(cosponsored by SSERC)
Department of Mathematics,
Arizona State University
Monday,
March 6, 2000, 12:00 noon in GWC Room 604
Omar Ghattas
Laboratory for Algorithms, Computing, and Mechanics, CMU
Large-Scale PDE-Constrained Optimization:
Parallel Algorithms and Applications to
Optimal Design, Optimal Control, and Parameter Estimation Problems
Abstract
Very large scale PDE-constrained optimization problems arise naturally
in many areas of science and engineering, and take the form of optimal
design, optimal control, and parameter estimation problems. The common
denominator is a nonlinear optimization problem that is constrained by
the PDEs that govern behavior of the physical system. Thus, solving
the PDEs is just a subproblem associated with optimization, which can
be orders of magnitude more challenging computationally. Despite its
importance, little attention has been focused on the design of
parallel algorithms for PDE-constrained optimization. This is
expected: it makes little sense to address the "inverse" problem until
the "forward" problem is well understood. However, recent advances in
parallel PDE solvers and the arrival of the teraflop computing era
provide the necessary ammunition. Large-scale PDE-constrained
optimization problems are intractable with current black-box
optimization technology. To render them tractable, we need to develop
parallel optimization algorithms that exploit the PDE nature of the
constraints, scale to the millions of constraints and variables that
arise upon discretization, and capitalize on emerging highly parallel
supercomputers.
I will give an overview of the TAOS (Terascale Algorithms for
Optimization of Simulations) Project at CMU, whose goals are to
develop the enabling parallel numerical algorithms for large-scale
simulation-based optimization, and to apply them to driving optimal
control, optimal design, and parameter estimation problems in
engineering and science. I will illustrate the talk with motivating
applications in each of the three classes: optimal design of
artificial heart devices, optimal boundary control of viscous flows,
and inverse earthquake ground motion modeling. I will describe a
family of parallel Newton-Krylov methods for solution of the
optimality system of PDE-constrained optimization problems. These
methods can be thought of as full-space Newton-SQP with
preconditioning by reduced-space limited memory quasi-Newton SQP and
approximate forward Jacobians. This combines the rapid convergence of
Newton methods with the low per-iteration cost of approximate
methods. I will present studies of parallel efficiency and scalability
of a PETSc-based implementation for the problem of optimal control of
a viscous incompressible fluid by suction/injection of fluid on its
boundary. Numerical experiments for problems of size up to a million
state variables and 40,000 control variables on up to 256 processors
of a Cray T3E-900 yield encouraging results.
The TAOS Project is jointly directed with Larry Biegler. The research
on Newton-Krylov optimization methods and optimal flow control is
joint work with PhD student George Biros; artificial heart design with
PhD student Ivan Malcevic and collaborators Jim Antaki and Greg
Burgreen (University of Pittsburgh Medical Center); and inverse
earthquake modeling with PhD students Volkan Akcelik and Yiannis
Epanomeritakis and collaborator Jacobo Bielak.
Speaker Bio:
Omar Ghattas is Associate Professor and Director of the Laboratory for
Algorithms, Computing, and Mechanics at Carnegie Mellon University. He
has appointments or affiliations with the Department of Civil and
Environmental Engineering, the Program in Biomedical and Health
Engineering, the Institute for Complex Engineered Systems, and the
School of Computer Science. He received his B.S. in civil engineering
in 1984, and his M.S. and Ph.D. in mechanics in 1986 and 1988, all
from Duke University. He joined CMU in 1989 after serving as a
postdoctoral research associate at Duke. He has general research
interests in high performance scientific computation, with particular
emphasis on simulation and optimization of complex systems governed by
fluid- and solid-mechanical phenomena.
For further information please contact:
mittelmann@asu.edu