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
Our work considers the optimization of the sum of a non-smooth convex function and a finite family of composite convex functions, each one of which is composed of a convex function and a bounded linear operator. This ...Our work considers the optimization of the sum of a non-smooth convex function and a finite family of composite convex functions, each one of which is composed of a convex function and a bounded linear operator. This type of problem is associated with many interesting challenges encoun- tered in the image restoration and image reconstruction fields. We developed a splitting primal-dual proximity algorithm to solve this problem. Furthermore, we propose a preconditioned method~ of which the iterative parameters are obtained without the need to know some particular operator norm in advance. Theoretical convergence theorems are presented. We then apply the proposed methods to solve a total variation regularization model, in which the L2 data error function is added to the L1 data error function. The main advantageous feature of this model is its capability to combine different loss functions. The numerical results obtained for computed tomography （CT） image recon- struction demonstrated the ability of the proposed algorithm to reconstruct an image with few and sparse projection views while maintaining the image quality.展开更多
This paper addresses distributed adaptive optimal resource allocation problems over weight-balanced digraphs.By leveraging state-of-the-art adaptive coupling designs for multiagent systems,two adaptive algorithms are ...This paper addresses distributed adaptive optimal resource allocation problems over weight-balanced digraphs.By leveraging state-of-the-art adaptive coupling designs for multiagent systems,two adaptive algorithms are proposed,namely a directed-spanning-tree-based algorithm and a node-based algorithm.The benefits of these algorithms are that they require neither sufficiently small or unitary step sizes,nor global knowledge of Laplacian eigenvalues,which are widely required in the literature.It is shown that both algorithms belong to a class of uncertain saddle-point dynamics,which can be tackled by repeatedly adopting the Peter-Paul inequality in the framework of Lyapunov theory.Thanks to this new viewpoint,global asymptotic convergence of both algorithms can be proven in a unified way.The effectiveness of the proposed algorithms is validated through numerical simulations and case studies in IEEE 30-bus and 118-bus power systems.展开更多
In this paper,a two-step semi-regularized Hermitian and skew-Hermitian splitting(SHSS)iteration method is constructed by introducing a regularization matrix in the(1,1)-block of the first iteration step,to solve the s...In this paper,a two-step semi-regularized Hermitian and skew-Hermitian splitting(SHSS)iteration method is constructed by introducing a regularization matrix in the(1,1)-block of the first iteration step,to solve the saddle-point linear system.By carefully selecting two different regularization matrices,two kinds of SHSS preconditioners are proposed to accelerate the convergence rates of the Krylov subspace iteration methods.Theoretical analysis about the eigenvalue distribution demonstrates that the proposed SHSS preconditioners can make the eigenvalues of the corresponding preconditioned matrices be clustered around 1 and uniformly bounded away from 0.The eigenvector distribution and the upper bound on the degree of the minimal polynomial of the SHSS-preconditioned matrices indicate that the SHSS-preconditioned Krylov subspace iterative methods can converge to the true solution within finite steps in exact arithmetic.In addition,the numerical example derived from the optimal control problem shows that the SHSS preconditioners can significantly improve the convergence speeds of the Krylov subspace iteration methods,and their convergence rates are independent of the discrete mesh size.展开更多
This paper is concerned with the general study in the existence,uniqueness and error estimationof finite element solutions for a larger class of 'saddle-point' schemes. The established theory inthe form of Lax...This paper is concerned with the general study in the existence,uniqueness and error estimationof finite element solutions for a larger class of 'saddle-point' schemes. The established theory inthe form of Lax-like equivalency theorem includes Brezzi’s theory that has been treated as a specialcase.Two criteria are presented so as to help the practical verification of S-Babuska condition.展开更多
In this paper, a generalized augmented Lagrangian-successive over-relaxation （GAL- SOR） iteration method is presented for solving saddle-point systems arising from distributed control problems. The convergence prope...In this paper, a generalized augmented Lagrangian-successive over-relaxation （GAL- SOR） iteration method is presented for solving saddle-point systems arising from distributed control problems. The convergence properties of the GAL-SOR method are studied in the spectral properties for the precondidetail. Moreover, when0 ≤ω≤ 1 and Q=1/γI , tioned matrix are analyzed. Numerical experiments show that if the mass matrix from the distributed control problems is not easy to inverse and the regularization parameter β is very small, the GAL-SOR iteration method can work well.展开更多
For stabilized saddle-point problems, we apply the two iteration parameters idea for regularized Hermitian and skew-Hermitian splitting (RHSS) method and establish accelerated RHSS (ARHSS) iteration method. Theoretica...For stabilized saddle-point problems, we apply the two iteration parameters idea for regularized Hermitian and skew-Hermitian splitting (RHSS) method and establish accelerated RHSS (ARHSS) iteration method. Theoretical analysis shows that the ARHSS method converges unconditionally to the unique solution of the saddle point problem. Finally, we use a numerical example to confirm the effectiveness of the method.展开更多
