Multigrid for High-Dimensional Elliptic Partial Differential Equations on Non-equidistant Grids

Abstract

Abstract. This work presents techniques, theory, and numbers for multigrid in a general ddimensional setting. The main focus of this paper is the multigrid convergence for high-dimensional partial differential equations on non-equidistant grids such as may be encountered in a sparse-grid solution. As a model problem we have chosen the anisotropic stationary diffusion equation on a rectangular hypercube. We present some techniques for building the general d-dimensional adaptations of the multigrid components and propose grid-coarsening strategies to handle anisotropies that are induced due to discretization on a non-equidistant grid. Apart from the practical formulas and techniques, we present-in detail-the smoothing analysis of the point ω-red-black Jacobi method for a general multidimensional case. We show how relaxation parameters may be evaluated efficiently and used for better convergence. This analysis incorporates full and partial doubling and quadrupling coarsening strategies as well as the second-and the fourth-order finite difference operators. Finally we present some results derived from numerical experiments based on the test problem.Key words. multigrid, high-dimensional PDEs, anisotropic diffusion equation, coarsening strategies, point-smoothing methods, relaxation parameters, Fourier-smoothing analysis AMS subject classifications. 65N55, 65F10, 65Y20 DOI. 10.1137/060665695 Introduction. Multidimensional partial differential equations have diverse applications in various fields of applied sciences, including financial engineering [8], molecular biology [3], and quantum dynamics [1,21]. There are quite a few fast and efficient solution techniques for partial differential equations (henceforth PDEs) of which multigrid ranks among the best. Multigrid is a well-known iterative procedure for the solution of large and sparse linear systems that arise from various kinds of PDE discretizations. The existing literature on the multigrid treatment of various problems, however, rarely explores issues that arise out of growth in the dimensionality of the problem. The implications of dimensionality growth include deterioration of the multigrid convergence rate, impractical storage requirements, and huge amounts of CPU time for single grid solution methods. Our main emphasis in this paper lies on the first challenge. We abbreviate multigrid for d-dimensional PDEs as d-multigrid. In this paper d represents both abstract dimensionality and dimensions. So, e.g., 3d is to be interpreted as 3-dimensional.The major hinderance in the numerical solution of multidimensional PDEs is the so-called curse of dimensionality, which implies that with the growth in dimensions we have an exponential growth in the number of grid points. This increases the

A Flexible Generalized Conjugate Residual Method with Inner Orthogonalization and Deflated Restarting

“…These sequences correspond to situations where only the righthand sides are changing for a given dimension d. An efficient solution method is of primary interest in certain applications related to e.g. financial engineering, molecular biology or quantum dynamics [5,6]. In the numerical experiments reported here (performed in Matlab) we have used second order finite difference discretization schemes leading to sparse matrices with at most 2d+ 1 nonzero elements per row.…”

Fourier Analysis of Periodic Stencils in Multigrid Methods

“…In the case that A is a constant stencil operator, define its Fourier representation A ∈ L(H h ; H h ) by (2) AF h = F h A .…”

Efficientd-multigrid preconditioners for sparse-grid solution of high-dimensional partial differential equations

“…In [19] we pointed out that for grids stretched along a single dimension, a method based on partial quadrupling as the coarsening strategy has an O(M) complexity even with W cycles; achieving this is not possible with methods based on partial doubling along one dimension. This fact makes it a strategy of choice for such grids, figure 6.…”

“…These techniques can be extended to arbitrary higher d. A relaxation method based on hyperplane relaxation has been proposed in [18], which is analogous to line-relaxation in two dimensions. Contrary to that approach, we proposed a method based on point smoothing and partial coarsening schemes in [19]. There we treat discrete anisotropies induced by nonequidistant grids.…”

“…With the coarsening strategy, the discrete coarse grid problems are defined, as in Example (17). The set-up stage ends with the construction of the transfer operators between the various coarse grids, as in (18), (19). They are also dictated by the coarse grids.…”