Report on the Copper Mountain Conference on Multigrid Methods

OAK B188 Report on the Copper Mountain Conference on Multigrid Methods. The Copper Mountain Conference on Multigrid Methods was held on April 11-16, 1999. Over 100 mathematicians from all over the world attended the meeting. The conference had two major themes: algebraic multigrid and parallel multigrid. During the five day meeting 69 talks on current research topics were presented as well as 3 tutorials. Talks with similar content were organized into sessions. Session topics included: Fluids; Multigrid and Multilevel Methods; Applications; PDE Reformulation; Inverse Problems; Special Methods; Decomposition Methods; Student Paper Winners; Parallel Multigrid; Parallel Algebraic Multigrid; and FOSLS

A multi-level preconditioned Krylov method for the efficient solution of algebraic tomographic reconstruction problems

Classical iterative methods for tomographic reconstruction include the class of Algebraic Reconstruction Techniques (ART). Convergence of these stationary linear iterative methods is however notably slow. In this paper we propose the use of Krylov solvers for tomographic linear inversion problems. These advanced iterative methods feature fast convergence at the expense of a higher computational cost per iteration, causing them to be generally uncompetitive without the inclusion of a suitable preconditioner. Combining elements from standard multigrid (MG) solvers and the theory of wavelets, a novel wavelet-based multi-level (WMG) preconditioner is introduced, which is shown to significantly speed-up Krylov convergence. The performance of the WMG-preconditioned Krylov method is analyzed through a spectral analysis, and the approach is compared to existing methods like the classical Simultaneous Iterative Reconstruction Technique (SIRT) and unpreconditioned Krylov methods on a 2D tomographic benchmark problem. Numerical experiments are promising, showing the method to be competitive with the classical Algebraic Reconstruction Techniques in terms of convergence speed and overall performance (CPU time) as well as precision of the reconstruction.Comment: Journal of Computational and Applied Mathematics (2014), 26 pages, 13 figures, 3 table

Local Fourier Analysis of the Complex Shifted Laplacian preconditioner for Helmholtz problems

In this paper we solve the Helmholtz equation with multigrid preconditioned Krylov subspace methods. The class of Shifted Laplacian preconditioners are known to significantly speed-up Krylov convergence. However, these preconditioners have a parameter beta, a measure of the complex shift. Due to contradictory requirements for the multigrid and Krylov convergence, the choice of this shift parameter can be a bottleneck in applying the method. In this paper, we propose a wavenumber-dependent minimal complex shift parameter which is predicted by a rigorous k-grid Local Fourier Analysis (LFA) of the multigrid scheme. We claim that, given any (regionally constant) wavenumber, this minimal complex shift parameter provides the reader with a parameter choice that leads to efficient Krylov convergence. Numerical experiments in one and two spatial dimensions validate the theoretical results. It appears that the proposed complex shift is both the minimal requirement for a multigrid V-cycle to converge, as well as being near-optimal in terms of Krylov iteration count.Comment: 20 page

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

Fast and efficient solution techniques are developed for high-dimensional parabolic partial differential equations (PDEs). In this paper we present a robust solver based on the Krylov subspace method Bi-CGSTAB combined with a powerful, and efficient, multigrid preconditioner. Instead of developing the perfect multigrid method, as a stand-alone solver for a single problem discretized on a certain grid, we aim for a method that converges well for a wide class of discrete problems arising from discretization on various anisotropic grids. This is exactly what we encounter during a sparse grid computation of a high-dimensional problem. Different multigrid components are discussed and presented with operator construction formulae. An option-pricing application is focused and presented with results computed with this method

PDE Solvers for Hybrid CPU-GPU Architectures

Many problems of scientific and industrial interest are investigated through numerically solving partial differential equations (PDEs). For some of these problems, the scope of the investigation is limited by the costs of computational resources. A new approach to reducing these costs is the use of coprocessors, such as graphics processing units (GPUs) and Many Integrated Core (MIC) cards, which can execute floating point operations at a higher rate than a central processing unit (CPU) of the same cost. This is achieved through the use of a large number of processors in a single device, each with very limited dedicated memory per thread. Codes for a number of continuum methods, such as boundary element methods (BEM), finite element methods (FEM) and finite difference methods (FDM) have already been implemented on coprocessor architectures. These methods were designed before the adoption of coprocessor architectures, so implementing them efficiently with reduced thread-level memory can be challenging. There are other methods that do operate efficiently with limited thread-level memory, such as Monte Carlo methods (MCM) and lattice Boltzmann methods (LBM) for kinetic formulations of PDEs, but they are not competitive on CPUs and generally have poorer convergence than the continuum methods. In this work, we introduce a class of methods in which the parallelism of kinetic formulations on GPUs is combined with the better convergence of continuum methods on CPUs. We first extend an existing Feynman-Kac formulation for determining the principal eigenpair of an elliptic operator to create a version that can retrieve arbitrarily many eigenpairs. This new method is implemented for multiple GPUs, and combined with a standard deflation preconditioner on multiple CPUs to create a hybrid concurrent method with superior convergence to that of the deflation preconditioner alone. The hybrid method exhibits good parallelism, with an efficiency of 80% on a problem with 300 million unknowns, run on a configuration of 324 CPU cores and 54 GPUs.Doctor of Philosoph

A Newton-Krylov Solution to the Coupled Neutronics-Porous Medium Equations.

