Recycling BiCG with an Application to Model Reduction

TL;DR

This paper introduces recycling BiCG, a method that reuses Krylov subspaces across sequences of dual linear systems, improving efficiency in applications like PDE solving and model reduction.

Contribution

Recycling BiCG extends BiCG by incorporating recycled subspaces, enabling efficient solutions of sequential dual linear systems with applications in PDEs and model reduction.

Findings

Up to 70% reduction in iterations for PDE problems.

Recycling BiCG is about 50% faster than standard BiCG in model reduction.

Recycling BiCG can be extended to other bi-Lanczos based methods.

Abstract

Science and engineering problems frequently require solving a sequence of dual linear systems. Besides having to store only few Lanczos vectors, using the BiConjugate Gradient method (BiCG) to solve dual linear systems has advantages for specific applications. For example, using BiCG to solve the dual linear systems arising in interpolatory model reduction provides a backward error formulation in the model reduction framework. Using BiCG to evaluate bilinear forms -- for example, in quantum Monte Carlo (QMC) methods for electronic structure calculations -- leads to a quadratic error bound. Since our focus is on sequences of dual linear systems, we introduce recycling BiCG, a BiCG method that recycles two Krylov subspaces from one pair of dual linear systems to the next pair. The derivation of recycling BiCG also builds the foundation for developing recycling variants of other bi-Lanczos based methods, such as CGS, BiCGSTAB, QMR, and TFQMR. We develop an augmented bi-Lanczos algorithm and a modified two-term recurrence to include recycling in the iteration. The recycle spaces are approximate left and right invariant subspaces corresponding to the eigenvalues closest to the origin. These recycle spaces are found by solving a small generalized eigenvalue problem alongside the dual linear systems being solved in the sequence. We test our algorithm in two application areas. First, we solve a discretized partial differential equation (PDE) of convection-diffusion type. Such a problem provides well-known test cases that are easy to test and analyze further. Second, we use recycling BiCG in the Iterative Rational Krylov Algorithm (IRKA) for interpolatory model reduction. IRKA requires solving sequences of slowly changing dual linear systems. We show up to 70% savings in iterations, and also demonstrate that for a model reduction problem BiCG takes (about) 50% more time than recycling BiCG.