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Robust multigrid methods for Discontinuous Galerkin Discretizations of an Elliptic Optimal Control Problem

COMPUTATIONAL methods IN APPLIED MATHEMATICS 2025年第1期25卷 133-151页

作者： Liu, Sijing Univ Connecticut Dept Math Storrs CT 06269 USA Brown Univ Inst Computat & Expt Res Math Providence RI 02912 USA

We consider discontinuous Galerkin methods for an elliptic distributed optimal control problem, and we propose multigrid methods to solve the discretized system. We prove that the W-cycle algorithm is uniformly convergent in the energy norm and is robust with respect to a regularization parameter on convex domains. Numerical results are shown for both W-cycle and V-cycle algorithms.

关键词： Elliptic Distributed Optimal Control Problems General State Equations multigrid methods Discontinuous Galerkin methods

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MATRIX-FREE MONOLITHIC multigrid methods FOR STOKES AND GENERALIZED STOKES PROBLEMS

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2024年第3期46卷 A1599-A1627页

作者： Jodlbauer, Daniel Langer, Ulrich Wick, Thomas Zulehner, Walter Austrian Acad Sci Johann Radon Inst Computat Math A-4040 Linz Austria Johannes Kepler Univ Linz Inst Computat Math A-4040 Linz Austria Leibniz Univ Hannover Inst Angew Math D-30167 Hannover Germany Leibniz Univ Hannover Cluster Excellence PhoenixD Photon Opt & Engn Inno Hannover Germany

We consider the widely used continuous Qk-Qk k-Q k- 1 quadrilateral or hexahedral Taylor-Hood elements for the finite element discretization of the Stokes and generalized Stokes systems in two and three spatial dimensions. For the fast solution of the corresponding symmetric, but indefinite system of finite element equations, we propose and analyze matrix-free monolithic geometric multigrid solvers that are based on appropriately scaled Chebyshev--Jacobi smoothers. The analysis is based on results by Scho"\berl and Zulehner (2003). We present and discuss several numerical results for typical benchmark problems.

关键词： Stokes and generalized Stokes problems mixed finite element discretization multigrid methods matrix-free implementation

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Symbol based convergence analysis in multigrid methods for saddle point problems

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LINEAR ALGEBRA AND ITS APPLICATIONS 2023年第1期671卷 67-108页

作者： Bolten, Matthias Donatelli, Marco Ferrari, Paola Furci, Isabella Univ Wuppertal Sch Math & Nat Sci Wuppertal Germany Univ Insubria Dept Sci & High Technol Como Italy

Saddle point problems arise in a variety of applications, e.g., when solving the Stokes equations. They can be formulated such that the system matrix is symmetric, but indefinite, so the variational convergence theory that is usually used to prove multigrid convergence cannot be applied. In a 2016 pa-per in Numerische Mathematik Notay has presented a differ-ent algebraic approach that analyzes properly preconditioned saddle point problems, proving convergence of the two-grid method. The present paper analyzes saddle point problems where the blocks are circulant within this framework. It con-tains sufficient conditions for convergence and optimal param-eters for the preconditioning of the unilevel and multilevel saddle point problem and for the point smoother that is used. The analysis is based on the generating symbols of the circu-lant blocks. Further, it is shown that the structure can be kept on the coarse level, allowing for a recursive application of the approach in a W-or V-cycle and studying the "level indepen-dency" property. Numerical results demonstrate the efficiency of the proposed method in the circulant and the Toeplitz case.(c) 2023 Elsevier Inc. All rights reserved.

关键词： multigrid methods Saddle -point systems Spectral symbol Toeplitz-like matrices

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Symbol based convergence analysis in block multigrid methods with applications for Stokes problems

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APPLIED NUMERICAL MATHEMATICS 2023年第1期193卷 109-130页

作者： Bolten, Matthias Donatelli, Marco Ferrari, Paola Furci, Isabella Univ Wuppertal Sch Math & Nat Sci Wuppertal Germany Univ Insubna Dept Sci & High Technol Como Italy Univ Genoa Dept Math Genoa Italy

The main focus of this paper is the study of efficient multigrid methods for large linear systems with a particular saddle-point structure. Indeed, when the system matrix is symmetric, but indefinite, the variational convergence theory that is usually used to prove multigrid convergence cannot be directly ***, different algebraic approaches analyze properly preconditioned saddle-point problems, proving convergence of the Two-Grid method. In particular, this is efficient when the blocks of the coefficient matrix possess a Toeplitz or circulant structure. Indeed, it is possible to derive sufficient conditions for convergence and provide optimal parameters for the preconditioning of the saddle-point problem in terms of the associated symbols. In this paper, we propose a symbol based convergence analysis for problems that have a hidden block Toeplitz structure. Then, they can be investigated focusing on the properties of the associated generating function f, which consequently is a matrix-valued function with dimension depending on the block size of the problem. As numerical tests we focus on the matrix sequence stemming from the finite element approximation of the Stokes problem. We show the efficiency of the methods studying the hidden 9-by-9 block multilevel structure of the obtained matrix sequence. Moreover we propose an efficient algebraic multigrid method with convergence rate independent of the matrix size. Finally, we present several numerical tests comparing the results with state-of-the-art strategies.& COPY;2023 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： multigrid methods Block matrix sequences Spectral symbol Stokes equation

