Title: The Power of Polynomial Approximation for Parameterized Matrix Equations
Speaker:
Paul Constantine, Stanford University
Date/Time:
Wednesday, March 11, 2009, 9:00am – 10:00am
Location: CSRI Building, Room 90 (Sandia NM)
Brief Abstract: The glitz and glamour of the phrase "polynomial chaos" has captured the attention of both junior and senior researchers as interest in uncertainty quantification has blossomed in the last decade. Buzzwords aside, we have seen the power of polynomial approximation methods for quantifying output uncertainties given input uncertainties. I will analyze the application of these methods to a very general model problem -- a linear system of equations where the elements of the matrix and the right hand side depend on a set of parameters -- and argue for the utility of this analysis for both algorithmic and software development.
The techniques for uncertainty propagation are generally classified as *intrusive* or *non-intrusive*, where non-intrusive methods take advantage of existing solvers for deterministic problems and intrusive methods typically require extensive code modification. Sandia has had considerable success employing non-intrusive methods in the DAKOTA framework but has invested less in intrusive methods due to their highly problem-specific implementation. Using the parameterized matrix analysis, I propose an alternative algorithmic paradigm dubbed *weakly intrusive* that offers a much-needed middle ground for code development in UQ. In analogy with matrix-free iterative solvers -- such as conjugate gradient -- the weakly intrusive paradigm only allows matrix-vector multiplies where the matrix is evaluated at a point in the parameter space.
The parameterized matrix analysis also yields a simple heuristic for discovering the most important parameters in a model with respect to the overall variability in the output. We take advantage of this heuristic to construct efficient anisotropic approximations for the solution of large-scale systems with multiple parameters. The upside of the curse of dimensionality is that simple reduction ideas can dramatically impact the efficiency of the approximation.
I will tie these ideas together with three examples of increasing complexity -- from a toy problem where all theoretical considerations are well-understood to an engineering example where they are not.
CSRI POC:
Ken Alvin, (505) 844-9329 |