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

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    Type
    Dissertation
    Authors
    Chavez Chavez, Gustavo Ivan cc
    Advisors
    Keyes, David E. cc
    Committee members
    Moshkov, Mikhail cc
    Ketcheson, David I. cc
    Turkiyyah, George
    Huang, Jingfang
    Program
    Computer Science
    KAUST Department
    Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division
    Date
    2017-07-10
    Permanent link to this record
    http://hdl.handle.net/10754/625172
    
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    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.
    Citation
    Chavez Chavez, G. I. (2017). Robust and scalable hierarchical matrix-based fast direct solver and preconditioner for the numerical solution of elliptic partial differential equations. KAUST Research Repository. https://doi.org/10.25781/KAUST-68230
    DOI
    10.25781/KAUST-68230
    ae974a485f413a2113503eed53cd6c53
    10.25781/KAUST-68230
    Scopus Count
    Collections
    PhD Dissertations; Computer Science Program; Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division

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