In this paper, a least-squares mixed finite element method for the solution of the primal saddle-point problem is developed. It is proved that the approximate problem is consistent ellipticity in the conforming finite...In this paper, a least-squares mixed finite element method for the solution of the primal saddle-point problem is developed. It is proved that the approximate problem is consistent ellipticity in the conforming finite element spaces with only the discrete BB-condition needed for a smaller auxiliary problem. The abstract error estimate is derived. [ABSTRACT FROM AUTHOR]展开更多
An effective algorithm for solving large saddle-point linear systems, presented by Krukier et al., is applied to the constrained optimization problems. This method is a modification of skew-Hermitian triangular splitt...An effective algorithm for solving large saddle-point linear systems, presented by Krukier et al., is applied to the constrained optimization problems. This method is a modification of skew-Hermitian triangular splitting iteration methods. We consider the saddle-point linear systems with singular or semidefinite （1, 1） blocks. Moreover, this method is applied to precondition the GMRES. Numerical results have confirmed the effectiveness of the method and showed that the new method can produce high-quality preconditioners for the Krylov subspace methods for solving large sparse saddle-point linear systems.展开更多
.In this paper,an augmented Lagrangian Uzawa iterative method is developed and analyzed for solving a class of double saddle-point systems with semidefinite(2,2)block.Convergence of the iterativemethod is proved under....In this paper,an augmented Lagrangian Uzawa iterative method is developed and analyzed for solving a class of double saddle-point systems with semidefinite(2,2)block.Convergence of the iterativemethod is proved under the assumption that the double saddle-point problem exists a unique solution.An application of the iterative method to the double saddle-point systems arising from the distributed Lagrange multiplier/fictitious domain(DLM/FD)finite element method for solving elliptic interface problems is also presented,in which the existence and uniqueness of the double saddle-point system is guaranteed by the analysis of the DLM/FD finite element method.Numerical experiments are conducted to validate the theoretical results and to study the performance of the proposed iterative method.展开更多
We study the propagator for an electron moving in a two-dimensional （2D） quadratic saddle-point potential, in the presence of a perpendicular uniform magnetic field. A closed-form expression for the propagator is de...We study the propagator for an electron moving in a two-dimensional （2D） quadratic saddle-point potential, in the presence of a perpendicular uniform magnetic field. A closed-form expression for the propagator is derived using the Feynmann path integrals.展开更多
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Supported by the NSFC(Grant Nos.11201216,11401293,11461046 and 11661056)the National Basic Research Program of China(Grant No.2013CB329404)the NSF of Jiangxi Province(Grant Nos.20151BAB211010and 20142BAB211016)
文摘Our work considers the optimization of the sum of a non-smooth convex function and a finite family of composite convex functions, each one of which is composed of a convex function and a bounded linear operator. This type of problem is associated with many interesting challenges encoun- tered in the image restoration and image reconstruction fields. We developed a splitting primal-dual proximity algorithm to solve this problem. Furthermore, we propose a preconditioned method~ of which the iterative parameters are obtained without the need to know some particular operator norm in advance. Theoretical convergence theorems are presented. We then apply the proposed methods to solve a total variation regularization model, in which the L2 data error function is added to the L1 data error function. The main advantageous feature of this model is its capability to combine different loss functions. The numerical results obtained for computed tomography （CT） image recon- struction demonstrated the ability of the proposed algorithm to reconstruct an image with few and sparse projection views while maintaining the image quality.
基金supported in part by the China Postdoctoral Science Foundation(BX2021064)the Fundamental Research Funds for the Central Universities(2242022R20030)+1 种基金the National Key R&D Program of China(2022YFE0198700)the Natural Science Foundation of China(62150610499,62073074,61833005)。
文摘This paper addresses distributed adaptive optimal resource allocation problems over weight-balanced digraphs.By leveraging state-of-the-art adaptive coupling designs for multiagent systems,two adaptive algorithms are proposed,namely a directed-spanning-tree-based algorithm and a node-based algorithm.The benefits of these algorithms are that they require neither sufficiently small or unitary step sizes,nor global knowledge of Laplacian eigenvalues,which are widely required in the literature.It is shown that both algorithms belong to a class of uncertain saddle-point dynamics,which can be tackled by repeatedly adopting the Peter-Paul inequality in the framework of Lyapunov theory.Thanks to this new viewpoint,global asymptotic convergence of both algorithms can be proven in a unified way.The effectiveness of the proposed algorithms is validated through numerical simulations and case studies in IEEE 30-bus and 118-bus power systems.
