Generalized Krylov recycling methods for solution of multiple related linear equation systems in electromagnetic analysis

Abstract

In this paper we propose methods for fast iterative solution of multiple related linear systems of equations. Such systems arise, for example, in building pattern libraries for interconnect parasitic extraction, parasitic extraction under process variation, and parameterized interconnect characterization. Our techniques are based on a generalized form of "recycled" Krylov subspace methods that use sharing of information between related systems of equations to accelerate the iterative solution. Experimental results on electromagnetics problems demonstrate that the proposed method can achieve a speed-up of 5X∼30X compared to direct GMRES applied sequentially to the individual systems. These methods are generic, fully treat nonlinear perturbations without approximation, and can be applied in a wide variety of application domains outside electromagnetics.

Incremental power network analysis using backward random walks

“…These suffer from the fact that they require the full system to be solved again, for the entire chip, and do not fully utilize the "closeness" properties above. An iterative solver is also described in [10], but operates on smaller, denser systems.…”

Parametric model order reduction accelerated by subspace recycling

“…Unfortunately, these methods require that the right-hand sides of all the linear systems are available simultaneously, whereas the linear systems considered here can only appear sequentially rather than simultaneously. Different algorithms [5], [8], [9], [10] have also been proposed to deal with the linear systems whose right-hand sides are not available simultaneously. However, the method in [9] is limited to solving linear systems with symmetric definite coefficient matrices, which is not applicable to the linear systems with general nonsymmetric coefficient matrices.…”

“…We propose a simplified version SimGCRO-DR based on GCRO-DR in [8] and apply it to the PMOR method in [1]. We also compare SimGCRO-DR with the newly proposed recycling method MKR-GMRES in [10]. The numerical simulation results in Section IV shows that SimGCRO-DR is more efficient than MKR-GMRES for very large-scale systems.…”

“…In Section III, we present the recycling method GCRO-DR in [8] based on which a simplified algorithm SimGCRO-DR is derived. The method MKR-GMRES in [10] is also briefly explained for comparison. SimGCRO-DR is compared numerically with both standard GMRES and MKR-GMRES in detail in Section IV.…”

“…The standard way of solving these linear systems in case sparse direct solvers are not feasible is to use conventional iterative methods such as GMRES or CG. In this paper, a fast recycling algorithm is applied to solve the whole sequence of linear systems and is shown to be much more efficient than the standard iterative solver GMRES as well as the newly proposed recycling method MKR-GMRES from [10]. As a result, the computation of the reduced-order model can be significantly accelerated.…”