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Publications (2)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;
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    ABSTRACT: Science and engineering problems frequently require solving a sequence of dual linear systems. Two examples are the Iterative Rational Krylov Algorithm (IRKA) for model reduction and Quantum Monte Carlo (QMC) methods in electronic structure calculations. This paper introduces Recycling BiCG, a BiCG method that recycles two Krylov subspaces from one pair of linear systems to the next pair. 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 close 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 of convection-diffusion type, because these are well-known model problems. Second, we use Recycling BiCG for the linear systems arising in IRKA for model reduction, which requires solving a sequence of slowly changing, dual linear systems. Our experiments with Recycling BiCG give good results.
    01/2010;

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