Convergence Analysis of Variable-Separation-Based Model Reduction for Linear Parametric Partial Differential Equations

Abstract

This paper analyzes the convergence of a variable-separation (VS) model reduction method that successively constructs separated approximations to the solutions of linear parametric partial differential equations (PDEs) by solving residual problems at sampled parameter values. We prove that the resulting VS approximations converge almost surely to the exact parametric solution in the parameter-averaged energy norm, under finite-dimensional approximation conditions on the error family and the assumptions of symmetry, uniform coercivity, and uniform boundedness of the governing bilinear forms. For parametric PDEs whose operators and right-hand sides admit finite-term separated representations, we construct finite-dimensional approximation spaces for the error family and bound their dimensions and approximation errors, thereby verifying these approximation conditions and obtaining an $O((\log N)^{-1})$ upper bound for the expected parameter-averaged error energy after $N$ iterations. When the solution families are contained in a fixed finite-dimensional space, we obtain the sharper $O(N^{-1})$ bound for the expected error energy and, if the operator is also parameter-independent, exact recovery for almost every parameter after an almost surely finite number of iterations. Numerical experiments illustrate the error decay of the VS approximations, as well as finite-step recovery in the parameter-independent case.