Efficient Shallow Ritz Method For 1D Diffusion Problems

TL;DR

This paper introduces an efficient shallow Ritz method combined with a damped block Newton approach for solving 1D diffusion problems, achieving high approximation accuracy and computational efficiency, especially for non-smooth cases.

Contribution

It develops a novel damped block Newton method and an adaptive enhancement technique to improve the accuracy and efficiency of shallow Ritz solutions for 1D diffusion problems.

Findings

Damped block Newton method achieves $O(n)$ computational cost per iteration.

Numerical experiments demonstrate effective mesh point optimization.

Adaptive damped Newton further enhances efficiency and accuracy.

Abstract

This paper studies the shallow Ritz method for solving the one-dimensional diffusion problem. It is shown that the shallow Ritz method improves the order of approximation dramatically for non-smooth problems. To realize this optimal or nearly optimal order of the shallow Ritz approximation, we develop a damped block Newton (dBN) method that alternates between updates of the linear and non-linear parameters. Per each iteration, the linear and the non-linear parameters are updated by exact inversion and one step of a modified, damped Newton method applied to a reduced non-linear system, respectively. The computational cost of each dBN iteration is $O(n)$. Starting with the non-linear parameters as a uniform partition of the interval, numerical experiments show that the dBN is capable of efficiently moving mesh points to nearly optimal locations. To improve efficiency of the dBN further, we propose an adaptive damped block Newton (AdBN) method by combining the dBN with the adaptive neuron enhancement (ANE) method [26].