基金the National Natural Science Foundation of China(No.12001048)R&D Program of Beijing Municipal Education Commission(No.KM202011232019),China.
文摘In this paper,a two-step semi-regularized Hermitian and skew-Hermitian splitting(SHSS)iteration method is constructed by introducing a regularization matrix in the(1,1)-block of the first iteration step,to solve the saddle-point linear system.By carefully selecting two different regularization matrices,two kinds of SHSS preconditioners are proposed to accelerate the convergence rates of the Krylov subspace iteration methods.Theoretical analysis about the eigenvalue distribution demonstrates that the proposed SHSS preconditioners can make the eigenvalues of the corresponding preconditioned matrices be clustered around 1 and uniformly bounded away from 0.The eigenvector distribution and the upper bound on the degree of the minimal polynomial of the SHSS-preconditioned matrices indicate that the SHSS-preconditioned Krylov subspace iterative methods can converge to the true solution within finite steps in exact arithmetic.In addition,the numerical example derived from the optimal control problem shows that the SHSS preconditioners can significantly improve the convergence speeds of the Krylov subspace iteration methods,and their convergence rates are independent of the discrete mesh size.
文摘This paper is concerned with the general study in the existence,uniqueness and error estimationof finite element solutions for a larger class of 'saddle-point' schemes. The established theory inthe form of Lax-like equivalency theorem includes Brezzi’s theory that has been treated as a specialcase.Two criteria are presented so as to help the practical verification of S-Babuska condition.
基金Acknowledgements. 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. This work was supported by the National Natural Science Foundation of China （No. 11271174 and No. 11511130015） and the Scientific Research Project of Putian University （No. 2015061）.
文摘In this paper, a generalized augmented Lagrangian-successive over-relaxation （GAL- SOR） iteration method is presented for solving saddle-point systems arising from distributed control problems. The convergence properties of the GAL-SOR method are studied in the spectral properties for the precondidetail. Moreover, when0 ≤ω≤ 1 and Q=1/γI , tioned matrix are analyzed. Numerical experiments show that if the mass matrix from the distributed control problems is not easy to inverse and the regularization parameter β is very small, the GAL-SOR iteration method can work well.
文摘For stabilized saddle-point problems, we apply the two iteration parameters idea for regularized Hermitian and skew-Hermitian splitting (RHSS) method and establish accelerated RHSS (ARHSS) iteration method. Theoretical analysis shows that the ARHSS method converges unconditionally to the unique solution of the saddle point problem. Finally, we use a numerical example to confirm the effectiveness of the method.
文摘In this paper, a least-squares mixed finite element method for the solution of the primal saddle-point problem is developed. It is proved that the approximate problem is consistent ellipticity in the conforming finite element spaces with only the discrete BB-condition needed for a smaller auxiliary problem. The abstract error estimate is derived. [ABSTRACT FROM AUTHOR]
文摘An effective algorithm for solving large saddle-point linear systems, presented by Krukier et al., is applied to the constrained optimization problems. This method is a modification of skew-Hermitian triangular splitting iteration methods. We consider the saddle-point linear systems with singular or semidefinite （1, 1） blocks. Moreover, this method is applied to precondition the GMRES. Numerical results have confirmed the effectiveness of the method and showed that the new method can produce high-quality preconditioners for the Krylov subspace methods for solving large sparse saddle-point linear systems.
基金supported by the 10 plus 10 project of Tongji University(No.4260141304/004/010).
文摘.In this paper,an augmented Lagrangian Uzawa iterative method is developed and analyzed for solving a class of double saddle-point systems with semidefinite(2,2)block.Convergence of the iterativemethod is proved under the assumption that the double saddle-point problem exists a unique solution.An application of the iterative method to the double saddle-point systems arising from the distributed Lagrange multiplier/fictitious domain(DLM/FD)finite element method for solving elliptic interface problems is also presented,in which the existence and uniqueness of the double saddle-point system is guaranteed by the analysis of the DLM/FD finite element method.Numerical experiments are conducted to validate the theoretical results and to study the performance of the proposed iterative method.
基金supported by the National Natural Science Foundation of China (Grant No. 10805029)the Zhejiang Natural Science Foundation,China (Grant No. R6090717)the K.C. Wong Magna Foundation of Ningbo University,China
文摘We study the propagator for an electron moving in a two-dimensional （2D） quadratic saddle-point potential, in the presence of a perpendicular uniform magnetic field. A closed-form expression for the propagator is derived using the Feynmann path integrals.
