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 approxim...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 function.Since 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.展开更多
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 op...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.展开更多
In this paper, by exploiting the special block and sparse structure of the coefficient matrix, we present a new preconditioning strategy for solving large sparse linear systems arising in the time-dependent distribute...In this paper, by exploiting the special block and sparse structure of the coefficient matrix, we present a new preconditioning strategy for solving large sparse linear systems arising in the time-dependent distributed control problem involving the heat equation with two different functions. First a natural order-reduction is performed, and then the reduced- order linear system of equations is solved by the preconditioned MINRES algorithm with a new preconditioning techniques. The spectral properties of the preconditioned matrix are analyzed. Numerical results demonstrate that the preconditioning strategy for solving the large sparse systems discretized from the time-dependent problems is more effective for a wide range of mesh sizes and the value of the regularization parameter.展开更多
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 Galerki...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.展开更多
In this paperwe consider PDE-constrained optimization problemswhich incorporate an H_(1)regularization control term.We focus on a time-dependent PDE,and consider both distributed and boundary control.The problems we c...In this paperwe consider PDE-constrained optimization problemswhich incorporate an H_(1)regularization control term.We focus on a time-dependent PDE,and consider both distributed and boundary control.The problems we consider include bound constraints on the state,and we use a Moreau-Yosida penalty function to handle this.We propose Krylov solvers and Schur complement preconditioning strategies for the different problems and illustrate their performance with numerical examples.展开更多
In this paper,the efficient preconditioned modified Hermitian and skew-Hermitian splitting(PMHSS)iteration method is further explored and it is extended to solve more general block two-by-two linear systems with diffe...In this paper,the efficient preconditioned modified Hermitian and skew-Hermitian splitting(PMHSS)iteration method is further explored and it is extended to solve more general block two-by-two linear systems with different and nonsymmetric off-diagonal blocks.With the aid of the singular value decomposition technique,the detailed analysis of the algebraic and convergence properties of the PMHSS iteration method demonstrates that it is still convergent unconditionally as when it is used to solve the well-studied case of block two-by-two linear systems with same and symmetric off-diagonal blocks.Moreover,the PMHSS preconditioned matrix is almost unitary diagonalizable with clustered eigenvalue distributions for this more general case.On account of the favorable spectral properties of the PMHSS preconditioned matrix,a parameter free Chebyshev accelerated PMHSS(CAPMHSS)method is established to further improve its convergence rate.Numerical experiments about Kroncker structured block two-by-two linear systems arising from a time-dependent PDE-constrained optimal control problem demonstrate quite satisfactory and competitive performance of the CAPMHSS method compared with some existing preconditioned Krylov subspace methods.展开更多
基金Supported by the National Natural Science Foundation of China(Grant Nos.1129014311471066+3 种基金11572081)the Fundamental Research of Civil Aircraft(Grant No.MJ-F-2012-04)the Fundamental Research Funds for the Central Universities(Grant No.DUT15LK44)the Scientific Research Funds of Inner Mongolia University for the Nationalities(Grant No.NMD1304)
文摘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 function.Since 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.
基金Acknowledgments. This work was supported by the National Natural Science Foundation of China（l1271174）. The authors are very much indebted to the referees for providing very valuable suggestions and comments, which greatly improved the original manuscript of this paper. The authors would also like to thank Dr. Zeng-Qi Wang for helping on forming the MATLAB data of the matrices.
文摘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.
基金The work was supported by the National Natural Science Foundation of China （11271174）. The authors would like to thank the referees for the comments and constructive suggestions, which are valuable in improving the quality of the manuscript.
文摘In this paper, by exploiting the special block and sparse structure of the coefficient matrix, we present a new preconditioning strategy for solving large sparse linear systems arising in the time-dependent distributed control problem involving the heat equation with two different functions. First a natural order-reduction is performed, and then the reduced- order linear system of equations is solved by the preconditioned MINRES algorithm with a new preconditioning techniques. The spectral properties of the preconditioned matrix are analyzed. Numerical results demonstrate that the preconditioning strategy for solving the large sparse systems discretized from the time-dependent problems is more effective for a wide range of mesh sizes and the value of the regularization parameter.
文摘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.
文摘In this paperwe consider PDE-constrained optimization problemswhich incorporate an H_(1)regularization control term.We focus on a time-dependent PDE,and consider both distributed and boundary control.The problems we consider include bound constraints on the state,and we use a Moreau-Yosida penalty function to handle this.We propose Krylov solvers and Schur complement preconditioning strategies for the different problems and illustrate their performance with numerical examples.
基金supported by the National Natural Science Foundation of China(Nos.11801242,11771193,and 11901267)the Fundamental Research Funds for the Central Universities(No.lzujbky-2022-05)the Natural Science Foundation of Gansu Province of China(Grant No.23JRRA1104).
文摘In this paper,the efficient preconditioned modified Hermitian and skew-Hermitian splitting(PMHSS)iteration method is further explored and it is extended to solve more general block two-by-two linear systems with different and nonsymmetric off-diagonal blocks.With the aid of the singular value decomposition technique,the detailed analysis of the algebraic and convergence properties of the PMHSS iteration method demonstrates that it is still convergent unconditionally as when it is used to solve the well-studied case of block two-by-two linear systems with same and symmetric off-diagonal blocks.Moreover,the PMHSS preconditioned matrix is almost unitary diagonalizable with clustered eigenvalue distributions for this more general case.On account of the favorable spectral properties of the PMHSS preconditioned matrix,a parameter free Chebyshev accelerated PMHSS(CAPMHSS)method is established to further improve its convergence rate.Numerical experiments about Kroncker structured block two-by-two linear systems arising from a time-dependent PDE-constrained optimal control problem demonstrate quite satisfactory and competitive performance of the CAPMHSS method compared with some existing preconditioned Krylov subspace methods.
