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dc.contributorVirginia Tech
dc.contributor.authorAhuja, K.
dc.contributor.authorde Sturler, E.
dc.contributor.authorGugercin, S.
dc.contributor.authorChang, E. R.
dc.date.accessioned2014-05-28T18:35:08Z
dc.date.available2014-05-28T18:35:08Z
dc.date.issued2012
dc.identifier.citationAhuja, K.; de Sturler, E.; Gugercin, S.; Chang, E. R., "Recycling BICG with an application to model reduction," SIAM J. Sci. Comput., 34(4), A1925-A1949, (2012). DOI: 10.1137/100801500
dc.identifier.issn1064-8275
dc.identifier.urihttp://hdl.handle.net/10919/48154
dc.description.abstractScience and engineering problems frequently require solving a sequence of dual linear systems. Besides having to store only a 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 analyze the generated recycle spaces and show up to 70% savings in iterations. For our model reduction test problem, we show that solving the problem without recycling leads to (about) a 50% increase in runtime.
dc.description.sponsorshipNSF NSF-EAR 0530643, NSF-DMS 1025327, NSF-DMS 0645347
dc.language.isoen_US
dc.publisherSiam Publications
dc.subjectkrylov subspace recycling
dc.subjectdeflation
dc.subjectbi-lanczos method
dc.subjectpetrov-galerkin
dc.subjectformulation
dc.subjectbicg
dc.subjectmodel reduction
dc.subjectrational krylov
dc.subjecth-2 approximation
dc.subjectnonsymmetric linear-systems
dc.subjectminimal residual algorithm
dc.subjectkrylov
dc.subjectsubspaces
dc.subjectdynamical-systems
dc.subjectapproximation
dc.subjectgmres
dc.subjecttomography
dc.subjectstrategies
dc.subjectfamilies
dc.subjectmathematics, applied
dc.titleRecycling BICG with an application to model reduction
dc.typeArticle
dc.identifier.urlhttp://epubs.siam.org/doi/abs/10.1137/100801500
dc.date.accessed2014-05-27
dc.title.serialSiam Journal on Scientific Computing
dc.identifier.doihttps://doi.org/10.1137/100801500


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