Full Record

Author | Zhang, Yabin |

Title | A fast direct solver for boundary value problems with locally-perturbed geometries |

URL | http://hdl.handle.net/1911/105467 |

Publication Date | 2017 |

Date Accessioned | 2019-05-16 20:02:18 |

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…