Numerical analysis of scattered point measurement-based regularization for backward problems for fractional wave equations

TL;DR

This paper develops a numerical method for reconstructing initial data in fractional wave equations from noisy scattered measurements, balancing errors and noise, with demonstrated efficiency and accuracy.

Contribution

It introduces a novel regularization framework on nonuniform grids for fractional wave equations with scattered noisy data, including optimal error estimates and parameter selection.

Findings

Effective reconstruction from noisy scattered data

Optimal error bounds balancing discretization and noise

Numerical experiments confirming method accuracy

Abstract

In this work, our aim is to reconstruct the unknown initial value from terminal data. We develop a numerical framework on nonuniform time grids for fractional wave equations under the lower regularity assumptions. Then, we introduce a regularization method that effectively handles scattered point measurements contaminated with stochastic noise. The optimal error estimates of stochastic convergence not only balance discretization errors, the noise, and the number of observation points, but also propose an a priori choice of regularization parameters. Finally, several numerical experiments are presented to demonstrate the efficiency and accuracy of the algorithm.