A flexible and adaptive Simpler GMRES with deflated restarting for shifted linear systems

In this paper, two efficient iterative algorithms based on the simpler GMRES method are proposed for solving shifted linear systems. To make full use of the shifted structure, the proposed algorithms utilizing the deflated restarting strategy and flexible preconditioning can significantly reduce the number of matrix-vector products and the elapsed CPU time. Numerical experiments are reported to illustrate the performance and effectiveness of the proposed algorithms.Comment: 17 pages. 9 Tables, 1 figure; Newly update: add some new numerical results and correct some typos and syntax error

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

Recycling Krylov Subspaces for Efficient Partitioned Solution of Aerostructural Adjoint Systems

Robust and efficient solvers for coupled-adjoint linear systems are crucial to successful aerostructural optimization. Monolithic and partitioned strategies can be applied. The monolithic approach is expected to offer better robustness and efficiency for strong fluid-structure interactions. However, it requires a high implementation cost and convergence may depend on appropriate scaling and initialization strategies. On the other hand, the modularity of the partitioned method enables a straightforward implementation while its convergence may require relaxation. In addition, a partitioned solver leads to a higher number of iterations to get the same level of convergence as the monolithic one. The objective of this paper is to accelerate the fluid-structure coupled-adjoint partitioned solver by considering techniques borrowed from approximate invariant subspace recycling strategies adapted to sequences of linear systems with varying right-hand sides. Indeed, in a partitioned framework, the structural source term attached to the fluid block of equations affects the right-hand side with the nice property of quickly converging to a constant value. We also consider deflation of approximate eigenvectors in conjunction with advanced inner-outer Krylov solvers for the fluid block equations. We demonstrate the benefit of these techniques by computing the coupled derivatives of an aeroelastic configuration of the ONERA-M6 fixed wing in transonic flow. For this exercise the fluid grid was coupled to a structural model specifically designed to exhibit a high flexibility. All computations are performed using RANS flow modeling and a fully linearized one-equation Spalart-Allmaras turbulence model. Numerical simulations show up to 39% reduction in matrix-vector products for GCRO-DR and up to 19% for the nested FGCRO-DR solver.Comment: 42 pages, 21 figure

Efficient variants of the CMRH method for solving a sequence of multi-shifted non-Hermitian linear systems simultaneously

Multi-shifted linear systems with non-Hermitian coefficient matrices arise in numerical solutions of time-dependent partial/fractional differential equations (PDEs/FDEs), in control theory, PageRank problems, and other research fields. We derive efficient variants of the restarted Changing Minimal Residual method based on the cost-effective Hessenberg procedure (CMRH) for this problem class. Then, we introduce a flexible variant of the algorithm that allows to use variable preconditioning at each iteration to further accelerate the convergence of shifted CMRH. We analyse the performance of the new class of methods in the numerical solution of PDEs and FDEs, also against other multi-shifted Krylov subspace methods.Comment: Techn. Rep., Univ. of Groningen, 34 pages. 11 Tables, 2 Figs. This manuscript was submitted to a journal at 20 Jun. 2016. Updated version-1: 31 pages, 10 tables, 2 figs. The manuscript was resubmitted to the journal at 9 Jun. 2018. Updated version-2: 29 pages, 10 tables, 2 figs. Make it concise. Updated version-3: 27 pages, 10 tables, 2 figs. Updated version-4: 28 pages, 10 tables, 2 fig

A perturbed two-level preconditioner for the solution of three-dimensional heterogeneous Helmholtz problems with applications to geophysics

Le sujet de cette thèse est le développement de méthodes itératives permettant la résolution de grands systèmes linéaires creux d'équations présentant plusieurs seconds membres simultanément. Ces méthodes seront en particulier utilisées dans le cadre d'une application géophysique : la migration sismique visant à simuler la propagation d'ondes sous la surface de la terre. Le problème prend la forme d'une équation d'Helmholtz dans le domaine fréquentiel en trois dimensions, discrétisée par des différences finies et donnant lieu à un système linéaire creux, complexe, non-symétrique, non-hermitien. De plus, lorsque de grands nombres d'onde sont considérés, cette matrice possède une taille élevée et est indéfinie. Du fait de ces propriétés, nous nous proposons d'étudier des méthodes de Krylov préconditionnées par des techniques hiérarchiques deux niveaux. Un tel pre-conditionnement s'est montré particulièrement efficace en deux dimensions et le but de cette thèse est de relever le défi de l'adapter au cas tridimensionel. Pour ce faire, des méthodes de Krylov sont utilisées à la fois comme lisseur et comme méthode de résolution du problème grossier. Ces derniers choix induisent l'emploi de méthodes de Krylov dites flexibles. ABSTRACT : The topic of this PhD thesis is the development of iterative methods for the solution of large sparse linear systems of equations with possibly multiple right-hand sides given at once. These methods will be used for a specific application in geophysics - seismic migration - related to the simulation of wave propagation in the subsurface of the Earth. Here the three-dimensional Helmholtz equation written in the frequency domain is considered. The finite difference discretization of the Helmholtz equation with the Perfect Matched Layer formulation produces, when high frequencies are considered, a complex linear system which is large, non-symmetric, non-Hermitian, indefinite and sparse. Thus we propose to study preconditioned flexible Krylov subspace methods, especially minimum residual norm methods, to solve this class of problems. As a preconditioner we consider multi-level techniques and especially focus on a two-level method. This twolevel preconditioner has shown efficient for two-dimensional applications and the purpose of this thesis is to extend this to the challenging three-dimensional case. This leads us to propose and analyze a perturbed two-level preconditioner for a flexible Krylov subspace method, where Krylov methods are used both as smoother and as approximate coarse grid solver