In this paper,we are presenting a proposal for new modified algorithms for RRGMRES and AGMRES.It is known that RRGMRES and AGMRES are viable methods for solving linear discrete ill-posed problems.In this paper we have...In this paper,we are presenting a proposal for new modified algorithms for RRGMRES and AGMRES.It is known that RRGMRES and AGMRES are viable methods for solving linear discrete ill-posed problems.In this paper we have focused on the residual norm and have come-up with two improvements where successive updates and the stabilization of decreases for the residual norm improve performance respectively.Our numerical experiments confirm that our improved algorithms are effective for linear discrete ill-posed problems.展开更多
A discrete differential evolution algorithm combined with the branch and bound method is developed to solve the integer linear bilevel programming problems, in which both upper level and lower level variables are forc...A discrete differential evolution algorithm combined with the branch and bound method is developed to solve the integer linear bilevel programming problems, in which both upper level and lower level variables are forced to be integer. An integer coding for upper level variables is adopted, and then a discrete differential evolution algorithm with an improved feasibility-based comparison is developed to directly explore the integer solution at the upper level. For a given upper level integer variable, the lower level integer programming problem is solved by the existing branch and bound algorithm to obtain the optimal integer solution at the lower level. In the same framework of the algorithm, two other constraint handling methods, i.e. the penalty function method and the feasibility-based comparison method are also tested. The experimental results demonstrate that the discrete differential evolution algorithm with different constraint handling methods is effective in finding the global optimal integer solutions, but the improved constraint handling method performs better than two compared constraint handling methods.展开更多
Let be a linear, closed, and densely defined unbounded operator, where X and Y are Hilbert spaces. Assume that A is not boundedly invertible. Suppose the equation Au=f is solvable, and instead of knowing exactly f onl...Let be a linear, closed, and densely defined unbounded operator, where X and Y are Hilbert spaces. Assume that A is not boundedly invertible. Suppose the equation Au=f is solvable, and instead of knowing exactly f only know its approximation satisfies the condition: In this paper, we are interested a regularization method to solve the approximation solution of this equation. This approximation is a unique global minimizer of the functional , for any , defined as follows: . We also study the stability of this method when the regularization parameter is selected a priori and a posteriori. At the same time, we give an application of this method to the weak derivative operator equation in Hilbert space.展开更多
The possibly most popular regularization method for solving the least squares problem rain ‖Ax - b‖2 with a highly ill-conditioned or rank deficient coefficient matrix A is the x Tikhonov regularization method. In ...The possibly most popular regularization method for solving the least squares problem rain ‖Ax - b‖2 with a highly ill-conditioned or rank deficient coefficient matrix A is the x Tikhonov regularization method. In this paper we present the explicit expressions of the normwise, mixed and componentwise condition numbers for the Tikhonov regularization when A has linear structures. The structured condition numbers in the special cases of nonlinear structure i.e. Vandermonde and Cauchy matrices are also considered. Some comparisons between structured condition numbers and unstructured condition numbers are made by numerical experiments. In addition, we also derive the normwise, mixed and componentwise condition numbers for the Tikhonov regularization when the coefficient matrix, regularization matrix and right-hand side vector are all perturbed, which generalize the results obtained by Chu et al. [Numer. Linear Algebra Appl., 18 （2011）, 87-103].展开更多
Iterative regularization multigrid methods have been successful applied to signal/image deblurring problems.When zero-Dirichlet boundary conditions are imposed the deblurring matrix has a Toeplitz structure and it is ...Iterative regularization multigrid methods have been successful applied to signal/image deblurring problems.When zero-Dirichlet boundary conditions are imposed the deblurring matrix has a Toeplitz structure and it is potentially full.A crucial task of a multilevel strategy is to preserve the Toeplitz structure at the coarse levels which can be exploited to obtain fast computations.The smoother has to be an iterative regularization method.The grid transfer operator should preserve the regularization property of the smoother.This paper improves the iterative multigrid method proposed in[11]introducing a wavelet soft-thresholding denoising post-smoother.Such postsmoother avoids the noise amplification that is the cause of the semi-convergence of iterative regularization methods and reduces ringing effects.The resulting iterative multigrid regularization method stabilizes the iterations so that and imprecise(over)estimate of the stopping iteration does not have a deleterious effect on the computed solution.Numerical examples of signal and image deblurring problems confirm the effectiveness of the proposed method.展开更多
The maximum principle is a basic qualitative property of the solution of second-order elliptic boundary value problems.The preservation of the qualitative characteristics,such as the maximum principle,in discrete mode...The maximum principle is a basic qualitative property of the solution of second-order elliptic boundary value problems.The preservation of the qualitative characteristics,such as the maximum principle,in discrete model is one of the key requirements.It is well known that standard linear finite element solution does not satisfy maximum principle on general triangular meshes in 2D.In this paper we consider how to enforce discrete maximum principle for linear finite element solutions for the linear second-order self-adjoint elliptic equation.First approach is based on repair technique,which is a posteriori correction of the discrete solution.Second method is based on constrained optimization.Numerical tests that include anisotropic cases demonstrate how our method works for problems for which the standard finite element methods produce numerical solutions that violate the discrete maximum principle.展开更多
