In recent contributions, algebraic multigrid methods have been designed and studied from the viewpoint of the spectral complementarity. In this note we focus our efforts on specific applications and, more precisely, on large linear systems arising from the approximation of weighted Laplacian with various boundary conditions. We adapt the multigrid idea to this specific setting and we present and critically discuss a wide numerical experimentation showing the potentiality of the considered approach.
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. Under suitable assumption, local Fourier analysis shows the proposed Vanka-type smoother achieves a uniform smoothing factor, which is verified by several numerical examples.
The multigrid-reduction-in-time (MGRIT) technique has proven to be successful in achieving higher run-time speedup by exploiting parallelism in time. The goal of this article is to develop and analyze a MGRIT algorithm, using FCF-relaxation with time-dependent time-grid propagators, to seek the finite element approximations of unsteady fractional Laplacian problems. The multigrid with line smoother proposed in [L. Chen, R. H. Nochetto, E. Ot{á}rola, A. J. Salgado, Math. Comp. 85 (2016) 2583--2607] is chosen to be the spatial solver. Motivated by [B. S. Southworth, SIAM J. Matrix Anal. Appl. 40…
Aspects of an adaptive finite element method for the fractional Laplacian: a priori and a posteriori error estimates, efficient implementation and multigrid solver
Brown University（布朗大学）；Oak Ridge National Laboratory（橡树岭国家实验室）
A Multigrid Optimization Algorithm for the Numerical Solution of Quasilinear Variational Inequalities Involving the $p$-Laplacian
Escuela Politécnica Nacional（厄瓜多尔国立理工学院）；Research Center on Mathematical Modeling (MODEMAT)（数学建模研究中心）；Departamento de Matemática - Escuela Politécnica Nacional（厄瓜多尔国立理工学院数学系）
Coupled multiphysics problems often give rise to interface conditions naturally formulated in fractional Sobolev spaces. Here, both positive and negative fractionality are common. When designing efficient solvers for discretizations of such problems it would then be useful to have a preconditioner for the fractional Laplacian, $(-Δ)^s$, with $s \in [-1,1]$. Previously, additive multigrid preconditioners for the case when $s \geq 0$ have been proposed. In this work we complement this construction with auxiliary space preconditioners suitable when $s \leq 0$. These preconditioners are shown to b…
Emory University（埃默里大学）；Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）
Multigrid methods are one of the most efficient techniques for solving linear systems arising from Partial Differential Equations (PDEs) and graph Laplacians from machine learning applications. One of the key components of multigrid is smoothing, which aims at reducing high-frequency errors on each grid level. However, finding optimal smoothing algorithms is problem-dependent and can impose challenges for many problems. In this paper, we propose an efficient adaptive framework for learning optimized smoothers from operator stencils in the form of convolutional neural networks (CNNs). The CNNs …
In this note we present a multigrid preconditioning method for solving quadratic optimization problems constrained by a fractional diffusion equation. Multigrid methods within the all-at-once approach to solve the first order-order optimality Karush-Kuhn-Tucker (KKT) systems are widely popular, but their development have relied on the underlying systems being sparse. On the other hand, for most discretizations, the matrix representation of fractional operators is expected to be dense. We develop a preconditioning strategy for our problem based on a reduced approach, namely we eliminate the sta…
In this paper we propose and analyze new efficient sparse approximate inverse (SPAI) smoothers for solving the two-dimensional (2D) and three-dimensional (3D) Laplacian linear system with geometric multigrid methods. Local Fourier analysis shows that our proposed SPAI smoother for 2D achieves a much smaller smoothing factor than the state-of-the-art SPAI smoother studied in [Bolten, M., Huckle, T.K. and Kravvaritis, C.D., 2016. Sparse matrix approximations for multigrid methods. Linear Algebra and its Applications, 502, pp.58-76.]. The proposed SPAI smoother for 3D cases provides smaller optim…
This note describes the full approximation storage (FAS) multigrid scheme for an easy one-dimensional nonlinear boundary value problem. The problem is discretized by a simple finite element (FE) scheme. We apply both FAS V-cycles and F-cycles, with a nonlinear Gauss-Seidel smoother, to solve the resulting finite-dimensional problem. The mathematics of the FAS restriction and prolongation operators, in the FE case, are explained. A self-contained Python program implements the scheme. Optimal performance, i.e. work proportional to the number of unknowns, is demonstrated for both kinds of cycles,…
Large linear systems of saddle-point type have arisen in a wide variety of applications throughout computational science and engineering. The discretizations of distributed control problems have a saddle-point structure. The numerical solution of saddle-point problems has attracted considerable interest in recent years. In this work, we propose a novel Braess-Sarazin multigrid relaxation scheme for finite element discretizations of the distributed control problems, where we use the stiffness matrix obtained from the five-point finite difference method for the Laplacian to approximate the inver…
We consider effective preconditioners for solving Laplacians of general weighted graphs. Theoretically, spectral sparsifiers (SSs) provide preconditioners of optimal computational complexity. However, they are not easy to use for real-world applications due to the implementation complications. Multigrid (MG) methods, on the contrary, are computationally efficient but lack of theoretical justifications. To bridge the gap between theory and practice, we adopt ideas of MG and SS methods and proposed preconditioners that can be used in practice with theoretical guarantees. We expand the original g…
