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Thesis Defense - Prudencio

Parallel Fully Coupled Lagrange-Newton-Krylov-Schwarz Algorithms and Software for Optimization Problems Constrained by Partial Differential Equations
Computer Science PhD Candidate
2/11/2005
10:00am-12:00pm

As the technology and size of parallel computing systems advance, so does the detail at which we can solve very complex numerical problems, including optimization problems constrained by nonlinear partial differential equations (PDEs). This trend of increasing computational complexity demands both the design of scalable parallel numerical algorithms and the adoption of modern software engineering techniques for the development of numerical libraries.

This thesis presents the development of robust parallel numerical methods and object-oriented software for solving PDE-constrained optimization problems. We propose a new class of fully coupled full space sequential quadratic programming algorithms, which we refer to as Lagrange-Newton-Krylov-Schwarz (LNKSz). In LNKSz, a Lagrangian functional is formed and differentiated to obtain an optimality system of nonlinear equations called Karush-Kuhn-Tucker (KKT) system, which is then solved with Newton-Krylov-Schwarz (NKSz) algorithms. Although NKSz has been applied to the solution of many simulation PDEs, until this thesis little is known about its suitability to KKT systems.

In order to implement and test multi-level LNKSz methods, we have developed, over the Portable, Extensible Toolkit for Scientific Computing library from Argonne National Laboratory, a parallel C++ software application called PDE Constrained Optimization Package (PCOP). Numerical experiments are performed on some boundary control problems of steady-state incompressible Navier-Stokes flows. For the case of relatively low computational complexity, i.e., small Reynolds number (Re), small mesh size and small number of processors, many good algorithms exist. LNKSz, however, remains robust as the computational complexity increases. We report the performance of LNKSz in terms of nonlinear and linear convergences, analyze its sensitivity with respect to the mesh size and Re, and analyze its scalability with respect to the number of processors. We also present the design of PCOP and discuss some interesting implementation issues.

Committee: Xiao-Chuan Cai, Professor (Chair)
Richard Byrd, Professor
Leopoldo Franca, University of Colorado at Denver
Elizabeth Jessup, Associate Professor
Henry Tufo, Associate Professor
Department of Computer Science
University of Colorado Boulder
Boulder, CO 80309-0430 USA
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