finite element and difference discretizations.
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
(2021)
Article
Mathematics, Applied
Theophile Chaumont-frelet, Martin Vohralik
Summary: We propose a local method to construct H(curl)-conforming piecewise polynomials that satisfy a given curl constraint. Starting from a piecewise polynomial that does not belong to the H(curl) space but satisfies certain orthogonality properties, we employ minimizations in vertex patches to achieve the same accuracy as the best approximations over the entire local versions of H(curl), up to a generic constant independent of the polynomial degree. This enables the design of guaranteed, fully computable, constant-free, and polynomial-degree-robust a posteriori error estimates for Neumann vector finite element approximations of the curl-curl problem, using a divergence-free decomposition of a divergence-free H(div)-conforming piecewise polynomial obtained through overconstrained minimizations in Raviart-Thomas spaces. Numerical experiments demonstrate the effectiveness of the theoretical developments.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Shuijin Zhang, Minbo Yang
Summary: This paper considers a curl-curl equation with nonlocal nonlinearity involving Riesz potential, and obtains a solution using specific assumptions and methods.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics, Applied
Fazal Ghaffar, Saif Ullah, Noor Badshah, Najeeb Alam Khan
Summary: In this paper, a higher-order compact finite difference scheme with multigrid algorithm is used to solve the one-dimensional time fractional diffusion equation. The scheme achieves eighth-order accuracy in space, and its convergence is proven through Fourier analysis and matrix analysis. Numerical experiments confirm the performance and accuracy of the proposed scheme.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Clemens Hofreither, Ludwig Mitter, Hendrik Speleers
Summary: This paper proposes local multigrid solvers for adaptively refined isogeometric discretizations using (truncated) hierarchical B-splines ((T)HB-splines). Smoothing is only performed in or near the refinement areas, leading to a computationally efficient solving strategy. The robust convergence of the proposed solvers is proven with respect to the number of levels and the mesh sizes of the hierarchical discretization space, assuming an admissibility condition is satisfied. Several numerical experiments are also provided.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2022)
Article
Mathematics, Applied
Andrew T. Barker, Tzanio Kolev
Summary: The paper discusses the preconditioning of a definite Maxwell operator at high polynomial order without assembling a matrix, showing how efficient H(curl) preconditioners can be constructed in an auxiliary space framework. By utilizing a sparsified H-1 solver constructed on a low-order mesh, the resulting H(curl) preconditioner is effective at very high polynomial orders for two-dimensional model problems.
NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Mostak Ahmed, Chengjian Zhang
Summary: This paper introduces a new multigrid method for solving 2D damped Helmholtz equations, which utilizes different finite difference schemes as coarse and fine grid operators to achieve a faster convergent rate than the regular method. The validity of the proposed method is confirmed by local mode Fourier analysis, and numerical results demonstrate its efficiency.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Yunhui He, Jun Liu
Summary: We propose and analyze a Vanka-type multigrid solver for solving a sequence of complex-shifted Laplacian systems arising in diagonalization-based parallel-in-time algorithms for evolutionary equations. The proposed Vanka-type smoother achieves a uniform smoothing factor, which is verified by several numerical examples.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics, Applied
Matthias Bolten, Marco Donatelli, Paola Ferrari, Isabella Furci
Summary: The main focus of this paper is the study of efficient multigrid methods for large linear systems with a particular saddle-point structure. The paper proposes a symbol based convergence analysis for problems that have a hidden block Toeplitz structure and provides optimal parameters for the preconditioning of the saddle-point problem. The efficiency and convergence rate of the proposed methods are demonstrated through numerical tests.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics
Jan Rozendaal
Summary: We show that the Hardy spaces for Fourier integral operators naturally serve as initial data spaces when applying $p$-decoupling inequalities to local smoothing for the wave equation. This yields new local smoothing estimates that quantitatively improve the bounds in the local smoothing conjecture on $\mathbb{R}^n$ for $p \geq \frac{2(n + 1)}{n - 1}$ and complements them for $2 < p < \frac{2(n + 1)}{n - 1}$. These estimates are invariant under the application of Fourier integral operators and are essentially sharp.
