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

Global Range Restricted GMRES for Linear Systems with Multiple Right Hand Sides

This work concerns the solution of non-symmetric, sparse linear systems with multiple right hand sides by iterative methods. Herein a global version of the range restricted generalized minimal residual method (RRGMRES) is proposed for solving this sort of problems. Numerical results confirm that this new algorithm is applicable

GMRES implementations and residual smoothing techniques for solving ill-posed linear systems

AbstractThere are verities of useful Krylov subspace methods to solve nonsymmetric linear system of equations. GMRES is one of the best Krylov solvers with several different variants to solve large sparse linear systems. Any GMRES implementation has some advantages. As the solution of ill-posed problems are important. In this paper, some GMRES variants are discussed and applied to solve these kinds of problems. Residual smoothing techniques are efficient ways to accelerate the convergence speed of some iterative methods like CG variants. At the end of this paper, some residual smoothing techniques are applied for different GMRES methods to test the influence of these techniques on GMRES implementations

Krylov subspace methods and their generalizations for solving singular linear operator equations with applications to continuous time Markov chains

Viele Resultate über MR- und OR-Verfahren zur Lösung linearer Gleichungssysteme bleiben (in leicht modifizierter Form) gültig, wenn der betrachtete Operator nicht invertierbar ist. Neben dem für reguläre Probleme charakteristischen Abbruchverhalten, kann bei einem singulären Gleichungssystem auch ein so genannter singulärer Zusammenbruch auftreten. Für beide Fälle werden verschiedene Charakterisierungen angegeben. Die Unterrauminverse, eine spezielle verallgemeinerte Inverse, beschreibt die Näherungen eines MR-Unterraumkorrektur-Verfahrens. Für Krylov-Unterräume spielt die Drazin-Inverse eine Schlüsselrolle. Bei Krylov-Unterraum-Verfahren kann a-priori entschieden werden, ob ein regulärer oder ein singulärer Abbruch auftritt. Wir können zeigen, dass ein Krylov-Verfahren genau dann für beliebige Startwerte eine Lösung des linearen Gleichungssystems liefert, wenn der Index der Matrix nicht größer als eins und das Gleichungssystem konsistent ist. Die Berechnung stationärer Zustandsverteilungen zeitstetiger Markov-Ketten mit endlichem Zustandsraum stellt eine praktische Aufgabe dar, welche die Lösung eines singulären linearen Gleichungssystems erfordert. Die Eigenschaften der Übergangs-Halbgruppe folgen aus einfachen Annahmen auf rein analytischem und matrixalgebrischen Wege. Insbesondere ist die erzeugende Matrix eine singuläre M-Matrix mit Index 1. Ist die Markov-Kette irreduzibel, so ist die stationäre Zustandsverteilung eindeutig bestimmt