Publications (1)0 Total impact
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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.
10/2010;
