Relaxation strategies for nested Krylov methods

Abstract

There are classes of linear problems for which the matrix-vector product is a time consuming operation because an expensive approximation method is required to compute it to a given accuracy. In recent years different authors have investigated the use of, what is called, relaxation strategies for various Krylov subspace methods. These relaxation strategies aim to minimize the amount of work that is spent in the computation of the matrix-vector product without compromising the accuracy of the method or the convergence speed too much. In order to achieve this goal, the accuracy of the matrix-vector product is decreased when the iterative process comes closer to the solution. In this paper we show that a further significant reduction in computing time can be obtained by combining a relaxation strategy with the nesting of inexact Krylov methods. Flexible Krylov subspace methods allow variable preconditioning and therefore can be used in the outer most loop of our overall method. We analyze for several flexible Krylov methods strategies for controlling the accuracy of both the inexact matrix-vector products and of the inner iterations. The results of our analysis will be illustrated with an example that models global ocean circulation.

Preconditioning Lanczos Approximations to the Matrix Exponential

“…The amount of work in the inversions of the linear systems, measured as the total number of applications of the ILU preconditioner, is reduced by about 30 to 40 percent. This is comparable with reductions that are seen in applications of inexact Krylov methods in other areas, see for a more detailed discussion [40,Section 3]. For the purpose of illustration we have included in Figure 6.2 and Figure 6.3 a visual representation of the results for τ = 1/10 and τ = 1/100 for both strategies.…”

Solutions of Large-Scale Electromagnetics Problems Using an Iterative Inner-Outer Scheme With Ordinary and Approximate Multilevel Fast Multipole Algorithms

“…Van den Eshof et al [25] showed that fixing ε in is nearly optimal if relaxation is not applied. However, even 0.1 (10%) residual error can cause a significant number of inner iterations for large-scale problems.…”

“…In addition to more efficient solutions, the inner-outer scheme prevents numerical errors that arise because of the deviations of the computed residual from the true residual by significantly decreasing the number of outer iterations. This is because the "residual gap," i.e., the difference between the true and computed residuals, increases with the number of iterations [25]. Another benefit of the reduction in iteration counts appears when the iterative solutions are performed with the generalized minimal residual (GMRES) algorithm, which is usually an optimal method for EFIE in terms of the processing time [14,18].…”

“…On the other hand, an innerouter scheme usually increases the total number of matrix-vector multiplications compared to the ordinary solutions [25]. In addition, an approximate solution instead of an ordinary solution can be sufficient to construct a robust preconditioner.…”

Avanços em Métodos de Krylov para Solução de Sistemas Lineares de Grande Porte

“…Baseados em [19], estratégias semelhantes foram aplicadas com sucesso na solução de problemas de difusão heterogênea usando decomposição de domínio [21], como precondicionadores de problemas de difusão de radiação [135], em problemas de eletromagnetismo [79], em cromodinâmica quântica [40] e em modelos de circulação oceânica de fluxos barotrópicos estáveis [130]. Passos significativos, em direção a uma explicação teórica do comportamento observado acima, foram propostos em [112], [129], e, mais recentemente, em [52].…”