JOURNAL OF FUNCTIONAL ANALYSIS
(2022)
Article
Computer Science, Interdisciplinary Applications
C. Nita, S. Vandewalle, J. Meyers
COMPUTERS & FLUIDS
(2016)
Article
Computer Science, Theory & Methods
Samuel Corveleyn, Stefan Vandewalle
FUZZY SETS AND SYSTEMS
(2017)
Article
Mathematics, Applied
Pieterjan Robbe, Dirk Nuyens, Stefan Vandewalle
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2017)
Article
Computer Science, Interdisciplinary Applications
C. Nita, S. Vandewalle, J. Meyers
JOURNAL OF COMPUTATIONAL PHYSICS
(2018)
Editorial Material
Chemistry, Inorganic & Nuclear
Ian S. Butler, Sahar I. Mostafa
INORGANICA CHIMICA ACTA
(2014)
Editorial Material
Cell Biology
Aurelia Santoro, Patrizia Brigidi, Efstathios S. Gonos, Vilhelm A. Bohr, Claudio Franceschi
MECHANISMS OF AGEING AND DEVELOPMENT
(2014)
Article
Mathematics, Interdisciplinary Applications
Andreas Van Barel, Stefan Vandewalle
SIAM-ASA JOURNAL ON UNCERTAINTY QUANTIFICATION
(2019)
Article
Mathematics, Applied
Pieterjan Robbe, Dirk Nuyens, Stefan Vandewalle
Summary: The article introduces a new MIMC method that reuses coarse solutions from MSG, providing unbiased estimation by learning the unknown distribution of sample numbers across all indices. Through numerical experiments, the cost and robustness of this new estimator in various anisotropic random fields are demonstrated, showing its superiority over unbiased MIMC without sample reuse.
NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Emil Lovbak, Giovanni Samaey, Stefan Vandewalle
Summary: This paper investigates the long-time behavior of particles in high-collisional regimes and proposes a multilevel Monte Carlo scheme to reduce bias by combining estimates from different time step sizes. The approach significantly reduces computational requirements for accurate simulations of the considered kinetic equations compared to classical Monte Carlo methods.
NUMERISCHE MATHEMATIK
(2021)
Article
Computer Science, Interdisciplinary Applications
Liang Fang, Stefan Vandewalle, Johan Meyers
Summary: This study proposes a new parallel-in-time multiple shooting algorithm for solving large scale optimal control problems governed by parabolic PDEs. The algorithm is validated and analyzed using different test cases, showing significant speed-ups and better performance for complex flow fields.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Nuclear Science & Technology
Henri Dolfen, Stefan Vandewalle, Joris Degroote
Summary: The design evaluation of nuclear components using numerical methods typically focuses on ideal conditions, but in reality, the geometry and operating conditions may differ. Understanding and ensuring the safety of nuclear energy systems requires investigating more realistic conditions, such as the deformation of fuel assemblies due to thermal and irradiation effects. A paradigm shift is needed to move from deterministic simulations to simulations involving stochastic processes.
NUCLEAR ENGINEERING AND DESIGN
(2023)
Article
Computer Science, Artificial Intelligence
Philippe Blondeel, Pieterjan Robbe, Cedric Van Hoorickx, Stijn Francois, Geert Lombaert, Stefan Vandewalle
Article
Mathematics, Applied
Pieterjan Robbe, Dirk Nuyens, Stefan Vandewalle
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2019)
Proceedings Paper
Engineering, Mechanical
P. Blondeel, P. Robbe, C. Van Hoorickx, G. Lombaert, S. Vandewalle
PROCEEDINGS OF INTERNATIONAL CONFERENCE ON NOISE AND VIBRATION ENGINEERING (ISMA2018) / INTERNATIONAL CONFERENCE ON UNCERTAINTY IN STRUCTURAL DYNAMICS (USD2018)
(2018)
Article
Mathematics, Interdisciplinary Applications
R. D. Falgout, S. Friedhoff, Tz. V. Kolev, S. P. MacLachlan, J. B. Schroder, S. Vandewalle
COMPUTING AND VISUALIZATION IN SCIENCE
(2017)
