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Thesis Defense - Murillo

Parallel Algorithms and Software for Time-Dependent Nonlinear Systems of Partial Differential Equations with an Application in Computational Biology
Maria Murillo
Computer Science PhD Candidate
11/12/2002
1:30pm-3:30pm

In this research we study the two dimensional bidomain equations which model the excitation process in the heart. We develop a robust, parallel, and scalable method using a fully implicit, domain decomposition based Newton-Krylov-Schwarz approach. The model consists of a coupled system of time-dependent nonlinear partial differential equations and ordinary differential equations, including both parabolic and hyperbolic types of equations. To solve the system of equations, we use a fully implicit backward Euler discretization scheme for the time variable, and the resulting large sparse nonlinear algebraic system is solved using a Newton-type algorithm at each time step. Within each Newton iteration, the Jacobian system of linear equations is solved inexactly using the restarted GMRES method with an additive Schwarz preconditioner. In order to reduce the storage and execution time, an incomplete factorization technique is applied to each of the subdomain systems of equations.

The numerical algorithms are implemented on various parallel platforms using the Portable, Extensible Toolkit for Scientific computation of Argonne National Laboratory. Computational and numerical factors are considered in order to study the mathematical and software performance of the algorithm. Our algorithm allows the use of time steps much larger than other existing methods, rapid convergence for the nonlinear portion, superlinear scalability on distributed parallel machines, and a drastic reduction in total CPU time.

Committee: Xiao-Chuan Cai, Associate Professor (Chair)
Bernard Bialecki, Colorado School of Mines
Elizabeth Bradley, Associate Professor
Richard Byrd, Professor
Robert (Bobby) Schnabel, Professor
Department of Computer Science
University of Colorado Boulder
Boulder, CO 80309-0430 USA
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