A framework for deflated and augmented Krylov subspace methods

We consider deflation and augmentation techniques for accelerating the convergence of Krylov subspace methods for the solution of nonsingular linear algebraic systems. Despite some formal similarity, the two techniques are conceptually different from preconditioning. Deflation (in the sense the term is used here) "removes" certain parts from the operator making it singular, while augmentation adds a subspace to the Krylov subspace (often the one that is generated by the singular operator); in contrast, preconditioning changes the spectrum of the operator without making it singular. Deflation and augmentation have been used in a variety of methods and settings. Typically, deflation is combined with augmentation to compensate for the singularity of the operator, but both techniques can be applied separately. We introduce a framework of Krylov subspace methods that satisfy a Galerkin condition. It includes the families of orthogonal residual (OR) and minimal residual (MR) methods. We show that in this framework augmentation can be achieved either explicitly or, equivalently, implicitly by projecting the residuals appropriately and correcting the approximate solutions in a final step. We study conditions for a breakdown of the deflated methods, and we show several possibilities to avoid such breakdowns for the deflated MINRES method. Numerical experiments illustrate properties of different variants of deflated MINRES analyzed in this paper.Comment: 24 pages, 3 figure

Subdomain deflation combined with local AMG: a case study using AMGCL library

The final publication is available at Springer via http://dx.doi.org/10.1134/S1995080220040071The paper proposes a combination of the subdomain deflation method and local algebraic multigrid as a scalable distributed memory preconditioner that is able to solve large linear systems of equations. The implementation of the algorithm is made available for the community as part of an open source AMGCL library. The solution targets both homogeneous (CPU-only) and heterogeneous (CPU/GPU) systems, employing hybrid MPI/OpenMP approach in the former and a combination of MPI, OpenMP, and CUDA in the latter cases. The use of OpenMP minimizes the number of MPI processes, thus reducing the communication overhead of the deflation method and improving both weak and strong scalability of the preconditioner. The examples of scalar (single degree of freedom per grid node), Poisson-like, systems as well as non-scalar problems, stemming out of the discretization of the Navier-Stokes equations, are considered in order to estimate performance of the implemented algorithm. A comparison with a traditional global AMG preconditioner based on a well-established Trilinos ML package is provided.Contribution of Dr. Demidov was funded by the state assignment to the Joint Supercomputer Center of theRussian Academy of Sciences for Scientific Research and Russian Foundation for Basic Research, grant no. 18-07-00964. Dr. Rossi acknowledges the financial support to CIMNE via the CERCA Programme/Generalitat de Catalunya and the support of the ExaQUte FetHPC, project GA 800898. The authors thankfully acknowledge the support of the PRACE program (project 2010PA4058), in providing access to the MareNostrum 4 and PizDaint clusters. Without such resources the testing would not have been possible. The help of Prof. Labarta of the POP Center of Excellence in improving the NUMA scalability of the solver is also gratefully acknowledged.Peer ReviewedPostprint (author's final draft

Deflation of the Finite Pointset Method

In this thesis a deflation method for the Finite Pointset Method (FPM) is presented. FPM is a particle method based on Lagrangian coordinates to solve problems in fluid dynamics. A strong formulation of the occuring differential equations is produced by FPM, and the linear system of equations obtained by an implicit approach is solved by an iterative method such as BiCGSTAB. To improve the convergence rate of BiCGSTAB, the computational domain is divided into a number of deflation cells and a projection between the deflated domain and the original domain is constructed with the help of different ansatz functions, either constant, linear or quadratic. Also, the Moore Penrose pseudoinverse of the projection is computed. Applying the projection and restriction to the linear FPM system, a deflated system is obtained which can easily be solved with a direct method. The deflated solution is then projected onto the full domain. The deflation is tested for a number of test cases in one and two dimensions. Constant ansatz functions provide acceptable results for Dirichlet problems, but give big errors when deflating problems with mixed boundary conditions. Linear ansatz functions provide good approximations which converge to the exact solution as the number of deflation cells increases. Quadratic ansatz functions provide deflated solutions as good as the exact solutions for all test cases but are computationally expensive. The BiCGSTAB convergence rate is improved when using a deflated solution as initial guess, compared to using the zero-vector. The size of the improvement varies between which ansatz functions are used. Overall, the proposed method provides an increased convergence rate in the BiCGSTAB algorithm for FPM. However, the computational effort in the deflation process should also be taken into account

