These notes were written for an introductory course on the application of multigrid methods to elliptic and hyperbolic partial differential equations for engineers, physicists and applied mathematicians. The use of more advanced mathematical tools, such as functional analysis, is avoided. The course is intended to be accessible to a wide audience of users of computational methods. We restrict ourselves to finite volume and finite difference discretization. The basic principles are given. Smoothing methods and Fourier smoothing analysis are reviewed. The fundamental multigrid algorithm is studied. The smoothing and coarse grid approximation properties are discussed. Multigrid schedules and structured programming of multigrid algorithms are treated. Robustness and efficiency are considered.

/pdf/an-introduction-to-multigrid-methods-3lcrqskq28.pdf

An Introduction to Multigrid Methods

We present IDR($s$), a new family of efficient, short-recurrence methods for large nonsymmetric systems of linear equations. The new methods are based on the induced dimension reduction (IDR) method proposed by Sonneveld in 1980. IDR($s$) generates residuals that are forced to be in a sequence of nested subspaces. Although IDR($s$) behaves like an iterative method, in exact arithmetic it computes the true solution using at most $N + N/s$ matrix-vector products, with $N$ the problem size and $s$ the codimension of a fixed subspace. We describe the algorithm and the underlying theory and present numerical experiments to illustrate the theoretical properties of the method and its performance for systems arising from different applications. Our experiments show that IDR($s$) is competitive with or superior to most Bi-CG-based methods and outperforms Bi-CGSTAB when $s > 1$.

IDR($s$): A Family of Simple and Fast Algorithms for Solving Large Nonsymmetric Systems of Linear Equations

In the present paper, a detailed analysis of a multigrid method with an ILU smoother applied to a singularly perturbed problem is given. Based on the analysis of a simple anisotropic model problem, a variant of the usual incomplete LU factorization is introduced, which is especially suited as robust smoother. For this variant a detailed analysis and a proof of robustness is given. Furthermore, some contradictions between the smoothing rates predicted by local Fourier analysis and the practically observed convergence factors are explained (see [W. Hackbusch, Multi-grid Methods and Applications, Springer-Verlag, Berlin, Heidelberg, 1985; R. Kettler, “Analysis and comparison of relaxation schemes in robust multi-grid and preconditioned conjugate gradient methods,” in Multi-grid Methods, Lecture Notes in Math. 960, Springer-Verlag, Berlin, 1982; C. A. Thole, Beitrage zur Fourieranalyse von Mehrgitterver fahren, Diplomarbeit, Universitat Bonn, 1983]. The theoretical results are confirmed by numerical tests.

On the Robustness of ILU Smoothing

Most of the fundamental problems in physics, chemistry and engineering involve computation too hard even for future supercomputers, if conventional mathematical approaches are used. The reason is always a product of several complexity factors associated with the wide range of space and time scales characteristic to such problems. Each of these complexity factors can in principle be removed by various multiscale algorithms, i.e., employing separate processing at each scale of the problem, combined with interscale iterative interactions. A wide range of multiscale computational methods is described, emphasizing main ideas and inter-relations between various fields. The reported areas include: top-efficiency multigrid methods in fluid dynamics; inverse PDE problems and data assimilation; feedback optimal control; PDE solvers on unbounded domains and on adaptable grids; wave/ray methods for highly indefinite equations; rigorous quantitative analysis of multigrid; manyeigenfunction problems and ab-initio quantum chemistry; fast evaluation of integral transforms on adaptive grids; multigrid Dirac solvers; fast inverse-matrix and determinant calculations and updates; multiscale Monte-Carlo methods in statistical physics, including the renormalization multigrid (RMG) methods; molecular mechanics (including fast force summation, fast macromolecular energy minimization, and Monte-Carlo methods at equilibrium, both for macromolecules and for large ensembles of small molecules); combination of small-scale equilibrium with large-scale dynamics; image processing (edge detection and picture segmentation); tomography (medical imaging and radar reconstruction); efficient, general and highly accurate algebraic multigrid (AMG) and homogenization schemes; fast practical graph algorithms; data clustering; and multiscale approaches to global optimization.

P. Sonneveld

Papers

Numerical experiments with a multiple grid and a preconditioned Lanczos type method