Nonlinear compressive reduced basis approximation for PDE’s

Abstract

Linear model reduction techniques design offline low-dimensional subspaces that are tailored to the approximation of solutions to a parameterized partial differential equation, for the purpose of fast online numerical simulations. These methods, such as the Proper Orthogonal Decomposition (POD) or Reduced Basis (RB) methods, are very effective when the family of solutions has fast-decaying Karhunen–Loève eigenvalues or Kolmogorov widths, reflecting the approximability by finite-dimensional linear spaces. On the other hand, they become ineffective when these quantities have a slow decay, in particular for families of solutions to hyperbolic transport equations with parameter-dependent shock positions. The objective of this work is to explore the ability of nonlinear model reduction to circumvent this particular situation. To this end, we first describe particular notions of nonlinear widths that have a substantially faster decay for the aforementioned families. Then, we discuss a systematic approach for achieving better performance via a nonlinear reconstruction from the first coordinates of a linear reduced model approximation, thus allowing us to stay in the same “classical” framework of projection-based model reduction. We analyze the approach and report on its performance for a simple and yet instructive univariate test case.

Keywords

• non linear reduced basis
• compressed sensing
• solution manifold
• machine learning
• $m$-width