Reduced-order modeling of parameterized PDEs using time-space-parameter principal component analysis

Abstract

This paper presents a methodology for constructing low-order surrogate models of finite element/finite
volume discrete solutions of parameterized steady-state partial differential equations. The construction
of proper orthogonal decomposition modes in both physical space and parameter space allows us to
represent high-dimensional discrete solutions using only a few coefficients. An incremental greedy approach
is developed for efficiently tackling problems with high-dimensional parameter spaces. For numerical
experiments and validation, several non-linear steady-state convection–diffusion–reaction problems are
considered: first in one spatial dimension with two parameters, and then in two spatial dimensions with
two and five parameters. In the two-dimensional spatial case with two parameters, it is shown that a 7×7
coefficient matrix is sufficient to accurately reproduce the expected solution, while in the five parameters
problem, a 13×6 coefficient matrix is shown to reproduce the solution with sufficient accuracy. The
proposed methodology is expected to find applications to parameter variation studies, uncertainty analysis,
inverse problems and optimal design

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