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Title A fast direct solver for boundary value problems with locally-perturbed geometries
Publication Date
Date Accessioned
Degree MA
Discipline/Department Engineering
Degree Level masters
University/Publisher Rice University
Abstract Many problems in science and engineering can be formulated as integral equations with elliptic kernels. In particular, in optimal control and design problems, the domain geometry evolves and results in a sequence of discretized linear systems to be constructed and inverted. While the systems can be constructed and inverted independently, the computational cost is relatively high. In the case where the change in the domain geometry for each new problem is only local, i.e. the geometry remains the same except within a small subdomain, we are able to reduce the cost of inverting the new system by reusing the pre-computed fast direct solvers of the original system. The resulting solver only requires inexpensive matrix-vector multiplications and matrix inversion of small size, thus dramatically reducing the cost of inverting the new linear system.
Subjects/Keywords fast direct solvers; boundary integral equations; local perturbation
Contributors Gillman, Adrianna (advisor)
Language en
Country of Publication us
Record ID handle:1911/105467
Repository rice
Date Indexed 2020-02-20
Grantor Rice University
Issued Date 2017-05-17 00:00:00

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…construct and solve this linear system as well as a more detailed discussion of one of the fast direct solvers, specifically, the HBS solver. 6 Chapter 2 Fast Direct Solvers for BIEs The solver proposed in this thesis uses a previously constructed fast

…equations (BIEs) and emphasizes the necessity of developing direct solvers for BIEs. Section 2.2 then provides a three-step outline to give the reader a very general idea of how a fast direct solver solves a discretized BIE. Section 2.3 describes…

…two preliminary tools that are popularly used in the construction of such solvers: the Nyström discretization and the ID decomposition. Finally, Section 2.4 covers the construction of an HBS solver in details. 2.1 History of Fast Solvers While the…

…generating highly-accurate solutions to a wider range of problems. In the last fifteen years, new fast direct solvers, such as HSS, HBS, and HODLR [20, 21, 22], took advantage of many of the developments and managed to reduce the cost to O(N…

…Outline of Fast Direct Solvers The previous section lists several recently-developed fast direct solvers for BIEs: HSS, HBS, and HODLR [20, 21, 22]. These variants are highly related to each other and follow the same outline to obtain a solution…

…x28;2.9). 2.3.2 The ID Decomposition for Rank-deficient Matrices The theory and practices of fast solving techniques for BIE, not limited to direct solvers, are based on the rank-deficiency of the off-diagonal blocks of the coefficient matrix…

…the fast direct solver previously-constructed for the old geometry and computes the solution to the locallyperturbed problem with a small number of new calculations. The cost of the new solver scales linearly with respect to the number of unknowns. The…

…only perform new calculations that are necessary to capture the changes. This thesis gives a detailed formulation of this technique for boundary integral equations and provides a fast direct solver that generates solutions to the locally-perturbed…