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Rice University

1.
Geldermans, Peter.
Accelerated PDE Constrained Optimization using *Direct* * Solvers*.

Degree: MA, Engineering, 2018, Rice University

URL: http://hdl.handle.net/1911/105485

In this thesis, I propose a method to reduce the cost of computing solutions to optimization problems governed by partial differential equations (PDEs). Standard second order methods such as Newton require the solution of two PDEs per iteration of the Newton system, which can be prohibitively expensive for iterative solvers. In
contrast, this work takes advantage a recently developed high order discretization method that comes with an efficient direct solver. The new technique precomputes a solution operator that can be reused for any body load, which is applied whenever
a PDE solve is required. Thus the precomputation cost is amortized over many PDE solves. This approach will make second order optimization algorithms computationally affordable for practical applications such as photoacoustic tomography and
optimal design problems.
*Advisors/Committee Members: Gillman, Adrianna (advisor), Heinkenschloss, Matthias (committee member).*

Subjects/Keywords: PDE constrained optimization; direct solvers; fast solvers

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Geldermans, P. (2018). Accelerated PDE Constrained Optimization using Direct Solvers. (Masters Thesis). Rice University. Retrieved from http://hdl.handle.net/1911/105485

Chicago Manual of Style (16^{th} Edition):

Geldermans, Peter. “Accelerated PDE Constrained Optimization using Direct Solvers.” 2018. Masters Thesis, Rice University. Accessed April 12, 2021. http://hdl.handle.net/1911/105485.

MLA Handbook (7^{th} Edition):

Geldermans, Peter. “Accelerated PDE Constrained Optimization using Direct Solvers.” 2018. Web. 12 Apr 2021.

Vancouver:

Geldermans P. Accelerated PDE Constrained Optimization using Direct Solvers. [Internet] [Masters thesis]. Rice University; 2018. [cited 2021 Apr 12]. Available from: http://hdl.handle.net/1911/105485.

Council of Science Editors:

Geldermans P. Accelerated PDE Constrained Optimization using Direct Solvers. [Masters Thesis]. Rice University; 2018. Available from: http://hdl.handle.net/1911/105485

King Abdullah University of Science and Technology

2.
Chavez Chavez, Gustavo Ivan.
Robust and scalable hierarchical matrix-based *fast* *direct* solver and preconditioner for the numerical solution of elliptic partial differential equations.

Degree: Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division, 2017, King Abdullah University of Science and Technology

URL: http://hdl.handle.net/10754/625172

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/Committee Members: Keyes, David E. (advisor), Moshkov, Mikhail (committee member), Ketcheson, David I. (committee member), Turkiyyah, George (committee member), Huang, Jingfang (committee member).*

Subjects/Keywords: hierarchical matrices; cyclic reduction; fast solvers; Direct solvers; preconditioning; Parallel Computing

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

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. (Thesis). King Abdullah University of Science and Technology. Retrieved from http://hdl.handle.net/10754/625172

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Chavez Chavez, Gustavo Ivan. “Robust and scalable hierarchical matrix-based fast direct solver and preconditioner for the numerical solution of elliptic partial differential equations.” 2017. Thesis, King Abdullah University of Science and Technology. Accessed April 12, 2021. http://hdl.handle.net/10754/625172.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Chavez Chavez, Gustavo Ivan. “Robust and scalable hierarchical matrix-based fast direct solver and preconditioner for the numerical solution of elliptic partial differential equations.” 2017. Web. 12 Apr 2021.

Vancouver:

Chavez Chavez GI. Robust and scalable hierarchical matrix-based fast direct solver and preconditioner for the numerical solution of elliptic partial differential equations. [Internet] [Thesis]. King Abdullah University of Science and Technology; 2017. [cited 2021 Apr 12]. Available from: http://hdl.handle.net/10754/625172.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Chavez Chavez GI. Robust and scalable hierarchical matrix-based fast direct solver and preconditioner for the numerical solution of elliptic partial differential equations. [Thesis]. King Abdullah University of Science and Technology; 2017. Available from: http://hdl.handle.net/10754/625172

Not specified: Masters Thesis or Doctoral Dissertation

3.
Zhang, Yabin.
A *fast* *direct* solver for boundary value problems with locally-perturbed geometries.

Degree: MA, Engineering, 2017, Rice University

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

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.
*Advisors/Committee Members: Gillman, Adrianna (advisor).*

Subjects/Keywords: fast direct solvers; boundary integral equations; local perturbation

…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… …wider range of problems.
In the last fifteen years, new *fast* *direct* *solvers*, such as HSS, HBS… …area in current
research.
10
2.2
Outline of *Fast* *Direct* *Solvers*
The previous section… …lists several recently-developed *fast* *direct* *solvers* for BIEs: HSS,
HBS, and HODLR [20…

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Zhang, Y. (2017). A fast direct solver for boundary value problems with locally-perturbed geometries. (Masters Thesis). Rice University. Retrieved from http://hdl.handle.net/1911/105467

Chicago Manual of Style (16^{th} Edition):

Zhang, Yabin. “A fast direct solver for boundary value problems with locally-perturbed geometries.” 2017. Masters Thesis, Rice University. Accessed April 12, 2021. http://hdl.handle.net/1911/105467.

MLA Handbook (7^{th} Edition):

Zhang, Yabin. “A fast direct solver for boundary value problems with locally-perturbed geometries.” 2017. Web. 12 Apr 2021.

Vancouver:

Zhang Y. A fast direct solver for boundary value problems with locally-perturbed geometries. [Internet] [Masters thesis]. Rice University; 2017. [cited 2021 Apr 12]. Available from: http://hdl.handle.net/1911/105467.

Council of Science Editors:

Zhang Y. A fast direct solver for boundary value problems with locally-perturbed geometries. [Masters Thesis]. Rice University; 2017. Available from: http://hdl.handle.net/1911/105467