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Generalizing smoothed aggregation-based algebraic multigrid

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Title: Generalizing smoothed aggregation-based algebraic multigrid
Author(s): Schroder, Jacob B.
Director of Research: Olson, Luke N.
Doctoral Committee Chair(s): Olson, Luke N.
Doctoral Committee Member(s): Gropp, William D.; Heath, Michael T.; Tuminaro, Raymond S.
Department / Program: Computer Science
Discipline: Computer Science
Degree Granting Institution: University of Illinois at Urbana-Champaign
Degree: Ph.D.
Genre: Dissertation
Subject(s): smoothed aggregation algebraic multigrid Helmholtz indefinite nonsymmetric algebraic coarsening discontinuous Galerkin high-order prolongation smoothing strength-of-connection
Abstract: Smoothed aggregation-based (SA) algebraic multigrid (AMG) is a popular and effective solver for systems of linear equations that arise from discretized partial differential equations. While SA has been effective over a broad class of problems, it has several limitations and weaknesses that this thesis is intended to address. This includes the development of a more robust strength-of-connection measure which guides coarsening and the choice of interpolation sparsity patterns. Unfortunately, the classic strength measure is only well-founded for M-matrices, leading us to develop a new measure based on local knowledge of both algebraically smooth error and the behavior of interpolation. Another limitation is that classic SA is only formally defined for Hermitian positive definite problems. For non-Hermitian operators, the operator-induced energy-norm does not exist, which impacts the complementary relationship between relaxation and interpolation. This requires a redesign of SA, such that restriction and prolongation operators approximate the left and right near null-spaces, respectively. As a result, we develop general SA setup algorithms for both the Hermitian positive-definite and the non-Hermitian cases. To realize these algorithms, we develop general prolongation smoothing methods so that restriction and prolongation target the left and right near null-spaces, respectively. Overall, the proposed methods do not assume any user-input beyond what standard SA does and the result is a new direction for multigrid methods for non-Hermitian systems. Several problem areas motivate our development. For example, rotated anisotropic diffusion and linearized elasticity problems using standard discretizations can easily generate non-M-matrices that prove difficult for standard SA and AMG. High- and low-order discontinuous Galerkin discretizations also generate difficult non-M-matrices for elliptic problems. Target non-Hermitian problems include flow problems and wave-like problems, e.g., Helmholtz. Additionally for wave-like problems, there is a rich non-standard wave-like near null-space, which must be captured by the coarse levels—a task beyond the scope of traditional AMG or SA coarsening techniques.
Issue Date: 2010-08-31
URI: http://hdl.handle.net/2142/17044
Rights Information: Copyright 2010 Jacob B. Schroder
Date Available in IDEALS: 2010-08-31
2012-09-07
Date Deposited: 2010-08
 

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