scholarly journals Recycling Krylov Subspaces and Truncating Deflation Subspaces for Solving Sequence of Linear Systems

2021 ◽  
Vol 47 (2) ◽  
pp. 1-30
Author(s):  
Hussam Al Daas ◽  
Laura Grigori ◽  
Pascal Hénon ◽  
Philippe Ricoux
Keyword(s):  
Linear Systems ◽  
Krylov Subspace ◽  
Real Life ◽  
Krylov Subspace Methods ◽  
Krylov Subspaces ◽  
Systems Of Equations ◽  
The Matrix ◽  
Value Decomposition ◽  
The Impact ◽  
Linear Systems Of Equations

This article presents deflation strategies related to recycling Krylov subspace methods for solving one or a sequence of linear systems of equations. Besides well-known strategies of deflation, Ritz-, and harmonic Ritz-based deflation, we introduce an Singular Value Decomposition based deflation technique. We consider the recycling in two contexts: recycling the Krylov subspace between the restart cycles and recycling a deflation subspace when the matrix changes in a sequence of linear systems. Numerical experiments on real-life reservoir simulation demonstrate the impact of our proposed strategy.

scholarly journals Some Hyperbolic Iterative Methods for Linear Systems

2020 ◽  
Vol 2020 ◽  
pp. 1-8 ◽  
Author(s):  
K. Niazi Asil ◽  
M. Ghasemi Kamalvand
Keyword(s):  
Linear Systems ◽  
Iterative Methods ◽  
Krylov Subspace ◽  
Polarized Light ◽  
Inner Product ◽  
Systems Of Equations ◽  
Theory Of Relativity ◽  
Indefinite Inner Product ◽  
Special Case ◽  
Linear Systems Of Equations

The indefinite inner product defined by J=diagj1,…,jn, jk∈−1,+1, arises frequently in some applications, such as the theory of relativity and the research of the polarized light. This indefinite scalar product is referred to as hyperbolic inner product. In this paper, we introduce three indefinite iterative methods: indefinite Arnoldi’s method, indefinite Lanczos method (ILM), and indefinite full orthogonalization method (IFOM). The indefinite Arnoldi’s method is introduced as a process that constructs a J-orthonormal basis for the nondegenerated Krylov subspace. The ILM method is introduced as a special case of the indefinite Arnoldi’s method for J-Hermitian matrices. IFOM is mentioned as a process for solving linear systems of equations with J-Hermitian coefficient matrices. Finally, by providing numerical examples, the FOM, IFOM, and ILM processes have been compared with each other in terms of the required time for solving linear systems and also from the point of the number of iterations.

Enlarged GMRES for solving linear systems with one or multiple right-hand sides

2018 ◽  
Vol 39 (4) ◽  
pp. 1924-1956 ◽  
Author(s):  
Hussam Al Daas ◽  
Laura Grigori ◽  
Pascal Hénon ◽  
Philippe Ricoux
Keyword(s):  
Linear Systems ◽  
Krylov Subspace ◽  
Real Life ◽  
Original Method ◽  
Gmres Method ◽  
Convection Diffusion ◽  
Right Hand ◽  
Test Matrices ◽  
Linear Systems Of Equations ◽  
Diffusion Problems

Abstract We propose a variant of the generalized minimal residual (GMRES) method for solving linear systems of equations with one or multiple right-hand sides. Our method is based on the idea of the enlarged Krylov subspace to reduce communication. It can be interpreted as a block GMRES method. Hence, we are interested in detecting inexact breakdowns. We introduce a strategy to perform the test of detection. Furthermore, we propose a technique for deflating eigenvalues that has two benefits. The first advantage is to avoid the plateau of convergence after the end of a cycle in the restarted version. The second is to have very fast convergence when solving the same system with different right-hand sides, each given at a different time (useful in the context of a constrained pressure residual preconditioner). We test our method with these deflation techniques on academic test matrices arising from solving linear elasticity and convection–diffusion problems as well as matrices arising from two real-life applications, seismic imaging and simulations of reservoirs. With the same memory cost we obtain a saving of up to $50 \%$ in the number of iterations required to reach convergence with respect to the original method.

Sign in / Sign up

Export Citation Format

Share Document