Second order asymptotical regularization methods for inverse problems in partial differential equations

TL;DR

This paper introduces Second Order Asymptotical Regularization (SOAR) methods for inverse source problems in elliptic PDEs, demonstrating convergence, acceleration, and numerical effectiveness compared to existing techniques.

Contribution

The paper develops a novel SOAR framework with convergence proofs, dynamic damping, and a symplectic numerical scheme for inverse PDE source problems, advancing regularization methods.

Findings

SOAR converges weakly to the true source as noise decreases.

Dynamic damping accelerates convergence compared to fixed damping.

Numerical results show SOAR's high accuracy and efficiency.

Abstract

We develop Second Order Asymptotical Regularization (SOAR) methods for solving inverse source problems in elliptic partial differential equations with both Dirichlet and Neumann boundary data. We show the convergence results of SOAR with the fixed damping parameter, as well as with a dynamic damping parameter, which is a continuous analog of Nesterov's acceleration method. Moreover, by using Morozov's discrepancy principle together with a newly developed total energy discrepancy principle, we prove that the approximate solution of SOAR weakly converges to an exact source function as the measurement noise goes to zero. A damped symplectic scheme, combined with the finite element method, is developed for the numerical implementation of SOAR, which yields a novel iterative regularization scheme for solving inverse source problems. Several numerical examples are given to show the accuracy and the acceleration effect of SOAR. A comparison with the state-of-the-art methods is also provided.