A priori model reduction through Proper Generalized Decomposition for solving time-dependent partial differential equations

Abstract

Over the past years, model reduction techniques have become a necessary path for the reduction of computational requirements in the numerical simulation of complex models. A family of a priori model reduction techniques, called Proper Generalized Decomposition (PGD) methods, are receiving a growing interest. These methods rely on the a priori construction of separated variables representations of the solution of models defined in tensor product spaces. They can be interpreted as generalizations of Proper Orthogonal Decomposition (POD) for the a priori construction of such separated representations. In this paper, we introduce and study different definitions of PGD for the solution of time-dependent partial differential equations. We review classical definitions of PGD based on Galerkin or Minimal Residual formulations and we propose and discuss several improvements for these classical definitions. We give an interpretation of optimal decompositions as the solution of pseudo-eigenproblems. We also introduce a new definition of PGD, called Minimax PGD, which can be interpreted as a Petrov–Galerkin model reduction technique, where test and trial reduced basis functions are related by an adjoint problem. This new definition improves convergence properties of separated representations with respect to a chosen metric. It coincides with a classical POD for degenerate time-dependent partial differential equations. For the numerical construction of each PGD, we propose algorithms inspired from the solution of eigenproblems. Several numerical examples illustrate and compare the different definitions of PGD on transient advection–diffusion–reaction equations.