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A SYMBOL-BASED ANALYSIS FOR multigrid methods FOR BLOCK-CIRCULANT AND BLOCK-TOEPLITZ SYSTEMS

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2022年第1期43卷 405-438页

作者： Bolten, Matthias Donatelli, Marco Ferrari, Paola Furci, Isabella Univ Wuppertal Dept Math & Informat Wuppertal Germany Univ Insubria Dept Sci & High Technol I-22100 Como Italy

In the literature, there exist several studies on symbol-based multigrid methods for the solution of linear systems having structured coefficient matrices. In particular, the convergence analysis for such methods has been obtained in an elegant form in the case of Toeplitz matrices generated by a scalar-valued function. In the block-To eplitz setting, that is, in the case where the matrix entries are small generic matrices instead of scalars, some algorithms have already been proposed regarding specific applications, and a first rigorous convergence analysis has been performed in [M. Donatelli et al., Numer. Linear Algebra Appl., 28 (2021), e2356]. However, with the existent symbol-based theoretical tools, it is still not possible to prove the convergence of many multigrid methods known in the literature. This paper aims to generalize the previous results, giving more general sufficient conditions on the symbol of the grid transfer operators. In particular, we treat matrix-valued trigonometric polynomials which can be nondiagonalizable and singular at all points, and we express the new conditions in terms of the eigenvectors associated with the ill-conditioned subspace. Moreover, we extend the analysis to the V-cycle method, proving a linear convergence rate under stronger conditions, which resemble those given in the scalar case. In order to validate our theoretical findings, we present a classical block structured problem stemming from an FEM approximation of a second order differential problem. We focus on two multigrid strategies that use the geometric and the standard bisection grid transfer operators and prove that both fall into the category of projectors satisfying the proposed conditions. In addition, using a tensor product argument, we provide a strategy to construct efficient V-cycle procedures in the block multilevel setting.

关键词： block-To eplitz matrices multigrid methods finite element methods

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multigrid methods for block-Toeplitz linear systems: convergence analysis and applications

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NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS 2021年第4期28卷 e2356-e2356页

作者： Donatelli, Marco Ferrari, Paola Furci, Isabella Serra-Capizzano, Stefano Sesana, Debora Insubria Univ Dept Sci & High Technol Como Italy Univ Wuppertal Dept Math & Informat Gaussstr 20 D-42119 Wuppertal Germany Insubria Univ Dept Humanities & Innovat Como Italy

In the past decades, multigrid methods for linear systems having multilevel Toeplitz coefficient matrices with scalar entries have been widely studied. On the other hand, only few papers have investigated the case of block entries, where the entries are small generic matrices of fixed size instead of scalars. In that case the efforts of the researchers have been mainly devoted to specific applications, focusing on algorithmic proposals, but with marginal theoretical results. In this paper, we propose a general two-grid convergence analysis, proving an optimal convergence rate independent of the matrix size, in the case of positive definite block-Toeplitz matrices with generic blocks. In particular, the proof of the approximation property is not a straightforward generalization of the scalar case and, in fact, we have to require a specific commutativity condition on the block symbol of the grid transfer operator. According to the analysis, we define a class of grid transfer operators satisfying the previous theoretical conditions and we propose a strategy to ensure fast multigrid convergence even for more than two grids. Among the numerous applications that lead to the block-Toeplitz structure, high-order Lagrangian finite element methods and staggered discontinuous Galerkin methods are considered in the numerical results, confirming the effectiveness of our proposal and the correctness of the proposed theoretical analysis.

关键词： block‐ Toeplitz sequences convergence analysis matrix valued symbol multigrid methods spectral analysis

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multigrid methods for an elliptic optimal control problem with pointwise state constraints?

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RESULTS IN APPLIED MATHEMATICS 2023年 17卷

作者： Brenner, Susanne C. Liu, Sijing Sung, Li-yeng Louisiana State Univ Dept Math Baton Rouge LA 70803 USA Louisiana State Univ Ctr Computat & Technol Baton Rouge LA 70803 USA Univ Connecticut Dept Math Storrs CT 06269 USA

We design multigrid methods for an elliptic distributed optimal control problem with pointwise state constraints. They are based on the P1 finite element method from Brenner et al. (2021), where the optimal control problem was reformulated as a variational inequality for the state variable. The performance of the multigrid methods is illustrated by numerical experiments. (c) 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://***/licenses/by-nc-nd/4.0/).

