Deflation of conjugate gradients with applications to boundary value problems
SIAM Journal on Numerical Analysis
On the conjugate gradient solution of the Schur complement system obtained from domain decomposition
SIAM Journal on Numerical Analysis
SIAM Journal on Scientific and Statistical Computing
Iterative solution methods
A Restarted GMRES Method Augmented with Eigenvectors
SIAM Journal on Matrix Analysis and Applications
Restarted GMRES preconditioned by deflation
Journal of Computational and Applied Mathematics
Analysis of Augmented Krylov Subspace Methods
SIAM Journal on Matrix Analysis and Applications
A multiscale finite element method for elliptic problems in composite materials and porous media
Journal of Computational Physics
Krylov-Secant methods for solving large-scale systems of coupled nonlinear parabolic equations
Krylov-Secant methods for solving large-scale systems of coupled nonlinear parabolic equations
An Adaptive Newton--Picard Algorithm with Subspace Iteration for Computing Periodic Solutions
SIAM Journal on Scientific Computing
Journal of Computational Physics
SIAM Journal on Scientific Computing
A Deflated Version of the Conjugate Gradient Algorithm
SIAM Journal on Scientific Computing
Implicitly Restarted GMRES and Arnoldi Methods for Nonsymmetric Systems of Equations
SIAM Journal on Matrix Analysis and Applications
Journal of Computational Physics
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
On the Construction of Deflation-Based Preconditioners
SIAM Journal on Scientific Computing
GMRES with Deflated Restarting
SIAM Journal on Scientific Computing
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Preconditioning via a Schur Complement Method: An Application in State Estimation
SIAM Journal on Scientific Computing
A Broyden Rank p+1 Update Continuation Method with Subspace Iteration
SIAM Journal on Scientific Computing
A Comparison of Deflation and Coarse Grid Correction Applied to Porous Media Flow
SIAM Journal on Numerical Analysis
Electrostatics and heat conduction in high contrast composite materials
Journal of Computational Physics
Towards a rigorously justified algebraic preconditioner for high-contrast diffusion problems
Computing and Visualization in Science
Restarted block-GMRES with deflation of eigenvalues
Applied Numerical Mathematics - 6th IMACS International symposium on iterative methods in scientific computing
Robust preconditioners for the high-contrast Stokes equation
Journal of Computational and Applied Mathematics
Hi-index | 0.00 |
Eigenvalues of smallest magnitude become a major bottleneck for iterative solvers especially when the underlying physical properties have severe contrasts. These contrasts are commonly found in many applications such as composite materials, geological rock properties and thermal and electrical conductivity. The main objective of this work is to construct a method as algebraic as possible. However, the underlying physics is utilized to distinguish between high and low degrees of freedom which is central to the construction of the proposed preconditioner. Namely, we propose an algebraic way of separating binary-like systems according to a given threshold into high- and low-conductivity regimes of coefficient size O(m) and O(1), respectively where m@?1. So, the proposed preconditioner is essentially physics-based because without the utilization of underlying physics such an algebraic distinction, hence, the construction of the preconditioner would not be possible. The condition number of the linear system depends both on the mesh size @Dx and the coefficient size m. For our purposes, we address only the m dependence since the condition number of the linear system is mainly governed by the high-conductivity sub-block. Thus, the proposed strategy is inspired by capturing the relevant physics governing the problem. Based on the algebraic construction, a two-stage preconditioning strategy is developed as follows: (1) a first stage that comprises approximation to the components of the solution associated to small eigenvalues and, (2) a second stage that deals with the remaining solution components with a deflation strategy (if ever needed). Due to its algebraic nature, the proposed approach can support a wide range of realistic geometries (e.g., layered and channelized media). Numerical examples show that the proposed class of physics-based preconditioners are more effective and robust compared to a class of Krylov-based deflation methods on highly heterogeneous media.