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Recycling BICG with an application to model reduction

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dc.contributor.authorAhuja, Kapilen
dc.contributor.authorde Sturler, Ericen
dc.contributor.authorGugercin, Serkanen
dc.contributor.authorChang, Eun R.en
dc.contributor.departmentMathematicsen
dc.date.accessed2014-05-27en
dc.date.accessioned2014-05-28T18:35:08Zen
dc.date.available2014-05-28T18:35:08Zen
dc.date.issued2012en
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.en
dc.description.sponsorshipNSF NSF-EAR 0530643, NSF-DMS 1025327, NSF-DMS 0645347en
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: 10.1137/100801500en
dc.identifier.doihttps://doi.org/10.1137/100801500en
dc.identifier.issn1064-8275en
dc.identifier.urihttp://hdl.handle.net/10919/48154en
dc.identifier.urlhttp://epubs.siam.org/doi/abs/10.1137/100801500en
dc.language.isoen_USen
dc.publisherSiam Publicationsen
dc.rightsIn Copyrighten
dc.rights.urihttp://rightsstatements.org/vocab/InC/1.0/en
dc.subjectkrylov subspace recyclingen
dc.subjectdeflationen
dc.subjectbi-lanczos methoden
dc.subjectpetrov-galerkinen
dc.subjectformulationen
dc.subjectbicgen
dc.subjectmodel reductionen
dc.subjectrational kryloven
dc.subjecth-2 approximationen
dc.subjectnonsymmetric linear-systemsen
dc.subjectminimal residual algorithmen
dc.subjectkryloven
dc.subjectsubspacesen
dc.subjectdynamical-systemsen
dc.subjectapproximationen
dc.subjectgmresen
dc.subjecttomographyen
dc.subjectstrategiesen
dc.subjectfamiliesen
dc.subjectmathematics, applieden
dc.titleRecycling BICG with an application to model reductionen
dc.title.serialSiam Journal on Scientific Computingen
dc.typeArticle - Refereeden

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