Publications (2)1.94 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 biLanczos based methods, such as CGS, BiCGSTAB, QMR, and TFQMR. We develop an augmented biLanczos algorithm and a modified twoterm 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 convectiondiffusion type. Such a problem provides wellknown 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.SIAM Journal on Scientific Computing 10/2010; 34(4). DOI:10.1137/100801500 · 1.94 Impact Factor 
Article: Recycling BiCG for Model Reduction
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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 biLanczos algorithm and a modified twoterm 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 convectiondiffusion type, because these are wellknown 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.
Publication Stats
9  Citations  
1.94  Total Impact Points  