Certified Rapid Solution of Parametrized Linear Elliptic Equations: Application to Parameter Estimation
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HPCES001.pdf
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Author(s)
Nguyen, N. C.
Liu, Guirong
Patera, Anthony T.
Date Issued
January 2005
Series/Report no.
High Performance Computation for Engineered Systems (HPCES);
Abstract
We present a technique for the rapid and reliable
evaluation of linear-functional output of elliptic partial differential equations with affine parameter dependence. The essential components are (i) rapidly uniformly convergent reduced-basis approximations — Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N (optimally) selected points in parameter space; (ii) a posteriori error estimation — relaxations of the residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs; and (iii) offline/online computational procedures — stratagems that exploit affine parameter dependence to de-couple the generation and projection stages of the approximation
process. The operation count for the online stage — in which, given a new parameter value, we calculate the output and associated error bound — depends only on N (typically small) and the parametric complexity of the problem. The method is thus ideally suited to the many-query and real-time contexts. In this paper, based on the technique we develop a robust inverse
computational method for very fast solution of inverse problems characterized by parametrized partial differential equations. The essential ideas are in three-fold: first, we apply the technique to the forward problem for the rapid certified evaluation of PDE input-output relations and associated rigorous error bounds; second, we incorporate the reduced-basis approximation and
error bounds into the inverse problem formulation; and third, rather than regularize the goodness-of-fit objective, we may instead identify all (or almost all, in the probabilistic sense)
system configurations consistent with the available experimental data — well-posedness is reflected in a bounded "possibility region" that furthermore shrinks as the experimental error is
decreased.
Subjects
Linear elliptic equations
Reduced-basis method
Reduced-basis approximation
A posteriori error estimation
Parameter estimation
Inverse computational method
Possibility region
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