Generalized Product-Type Variants of RBiCG for Solving Families of Linear Systems

Abstract

Families of slowly changing nonsingular large sparse linear systems arise frequently in many simulation problems in science and engineering. We consider iterative solution with recycling techniques for a general case where both left-hand sides and right-hand sides of the systems change from one family to the next. We firstly develop a generalized product-type method in the framework of recycling biconjugate gradient method (RBiCG), referred to as RGPBiCG, which can also be considered as a recycling variant of GPBiCG. However, as the same situation in RBiCG stabilized method (RBiCGSTAB), the construction of recycling spaces in RGPBiCG requires expensive computational costs due to invoking other algorithms (like RBiCG) to compute approximate eigenspaces. In order to further reduce such computational costs, we alternatively form the recycling spaces in RGPBiCG with difference vectors of approximate solutions, as employed for loose GMRES (LGMRES), resulting in a more promising algorithm termed as LR-GPBiCG. Numerical experiments on both a set of academic problems and engineering simulation problems demonstrate the efficiency of our proposed algorithms.