Preconditioned Iterative Method of Crank-Nicolson Scheme for a Parabolic Optimal Control Problem
For the distributed optimal control problem constrained by parabolic partial differential equations(PDE),a set of coupled initial and final value problems are obtained by the first-order optimality conditions of the problem.The coupled problems are discretized by the Crank-Nicolson scheme,and the block triangle preconditioned strategy is adopted to solve the 2*2 linear sys-tem.Then,the approximation of Schur complement based on Kronecker products is established by the matching strategy and split-ting technique based on Kronecker products.Based on this,a preconditioned iterative method based on Crank-Nicolson scheme is proposed.Finally,numerical experiments demonstrate the accuracy and efficiency of the proposed method.The results show that the approximation of Schur complement with the structural form of Kronecker products can effectively implement the precondi-tioning condition.The method can greatly help to reduce the number of iterations,save computing time,and reduce computing costs,with good results.The research results have application value in solving related optimal control problems in engineering technology and social sciences.
preconditioningCrank-Nicolson schemeinitial/final value problemPDE constrained optimization
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