Robust and scalable hierarchical matrix-based fast direct solver and preconditioner for the numerical solution of elliptic partial differential equations

Handle URI:
http://hdl.handle.net/10754/625172
Title:
Robust and scalable hierarchical matrix-based fast direct solver and preconditioner for the numerical solution of elliptic partial differential equations
Authors:
Chavez, Gustavo Ivan ( 0000-0002-7593-505X )
Abstract:
This dissertation introduces a novel fast direct solver and preconditioner for the solution of block tridiagonal linear systems that arise from the discretization of elliptic partial differential equations on a Cartesian product mesh, such as the variable-coefficient Poisson equation, the convection-diffusion equation, and the wave Helmholtz equation in heterogeneous media. The algorithm extends the traditional cyclic reduction method with hierarchical matrix techniques. The resulting method exposes substantial concurrency, and its arithmetic operations and memory consumption grow only log-linearly with problem size, assuming bounded rank of off-diagonal matrix blocks, even for problems with arbitrary coefficient structure. The method can be used as a standalone direct solver with tunable accuracy, or as a black-box preconditioner in conjunction with Krylov methods. The challenges that distinguish this work from other thrusts in this active field are the hybrid distributed-shared parallelism that can demonstrate the algorithm at large-scale, full three-dimensionality, and the three stressors of the current state-of-the-art multigrid technology: high wavenumber Helmholtz (indefiniteness), high Reynolds convection (nonsymmetry), and high contrast diffusion (inhomogeneity). Numerical experiments corroborate the robustness, accuracy, and complexity claims and provide a baseline of the performance and memory footprint by comparisons with competing approaches such as the multigrid solver hypre, and the STRUMPACK implementation of the multifrontal factorization with hierarchically semi-separable matrices. The companion implementation can utilize many thousands of cores of Shaheen, KAUST's Haswell-based Cray XC-40 supercomputer, and compares favorably with other implementations of hierarchical solvers in terms of time-to-solution and memory consumption.
Advisors:
Keyes, David Elliot ( 0000-0002-4052-7224 )
Committee Member:
Moshkov, Mikhail ( 0000-0003-0085-9483 ) ; Ketcheson, David I. ( 0000-0002-1212-126X ) ; Turkiyyah, George; Huang, Jingfang
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Program:
Computer Science
Issue Date:
10-Jul-2017
Type:
Dissertation
Appears in Collections:
Dissertations

Full metadata record

DC FieldValue Language
dc.contributor.advisorKeyes, David Ellioten
dc.contributor.authorChavez, Gustavo Ivanen
dc.date.accessioned2017-07-10T09:26:53Z-
dc.date.available2017-07-10T09:26:53Z-
dc.date.issued2017-07-10-
dc.identifier.urihttp://hdl.handle.net/10754/625172-
dc.description.abstractThis dissertation introduces a novel fast direct solver and preconditioner for the solution of block tridiagonal linear systems that arise from the discretization of elliptic partial differential equations on a Cartesian product mesh, such as the variable-coefficient Poisson equation, the convection-diffusion equation, and the wave Helmholtz equation in heterogeneous media. The algorithm extends the traditional cyclic reduction method with hierarchical matrix techniques. The resulting method exposes substantial concurrency, and its arithmetic operations and memory consumption grow only log-linearly with problem size, assuming bounded rank of off-diagonal matrix blocks, even for problems with arbitrary coefficient structure. The method can be used as a standalone direct solver with tunable accuracy, or as a black-box preconditioner in conjunction with Krylov methods. The challenges that distinguish this work from other thrusts in this active field are the hybrid distributed-shared parallelism that can demonstrate the algorithm at large-scale, full three-dimensionality, and the three stressors of the current state-of-the-art multigrid technology: high wavenumber Helmholtz (indefiniteness), high Reynolds convection (nonsymmetry), and high contrast diffusion (inhomogeneity). Numerical experiments corroborate the robustness, accuracy, and complexity claims and provide a baseline of the performance and memory footprint by comparisons with competing approaches such as the multigrid solver hypre, and the STRUMPACK implementation of the multifrontal factorization with hierarchically semi-separable matrices. The companion implementation can utilize many thousands of cores of Shaheen, KAUST's Haswell-based Cray XC-40 supercomputer, and compares favorably with other implementations of hierarchical solvers in terms of time-to-solution and memory consumption.en
dc.language.isoenen
dc.subjecthierarchical matricesen
dc.subjectcyclic reductionen
dc.subjectfast solversen
dc.subjectDirect solversen
dc.subjectpreconditioningen
dc.subjectParallel Computingen
dc.titleRobust and scalable hierarchical matrix-based fast direct solver and preconditioner for the numerical solution of elliptic partial differential equationsen
dc.typeDissertationen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
thesis.degree.grantorKing Abdullah University of Science and Technologyen_GB
dc.contributor.committeememberMoshkov, Mikhailen
dc.contributor.committeememberKetcheson, David I.en
dc.contributor.committeememberTurkiyyah, Georgeen
dc.contributor.committeememberHuang, Jingfangen
thesis.degree.disciplineComputer Scienceen
thesis.degree.nameDoctor of Philosophyen
dc.person.id101968en
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