Implementation of the deflated variants of the conjugate gradient method

Sdružené gradienty jsou jednou z nejpoužívanějších metod pro řešení rozsáhlých soustav lineárních rovnic se symetrickou pozitivně-semidefinitní maticí. Jeden ze způsobů urychlení konvergence metody je deflace. Principem deflace je skrývání té části spektra matice, která způsobuje zpomalení konvergence. Tato diplomová práce se zabývá efektivní implementací různých deflated verzí sdružených gradientů. Velká pozornost je také věnována teorii a volbě deflačního prostoru. Možnosti implementace jsou demonstrovány na rozsáhlém množství příkladůThe conjugate gradient algorithm is one of the most popular methods for the solution of large systems of linear equations with symmetric positive semi-definite matrix. One of the schemes accelerating the convergence of conjugate gradients is deflation which effectively hides parts of the matrix spectrum that slows down the convergence. This master's thesis deals with efficient parallel implementation of the deflated conjugate gradient method with various modifications. Detailed theoretical considerations and the crucial choice of the deflation space are also discussed. The implementation is showcased on a wide range of benchmarks9600 - IT4Innovationsvýborn

A General Algorithm for Reusing Krylov Subspace Information. I. Unsteady Navier-Stokes

A general algorithm is developed that reuses available information to accelerate the iterative convergence of linear systems with multiple right-hand sides A x = b (sup i), which are commonly encountered in steady or unsteady simulations of nonlinear equations. The algorithm is based on the classical GMRES algorithm with eigenvector enrichment but also includes a Galerkin projection preprocessing step and several novel Krylov subspace reuse strategies. The new approach is applied to a set of test problems, including an unsteady turbulent airfoil, and is shown in some cases to provide significant improvement in computational efficiency relative to baseline approaches

Schnelle Löser für Partielle Differentialgleichungen

The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch (Leipzig), and Gabriel Wittum (Frankfurt am Main), was held May 22nd–May 28th, 2011. This meeting was well attended by 54 participants with broad geographic representation from 7 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds

Domain decomposition preconditioning for the Helmholtz equation: a coarse space based on local Dirichlet-to-Neumann maps

In this thesis, we present a two-level domain decomposition method for the iterative solution of the heterogeneous Helmholtz equation. The Helmholtz equation governs wave propagation and scattering phenomena arising in a wide range of engineering applications. Its discretization with piecewise linear finite elements results in typically large, ill-conditioned, indefinite, and non- Hermitian linear systems of equations, for which standard iterative and direct methods encounter convergence problems. Therefore, especially designed methods are needed. The inherently parallel domain decomposition methods constitute a promising class of preconditioners, as they subdivide the large problems into smaller subproblems and are hence able to cope with many degrees of freedom. An essential element of these methods is a good coarse space. Here, the Helmholtz equation presents a particular challenge, as even slight deviations from the optimal choice can be fatal. We develop a coarse space that is based on local eigenproblems involving the Dirichlet-to-Neumann operator. Our construction is completely automatic, ensuring good convergence rates without the need for parameter tuning. Moreover, it naturally respects local variations in the wave number and is hence suited also for heterogeneous Helmholtz problems. Apart from the question of how to design the coarse space, we also investigate the question of how to incorporate the coarse space into the method. Also here the fact that the stiffness matrix is non-Hermitian and indefinite constitutes a major challenge. The resulting method is parallel by design and its efficiency is investigated for two- and three-dimensional homogeneous and heterogeneous numerical examples