Nonlinear compressive reduced basis approximation for multi-parameter elliptic problem

TL;DR

This work tackles nonlinear compressive reduced basis methods for multi-parameter elliptic PDEs, focusing on a multiparameter thermal-fin design problem. It investigates the gap between Kolmogorov width and manifold complexity, and introduces a nonlinear decoding strategy guided by sensing numbers to recover high RB modes from a small set of low-mode coefficients. Through P=1–2 experiments, spline-based and other regression approaches demonstrate substantial online efficiency gains and accuracy, often matching or approaching full RB performance with many fewer degrees of freedom. The study highlights parameter-count as a key driver of complexity and identifies the need for robust regression techniques as dimensionality grows, outlining avenues for future algorithmic enhancements.

Abstract

Reduced basis methods for approximating the solutions of parameter-dependant partial differential equations (PDEs) are based on learning the structure of the set of solutions - seen as a manifold ${\mathcal S}$ in some functional space - when the parameters vary. This involves investigating the manifold and, in particular, understanding whether it is close to a low-dimensional affine space. This leads to the notion of Kolmogorov $N$-width that consists of evaluating to which extent the best choice of a vectorial space of dimension $N$ approximates ${\mathcal S}$ well enough. If a good approximation of elements in ${\mathcal S}$ can be done with some well-chosen vectorial space of dimension $N$ -- provided $N$ is not too large -- then a ``reduced'' basis can be proposed that leads to a Galerkin type method for the approximation of any element in ${\mathcal S}$. In many cases, however, the Kolmogorov $N$-width is not so small, even if the parameter set lies in a space of small dimension yielding a manifold with small dimension. In terms of complexity reduction, this gap between the small dimension of the manifold and the large Kolmogorov $N$-width can be explained by the fact that the Kolmogorov $N$-width is linear while, in contrast, the dependency in the parameter is, most often, non-linear. There have been many contributions aiming at reconciling these two statements, either based on deterministic or AI approaches. We investigate here further a new paradigm that, in some sense, merges these two aspects: the nonlinear compressive reduced basisapproximation. We focus on a simple multiparameter problem and illustrate rigorously that the complexity associated with the approximation of the solution to the parameter dependant PDE is directly related to the number of parameters rather than the Kolmogorov $N$-width.

Figures (9)

• Figure 1: Schematic of a thermal fin showing the structure and important elements, see also rbpp.
• Figure 2: Number of reduced basis functions as a function of the parameter dimension for different accuracies
• Figure 3: Mean relative errors vs number of POD modes for $P = 1$(left) and $P = 9$(right)
• Figure 4: The Biot number as a function of the first RB coefficient
• Figure 5: Higher coefficients as a function of the first coefficient for $P=1$ and $n=1$