Trust-Region Methods for Nonlinear Elliptic Equations with Radial Basis Functions

TL;DR

This paper introduces a trust-region method for solving nonlinear elliptic PDEs using radial basis functions, deriving explicit Jacobian and Hessian formulas, and demonstrating superior performance over previous linearization-based methods.

Contribution

The paper develops a novel trust-region approach with explicit Jacobian and Hessian formulas for radial basis function collocation, improving solution accuracy and efficiency.

Findings

Outperforms previous linearization-based methods

Successfully solves complex nonlinear PDEs like Monge-Ampère

Demonstrates high accuracy on various elliptic problems

Abstract

We consider the numerical solution of nonlinear elliptic boundary value problems with Kansa's method. We derive analytic formulas for the Jacobian and Hessian of the resulting nonlinear collocation system and exploit them within the framework of the trust-region algorithm. This ansatz is tested on semilinear, quasilinear and fully nonlinear elliptic PDEs (including Plateau's problem, Hele-Shaw flow and the Monge-Amp\`ere equation) with excellent results. The new approach distinctly outperforms previous ones based on linearization or finite-difference Jacobians.