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BLOCK-SYMMETRIC AND BLOCK-LOWER-TRIANGULAR PRECONDITIONERS FOR pde-constrained optimization PROBLEMS＊

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Journal of Computational Mathematics 2013年第4期31卷 370-381页

作者： Guofeng Zhang Zhong Zheng School of Mathematics and Statistics Lanzhou University Lanzhou 730000 China

optimization problems with partial differential equations as constraints arise widely in many areas of science and engineering, in particular in problems of the design. The solution of such class of pde-constrained optimization problems is usually a major computational task. Because of the complexion for directly seeking the solution of pde-constrained op- timization problem, we transform it into a system of linear equations of the saddle-point form by using the Galerkin finite-element discretization. For the discretized linear system, in this paper we construct a block-symmetric and a block-lower-triangular preconditioner, for solving the pde-constrained optimization problem. Both preconditioners exploit the structure of the coefficient matrix. The explicit expressions for the eigenvalues and eigen- vectors of the corresponding preconditioned matrices are derived. Numerical implementa- tions show that these block preconditioners can lead to satisfactory experimental results for the preconditioned GMRES methods when the regularization parameter is suitably small.

关键词： Saddle-point matrix Preconditioning pde-constrained optimization Eigen-value and eigenvector Regularization parameter.

ON BLOCK PRECONDITIONERS FOR pde-constrained optimization PROBLEMS

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Journal of Computational Mathematics 2014年第3期32卷 272-283页

作者： Xiaoying Zhang Yumei Huang School of Mathematics and Statistics Lanzhou University Lanzhou 730000 China

Recently, Bal proposed a block-counter-diagonal and a block-counter-triangular precon- ditioning matrices to precondition the GMRES method for solving the structured system of linear equations arising from the Galerkin finite-element discretizations of the distributed control problems in （Computing 91 （2011） 379-395）. He analyzed the spectral properties and derived explicit expressions of the eigenvalues and eigenvectors of the preconditioned matrices. By applying the special structures and properties of the eigenvector matrices of the preconditioned matrices, we derive upper bounds for the 2-norm condition numbers of the eigenvector matrices and give asymptotic convergence factors of the preconditioned GMRES methods with the block-counter-diagonal and the block-counter-triangular pre- conditioners. Experimental results show that the convergence analyses match well with the numerical results.

关键词： pde-constrained optimization GMRES method Preconditioner Condition number Asymptotic convergence factor.

AFast Solver for an H_(1) Regularized pde-constrained optimization Problem

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Communications in Computational Physics 2016年第1期19卷 143-167页

作者： Andrew T.Barker Tyrone Rees Martin Stoll Center for Applied Scientific Computing Lawrence Livermore National LaboratoryMail Stop L-561LivermoreCA 94551USA Numerical Analysis Group Scientific Computing DepartmentRutherford Appleton LaboratoryChiltonDidcotOxfordshireOX110QXUnited Kingdom Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical SystemsSandtorstr.139106MagdeburgGermany

In this paperwe consider pde-constrained optimization problemswhich incorporate an H_(1)regularization control *** focus on a time-dependent pde,and consider both distributed and boundary *** problems we consider include bound constraints on the state,and we use a Moreau-Yosida penalty function to handle *** propose Krylov solvers and Schur complement preconditioning strategies for the different problems and illustrate their performance with numerical examples.

关键词： Preconditioning Krylov methods pde-constrained optimization optimal control of pdes

The pde-constrained optimization Method Based on MFS for Solving Inverse Heat Conduction Problems

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Journal of Mathematical Research with Applications 2018年第3期38卷 303-330页

作者： Yongfu ZHANG Chongjun LI School of Mathematical Sciences Dalian University of Technology College of Mathematics Inner Mongolia University for Nationalities

In this paper, we present an effective meshless method for solving the inverse heat conduction problems, with the Neumann boundary condition. A pde-constrained optimization method is developed to get a global approximation scheme in both spatial and temporal domains, by using the fundamental solution of the governing equation as the basis *** the initial measured data contain some noises, and the resulting systems of equations are usually ill-conditioned, the Tikhonov regularization technique with the generalized crossvalidation criterion is applied to obtain more stable numerical solutions. It is shown that the proposed schemes are effective by some numerical tests.

关键词： inverse heat conduction problem pde-constrained optimization method of fundamental solutions time-dependent heat source term Tikhonov regularization method

Optimal boundary control of 1D colloid transport in a dead-end channel 36

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Optimal boundary control of 1D colloid transport in a dead-e...