关键词： Elliptic distributed optimal control problems Pointwise state constraints multigrid methods P 1 finite element method Primal-dual active set algorithm

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EvoStencils: a grammar-based genetic programming approach for constructing efficient geometric multigrid methods

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GENETIC PROGRAMMING AND EVOLVABLE MACHINES 2021年第4期22卷 511-537页

作者： Schmitt, Jonas Kuckuk, Sebastian Koestler, Harald Friedrich Alexander Univ Erlangen Nurnberg Chair Syst Simulat Erlangen Germany

For many systems of linear equations that arise from the discretization of partial differential equations, the construction of an efficient multigrid solver is challenging. Here we present EvoStencils, a novel approach for optimizing geometric multigrid methods with grammar-guided genetic programming, a stochastic program optimization technique inspired by the principle of natural evolution. A multigrid solver is represented as a tree of mathematical expressions that we generate based on a formal grammar. The quality of each solver is evaluated in terms of convergence and compute performance by automatically generating an optimized implementation using code generation that is then executed on the target platform to measure all relevant performance metrics. Based on this, a multi-objective optimization is performed using a non-dominated sorting-based selection. To evaluate a large number of solvers in parallel, they are distributed to multiple compute nodes. We demonstrate the effectiveness of our implementation by constructing geometric multigrid solvers that are able to outperform hand-crafted methods for Poisson's equation and a linear elastic boundary value problem with up to 16 million unknowns on multi-core processors with Ivy Bridge and Broadwell microarchitecture.

关键词： Grammar-guided genetic programming multigrid methods Algorithm design Code generation Distributed computing

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Communication-Computation Overlapping for Parallel multigrid methods

Communication-Computation Overlapping for Parallel Multigrid...

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IEEE International Symposium on Parallel and Distributed Processing Workshops and Phd Forum (IPDPSW)

作者： Kengo Nakajima Information Technology Center/R-CCS The University of Tokyo/RIKEN Tokyo Japan

ISBN: (数字)9798350364606

ISBN: (纸本)9798350364613

Preconditioned iterative methods based on the Krylov subspace technique are widely employed in various scientific and technical computing. When utilizing large-scale parallel computing systems, the communication overhead tends to increase with the growth in the number of nodes, making its reduction a crucial challenge. In parallel finite element methods (FEM) and finite volume methods (FVM), halo communication and computation overlapping (CC-Overlapping) are commonly employed, often in conjunction with the dynamic loop scheduling feature of OpenMP. This approach has been primarily applied to sparse matrix-vector products (SpMV) and explicit solvers. Previous studies by the author have proposed reordering techniques for applying CC-Overlapping to processes involving global data dependencies, such as the Conjugate Gradient method preconditioned by Incomplete Cholesky Factorization (ICCG). Successful implementations on massively parallel supercomputers demonstrated high parallel performance, but the application of CC-Overlapping was limited to SpMV. In the present work, the author proposes a method to apply CC-Overlapping to the forward and backward substitutions of the IC(0) smoother of the parallel Conjugate Gradient method preconditioned by multigrid (MGCG). Using up to 4,096 nodes on Wisteria/BDEC-01 (Odyssey) with A64FX, performance improvement of approximately 40+% was achieved compared to the original implementation, while improvement was 20+% on 1,024 nodes of Oakbridge-CX system with Intel Xeon CPU's.

关键词： Gradient methods Processor scheduling multigrid methods Parallel processing Dynamic scheduling Supercomputers Finite element analysis

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On the robust solution of an isogeometric discretization of bilaplacian equation by using multigrid methods

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2020年第2期80卷 386-394页

作者： Pe de la Riva, A. Gaspar, F. J. Rodrigo, C. Univ Zaragoza IUMA Dept Matemat Aplicada Pedro Cerbuna 12 Zaragoza 50009 Spain Univ Zaragoza IUMA Dept Matemat Aplicada Maria de Luna 3 Zaragoza 50018 Spain

In this paper, we propose a multigrid method for the solution of the biharmonic problem. Isogeometric Analysis (IGA) is considered in order to easily obtain H-2 -conforming discretizations of the bilaplacian equation, which are difficult to get by means of standard finite element methods. Typically, the design of solvers for isogeometric discretizations that are robust with respect to the polynomial degree is a challenging task. Here, we achieve such robustness by using multiplicative Schwarz methods as smoothers within the multigrid algorithm. The design of the proposed solver is also supported by a local Fourier analysis (LFA), which allows us to choose appropriately the size of the block in the smoother depending on the polynomial degree of the discretization. The robustness and efficiency of the proposed multigrid method is demonstrated through numerical experiments in one- and two-dimensional cases. (C) 2019 Elsevier Ltd. All rights reserved.

关键词： Isogeometric analysis Bilaplacian equation multigrid methods Local Fourier analysis Overlapping multiplicative Schwarz methods

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