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.

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An Introduction to Multigrid Methods

Matching of locally variant data to an explicit 3-dimensional pictorial model is developed for X-ray computed tomography scans of the human brain, where the model is a voxel representation of an anatomical human brain atlas. The matching process is 3-dimensional without any preference given to the slicing plane. After global alignment the brain atlas is deformed like a piece of rubber, without tearing or folding. Deformation proceeds step-by-step in a coarse-to-fine strategy, increasing the local similarity and global coherence. The assumption underlying this approach is that all normal brains, at least at a certain level of representation, have the same topological structure, but may differ in shape details. Results show that we can account for these differences.

Multiresolution elastic matching

Multigrid methods: Fundamental algorithms, model problem analysis and applications

Dense and accurate motion tracking is an important requirement for many video feature extraction algorithms. In this paper we provide a method for computing point trajectories based on a fast parallel implementation of a recent optical flow algorithm that tolerates fast motion. The parallel implementation of large displacement optical flow runs about 78× faster than the serial C++ version. This makes it practical to use in a variety of applications, among them point tracking. In the course of obtaining the fast implementation, we also proved that the fixed point matrix obtained in the optical flow technique is positive semi-definite. We compare the point tracking to the most commonly used motion tracker - the KLT tracker - on a number of sequences with ground truth motion. Our resulting technique tracks up to three orders of magnitude more points and is 46% more accurate than the KLT tracker. It also provides a tracking density of 48% and has an occlusion error of 3% compared to a density of 0.1% and occlusion error of 8% for the KLT tracker. Compared to the Particle Video tracker, we achieve 66% better accuracy while retaining the ability to handle large displacements while running an order of magnitude faster.

https://link.springer.com/content/pdf/10.1007%2F978-3-642-15549-9_32.pdf

Dense point trajectories by GPU-accelerated large displacement optical flow

Algebraic multigrid (AMG): experiences and comparisons

A note on MGR methods

Publisher Summary This chapter examines the nonstandard multigrid (MG) techniques using checkered relaxation and intermediate grids. The MG techniques offer sensational perspectives in the numerical treatment of partial differential equations. There are close connections between the MG and the total reduction ideas suggesting certain combinations leading to the so called MGR methods. The quantitative results refer to model problems in the unit square with Dirichlet or Neumann boundary conditions, respectively. With respect to both theoretical rate of convergence, and the computational effort, MG methods turn out to be considerably superior to standard MG methods. MGR-CH-1 and MGR-CH-2 turn out to be the most efficient methods. It is observed that as only five-point operators are used in these methods, they can in principle be applied to very general problems. It is found that checkered relaxation techniques yield considerable improvements also if no intermediate grids are used explicitly.

Non-standard multigrid techniques using checkered relaxation and intermediate grids

We report on numerical investigations which were made in the development of the code SIHEM (SImulation of HEating processes in Metals) In SIHEM the associated nonlinear parabolic problem (2D-space variables) is treated by a combination of simple (low order) space and implicit time discretizations (Crank-Nicolson) with a time step size control, Newton's or a Newton-like linearization and the use of Fast Elliptic Solvers for the large linear systems In this report, the emphasis is laid upon systematic investigations and comparisons involving the use of typical Fast Solvers and well-known classical methods As the solvers are applied at each time and each linearization step the total computing time considerably depends on their efficiency As a result, we show that the use of a special Multigrid Solver in connection with the time step size control yields a very efficient composite algorithm

Ulrich Trottenberg

Papers

Multigrid methods: Fundamental algorithms, model problem analysis and applications

A note on MGR methods

Non-standard multigrid techniques using checkered relaxation and intermediate grids

Efficient solution of a nonlinear heat conduction problem by use of fast elliptic reduction and multigrid methods

On faces and W-irreducible operators in finite-dimensional linear spaces