Deflation in Preconditioned Conjugate Gradient Methods for Finite Element Problems

Abstract

We investigate the influence of the value of deflation vectors at interfaces on the rate of convergence of preconditioned conjugate gradient methods applied to a Finite Element discretization for an elliptic equation. Our set-up is a Poisson problem in two dimensions with continuous or discontinuous coefficients that vary in several orders of magnitude. In the continuous case we are interested in the convergence acceleration of deflation on block preconditioners. For the discontinuous case the Finite Element discretization gives a symmetric matrix with very large condition number and hence many iterations are needed to obtain a solution using conjugate gradients. We use an incomplete Choleski decomposition for the symmetric discretization matrix. Subsequently, deflation is applied to eliminate the disadvantageous effects to convergence caused by the remaining small eigenvalues. Here we investigate the effects of several variants of deflation and we propose an optimal choice for the deflation vectors. Finally, we apply deflation to a parallel preconditioned conjugate method to accellerate the calculation.

A Comparison of Deflation and Coarse Grid Correction Applied to Porous Media Flow

“…Thereafter this method has been generalized to other choices of the deflation vectors [26,27]. Finally an analysis of the effective condition number and a parallel implementation is given in [7,24].…”

Deflated preconditioned conjugate gradient solvers for the Pressure–Poisson equation

“…It is shown that if grid refinement is performed, keeping the subdomain grid resolution fixed, the condition number is insensitive to the grid size. Other possibilities for choosing the deflation vectors are proposed in [43]. Deflation is applied to magnetostatic problems in [9,10], where the approximate eigenvectors used to the deflation subspace mimic the flux patterns commonly sketched by engineers intuitively.…”

“…In [35,4] the projector relying on the deflation subspace is explicitely build from the structure of the coefficient matrix of the system to be solved. Vuik and coworkers [43,14,44,45] applied the deflated technique in various contexts, including problems characterized by layers of large contrast in the porosity [44,45], where the approximated eigenvectors verify the partial differential equation on each subdomain. In [14] some numerical experiments of deflated methods running on parallel machines are reported, as well as new bounds on the effective condition number of deflated and preconditioned deflated conjugate gradients.…”

Deflated preconditioned conjugate gradient solvers for linear elasticity

“…In this section, we rely on the approach proposed in 31, which may be more amenable to extensions in the non‐symmetric case and is fully equivalent to the one proposed in 1, 17 if no additional preconditioner is applied. Given a deflation space W , let us define the projector P as: P is an A −1 ‐orthogonal projector onto W ⟂ along span { AW } as: Its transpose is and is an A ‐orthogonal projector onto W along span { W } as: It is easily verified that P and P T are projectors: As A is symmetric, the following relation holds true: The solution x to the linear system is obtained as 32 with such that and In detail, x 2 is computed as and In order to obtain x , x 1 is multiplied by P T and is added to x 2 to form the solution: In this form, the deflated preconditioned conjugate gradient looks slightly different than the algorithm presented in 17. Furthermore, this form may not be of the most optimized one from a computational viewpoint.…”