Structured matrix algorithms for solving Helmholtz equation

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dc.contributor.advisorde Hoop, Maarten Ven_US
dc.creatorLiu, Xiaoen_US
dc.date.accessioned2019-05-16T18:52:49Zen_US
dc.date.available2019-05-16T18:52:49Zen_US
dc.date.created2019-05en_US
dc.date.issued2019-04-12en_US
dc.date.submittedMay 2019en_US
dc.date.updated2019-05-16T18:52:49Zen_US
dc.description.abstractIn this dissertation, novel solution algorithms are developed for large structured linear systems such as the discretized Helmholtz equation. Firstly, we develop two parallel randomized algorithms for constructing hierarchically semiseparable (HSS) matrices. Randomized sampling reduces the computational complexity and the communication cost simultaneously, and the resulting method is suitable for solving dense systems arising from multiple scattering problems. Secondly, a direct factorization update algorithm is proposed for solving the Helmholtz equation with changing wavespeed. The data dependency among a set of interior and exterior sub-problems is exploited to maximize the data reuse and to minimize the propagation of changes. Thirdly, interconnected HSS structures are designed to improve the efficiency of rank-structured factorization of PDE problems. Intermediate Schur complements at different levels share the same set of HSS bases during the factorization. Finally, we propose a contour integration preconditioner for solving 3D high-frequency Helmholtz equation. By solving systems with complex shifts, the problem is projected in subspaces with faster GMRES convergence. The shifted problems are solved by a polynomial fixed-point iteration, which is robust when the shift changes.en_US
dc.format.mimetypeapplication/pdfen_US
dc.identifier.citationLiu, Xiao. "Structured matrix algorithms for solving Helmholtz equation." (2019) Diss., Rice University. <a href="https://hdl.handle.net/1911/105384">https://hdl.handle.net/1911/105384</a>.en_US
dc.identifier.urihttps://hdl.handle.net/1911/105384en_US
dc.language.isoengen_US
dc.rightsCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.en_US
dc.subjectHelmholtz equationen_US
dc.subjecthierarchically semiseparable matrixen_US
dc.subjectrandomized samplingen_US
dc.subjectfactorization updateen_US
dc.subjectexterior boundary value problemen_US
dc.subjectcontour integrationen_US
dc.subjectmatrix exponentialen_US
dc.titleStructured matrix algorithms for solving Helmholtz equationen_US
dc.typeThesisen_US
dc.type.materialTexten_US
thesis.degree.departmentComputational and Applied Mathematicsen_US
thesis.degree.disciplineEngineeringen_US
thesis.degree.grantorRice Universityen_US
thesis.degree.levelDoctoralen_US
thesis.degree.nameDoctor of Philosophyen_US
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