The solution of the coupled field equations for nuclear reactor analysis has typically been performed by solving separately the individual field equations and transferring information between fields. This has generally been referred to as “operating splitting” and has been applied to a wide range of reactor steady-state and transient problems. Although this approach has generally been successful, it has been computationally inefficient and imposed some limitations on the range of problems considered. The research here investigated fully implicit methods which do not split the coupled field operators and the solution of the coupled equations using Neutron-Krylov methods. The focus of the work here was on the solution of the coupled neutron and temperature/fluid field equations for the specific application to the high temperature gas reactor. The solution of the neutron field equations was restricted to the steady-state multi-group neutron diffusion equations and the temperature fluid solution for the gas reactor involved the solution of the solid energy, fluid energy, and the single phase mass-momentum equations. In the research performed here, several Newton-Krylov solution approaches have been employed to improve the behavior and performance of the coupled neutronics / porous medium equations as implemented in the PARCS/AGREE code system. The Exact and Inexact Newton's method were employed first, using an analytical Jacobian, followed by a finite difference based Jacobian, and lastly a Jacobian-Free method was employed for the thermal-fluids. Results in the thermal fluids indicate that the Exact Newton's method outperformed the other methods, including the current operator split solution. Finite difference Jacobian and Jacobian-Free were slighty slower than the current solution, though fewer outer iterations were required. In the coupled solution, the exact Newton method performed the best. The finite difference Jacobian with optimized perturbation integrated into the GMRES solve also performed very well, which represented the best iterative solution to the coupled problem. Future analysis will consider the transient problem.Ph.D.Nuclear Engineering & Radiological SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/91609/1/wardam_1.pd

Fourier Analysis of GMRES(m) preconditioned by multigrid

: This paper deals with convergence estimations of a preconditioned GMRES(m) method [6], where multigrid ([1], [8]) is used as the preconditioner. Fourier analysis is a well-known useful tool in the multigrid community for the prediction of two-grid convergence rates ([1], [8]). This analysis is generalized to the situation where multigrid is a preconditioner. 1 Introduction Nowadays, it has become popular to study the convergence of multilevel methods and use them in combination with a Krylov subspace acceleration method. Recently, also standard (multiplicative) multigrid methods are used as a preconditioner. This application of standard multigrid is beneficial in situations where standard multigrid alone does not converge fully satisfactorily, because certain error frequencies are not reduced well enough, which mainly occurs when complicated PDEs are solved. In general, it is difficult to construct robust multigrid solvers for large classes of problems. Among many other papers, thi..

Contribution to the study of efficient iterative methods for the numerical solution of partial differential equations

Multigrid and domain decomposition methods provide efficient algorithms for the numerical solution of partial differential equations arising in the modelling of many applications in Computational Science and Engineering. This manuscript covers certain aspects of modern iterative solution methods for the solution of large-scale problems issued from the discretization of partial differential equations. More specifically, we focus on geometric multigrid methods, non-overlapping substructuring methods and flexible Krylov subspace methods with a particular emphasis on their combination. Firstly, the combination of multigrid and Krylov subspace methods is investigated on a linear partial differential equation modelling wave propagation in heterogeneous media. Secondly, we focus on non-overlapping domain decomposition methods for a specific finite element discretization known as the hp finite element, where unrefinement/refinement is allowed both by decreasing/increasing the step size h or by decreasing/increasing the polynomial degree p of the approximation on each element. Results on condition number bounds for the domain decomposition preconditioned operators are given and illustrated by numerical results on academic problems in two and three dimensions. Thirdly, we review recent advances related to a class of Krylov subspace methods allowing variable preconditioning. We examine in detail flexible Krylov subspace methods including augmentation and/or spectral deflation, where deflation aims at capturing approximate invariant subspace information. We also present flexible Krylov subspace methods for the solution of linear systems with multiple right-hand sides given simultaneously. The efficiency of the numerical methods is demonstrated on challenging applications in seismics requiring the solution of huge linear systems of equations with multiple right-hand sides on parallel distributed memory computers. Finally, we expose current and future prospectives towards the design of efficient algorithms on extreme scale machines for the solution of problems coming from the discretization of partial differential equations