Inexact primal-dual active set iteration for optimal distribution control of stationary heat or cold source

This paper focuses on efficient numerical methods for optimal distribution control problem of stationary heat or cold source. With the application of finite element method to discretize the model problem, we aim to take advantage of the benefits of primal-dual active set method and develop an inexact iteration strategy for approximating the optimal solution. In addition to the iteration error, the discretization error accounts for the significant portion of the total error when utilizing the numerical scheme to solve the problem. From this perspective, we present the error analysis that mingles both the discretization error and iteration error together. Based on our analysis, an adequate criterion is tailored for discretization mesh sizes to terminate the iteration, and the approximate solutions can achieve the acceptable precision consistent with discretization level. Numerical experiments are performed to verify the efficiency of the proposed method.