A novel parallel approach for solving some free boundary value problems

Abstract

This paper introduces a novel, efficient class of parallel direct and indirect iterative schemes to solve general obstacle and free boundary value problems. The uniqueness of the solution for the direct parallel method is established under the assumption that the model problem yields an $M$-matrix. The convergence analysis of the indirect approach is predicated on minimizing the discrete energy functional at each iterative stage of the linear approximation process. Under the aforementioned setting, we establish theoretical convergence results for both smooth and nonsmooth energy functionals defined over a convex set, in the sense of Ferris and Mangasarian \cite{ferris1994parallel}. For the numerical computation of the one-dimensional obstacle problem, we adopt a direct generalized parallel approach based on the SPIKE algorithm \cite{polizzi2006parallel}. To accelerate convergence and reduce computational complexity, a fast recursive version of the scheme is generalized to the nonsmooth case. For two- and three-dimensional obstacle problems, we employ a directional splitting method that treats each directional subproblem as a one-dimensional obstacle minimization problem. By framing the energy functional minimization as a parabolic time-dependent problem and utilizing an Armijo time-stepping rule during the solution update process, we successfully obtain results for higher-dimensional obstacles. Additionally, this study explores the possibility of extending the algorithm into a constrained quadratic programming optimization solver. We also evaluate the framework on image deblurring phenomena under the aforementioned setting, successfully recovering the original images. Finally, numerical illustrations are provided to validate the theoretical results.