2019/06/02 by Enrique Otárola, Otarola, Enrique
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.1906.00540
openalex publication_date 2019/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We propose and analyze an a posteriori error estimator for a PDE-constrained\noptimization problem involving a nondifferentiable cost functional, fractional\ndiffusion, and control-constraints. We realize fractional diffusion as the\nDirichlet-to-Neumann map for a nonuniformly PDE and propose an equivalent\noptimal control problem with a local state equation. For such an equivalent\nproblem, we design an a posteriori error estimator which can be defined as the\nsum of four contributions: two contributions related to the approximation of\nthe state and adjoint equations and two contributions that account for the\ndiscretization of the control variable and its associated subgradient. The\ncontributions related to the discretization of the state and adjoint equations\nrely on anisotropic error estimators in weighted Sobolev spaces. We prove that\nthe proposed a posteriori error estimator is locally efficient and, under\nsuitable assumptions, reliable. We design an adaptive scheme that yields, for\nthe examples that we perform, optimal experimental rates of convergence.\n