Second order asymptotical regularization methods for inverse problems in partial differential equations
This work provides a novel regularization framework for inverse problems in PDEs, offering improved convergence and acceleration for practitioners in applied mathematics and engineering.
The authors develop Second Order Asymptotical Regularization (SOAR) methods for inverse source problems in elliptic PDEs, proving convergence with fixed and dynamic damping parameters. Numerical examples demonstrate accuracy and acceleration, with comparisons to state-of-the-art methods.
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.