Scattered point measurement-based regularization for backward problems for fractional wave equations

TL;DR

This paper develops a regularization method for reconstructing initial data in fractional wave equations from scattered noisy measurements, using Mittag-Leffler functions and iterative algorithms, with proven stochastic convergence and numerical validation.

Contribution

It introduces a novel regularization approach for backward fractional wave problems with scattered data, including convergence analysis and an efficient iterative algorithm.

Findings

Regularization method effectively reconstructs initial data from noisy scattered measurements.

Stochastic convergence of the proposed method is established.

Numerical experiments demonstrate high accuracy and efficiency.

Abstract

In this work, we are devoted to the reconstruction of an unknown initial value from the terminal data. The asymptotic and root-distribution properties of Mittag-Leffler functions are used to establish stability of the backward problem. Furthermore, we introduce a regularization method that effectively handles scattered point measurements contaminated with stochastic noise. Furthermore, we prove the stochastic convergence of our proposed regularization and provide an iterative algorithm to find the optimal regularization parameter. Finally, several numerical experiments are presented to demonstrate the efficiency and accuracy of the algorithm.