Convergence analysis for a tree-based nonlinear reduced basis method

TL;DR

This paper introduces a tree-based nonlinear reduced basis method for parametrized elliptic PDEs, providing a convergence analysis and demonstrating its efficiency and accuracy through numerical experiments.

Contribution

It develops a novel tree-partitioned nonlinear RB method with a rigorous convergence analysis and demonstrates improved performance over existing methods.

Findings

Achieves explicit bounds on subdomain number for desired accuracy.

Numerical results confirm theoretical convergence rates.

Outperforms existing nonlinear RB methods in several cases.

Abstract

We develop and analyze a nonlinear reduced basis (RB) method for parametrized elliptic partial differential equations based on a binary-tree partition of the parameter domain into tensor-product structured subdomains. Each subdomain is associated with a local RB space of prescribed dimension, constructed via a greedy algorithm. A splitting strategy along the longest edge of the parameter subdomains ensures geometric control of the subdomains and enables a rigorous convergence analysis. Under the assumption that the parameter-to-solution map admits a holomorphic extension and that the resulting domain partition is quasi-uniform, we establish explicit bounds on the number of subdomains required to achieve a given tolerance for arbitrary parameter domain dimension and RB spaces size. Numerical experiments for diffusion and convection-diffusion problems confirm the theoretical predictions, demonstrating that the proposed approach, which has low storage requirements, achieves the expected convergence rates and in several cases outperforms an existing nonlinear RB method.