Core Concepts
The authors propose a novel approach to parameter identification in PDEs using monotone inclusion problems, demonstrating well-posedness and convergence of the regularization method.
Abstract
Parameter identification in partial differential equations (PDEs) is addressed through a total variation based regularization method. The inverse problem of reconstructing the source term from noisy data is discussed, emphasizing the need for regularization due to ill-posedness. Various regularization approaches like Tikhonov and iterative methods are compared, with a focus on Lavrentiev regularization for monotone problems. The study highlights numerical algorithms and inertial techniques for solving inclusion problems efficiently. Primal-dual splitting algorithms with inertial effects are explored, showcasing advancements in solving complex monotone inclusion problems.
Stats
A solution algorithm for the numerical solution of inclusion problems is discussed.
Regularization parameters are chosen appropriately to ensure convergence to the true solution.
Total variation and Sobolev norms are combined in the regularization method.
The subdifferential calculus results are applied to analyze the proximal operator.
The convergence speed of algorithms is enhanced by incorporating inertial terms.