This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of L1-Galerkin finite element methods.The analysis of L1 methods for time-fractional nonlinear problems is li...This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of L1-Galerkin finite element methods.The analysis of L1 methods for time-fractional nonlinear problems is limited mainly due to the lack of a fundamental Gronwall type inequality.In this paper,we establish such a fundamental inequality for the L1 approximation to the Caputo fractional derivative.In terms of the Gronwall type inequality,we provide optimal error estimates of several fully discrete linearized Galerkin finite element methods for nonlinear problems.The theoretical results are illustrated by applying our proposed methods to the time fractional nonlinear Huxley equation and time fractional Fisher equation.展开更多
New conditions are derived for the l2-stability of time-varying linear and nonlinear discrete-time multiple-input multipleoutput （MIMO） systems, having a linear time time-invariant block with the transfer function F...New conditions are derived for the l2-stability of time-varying linear and nonlinear discrete-time multiple-input multipleoutput （MIMO） systems, having a linear time time-invariant block with the transfer function F（z）, in negative feedback with a matrix of periodic/aperiodic gains A（k）, k = 0,1, 2,... and a vector of certain classes of non-monotone/monotone nonlinearities φp（-）, without restrictions on their slopes and also not requiring path-independence of their line integrals. The stability conditions, which are derived in the frequency domain, have the following features： i） They involve the positive definiteness of the real part （as evaluated on ｜z｜ = 1） of the product of Г （z） and a matrix multiplier function of z. ii） For periodic A（k）, one class of multiplier functions can be chosen so as to impose no constraint on the rate of variations A（k）, but for aperiodic A（k）, which allows a more general multiplier function, constraints are imposed on certain global averages of the generalized eigenvalues of （A（k ＋ 1）,A（k））, k = 1, 2 iii） They are distinct from and less restrictive than recent results in the literature.展开更多
Unbounded operators can transform arbitrarily small vectors into arbitrarily large vectors—a phenomenon known as instability. Stabilization methods strive to approximate a value of an unbounded operator by applying a...Unbounded operators can transform arbitrarily small vectors into arbitrarily large vectors—a phenomenon known as instability. Stabilization methods strive to approximate a value of an unbounded operator by applying a family of bounded operators to rough approximate data that do not necessarily lie within the domain of unbounded operator. In this paper we shall be concerned with the stable method of computing values of unbounded operators having perturbations and the stability is established for this method.展开更多
文摘In this paper,we are presenting a proposal for new modified algorithms for RRGMRES and AGMRES.It is known that RRGMRES and AGMRES are viable methods for solving linear discrete ill-posed problems.In this paper we have focused on the residual norm and have come-up with two improvements where successive updates and the stabilization of decreases for the residual norm improve performance respectively.Our numerical experiments confirm that our improved algorithms are effective for linear discrete ill-posed problems.
基金supported by the Natural Science Basic Research Plan in Shaanxi Province of China(2013JM1022)the Fundamental Research Funds for the Central Universities(K50511700004)
文摘A discrete differential evolution algorithm combined with the branch and bound method is developed to solve the integer linear bilevel programming problems, in which both upper level and lower level variables are forced to be integer. An integer coding for upper level variables is adopted, and then a discrete differential evolution algorithm with an improved feasibility-based comparison is developed to directly explore the integer solution at the upper level. For a given upper level integer variable, the lower level integer programming problem is solved by the existing branch and bound algorithm to obtain the optimal integer solution at the lower level. In the same framework of the algorithm, two other constraint handling methods, i.e. the penalty function method and the feasibility-based comparison method are also tested. The experimental results demonstrate that the discrete differential evolution algorithm with different constraint handling methods is effective in finding the global optimal integer solutions, but the improved constraint handling method performs better than two compared constraint handling methods.
文摘Let be a linear, closed, and densely defined unbounded operator, where X and Y are Hilbert spaces. Assume that A is not boundedly invertible. Suppose the equation Au=f is solvable, and instead of knowing exactly f only know its approximation satisfies the condition: In this paper, we are interested a regularization method to solve the approximation solution of this equation. This approximation is a unique global minimizer of the functional , for any , defined as follows: . We also study the stability of this method when the regularization parameter is selected a priori and a posteriori. At the same time, we give an application of this method to the weak derivative operator equation in Hilbert space.
基金The authors would like to thank the anonymous referees for their valu- able suggestions and comments. This work was supported by the National Natural Science Foundation of China （No. 11571004 and No. 11171371）.
文摘The possibly most popular regularization method for solving the least squares problem rain ‖Ax - b‖2 with a highly ill-conditioned or rank deficient coefficient matrix A is the x Tikhonov regularization method. In this paper we present the explicit expressions of the normwise, mixed and componentwise condition numbers for the Tikhonov regularization when A has linear structures. The structured condition numbers in the special cases of nonlinear structure i.e. Vandermonde and Cauchy matrices are also considered. Some comparisons between structured condition numbers and unstructured condition numbers are made by numerical experiments. In addition, we also derive the normwise, mixed and componentwise condition numbers for the Tikhonov regularization when the coefficient matrix, regularization matrix and right-hand side vector are all perturbed, which generalize the results obtained by Chu et al. [Numer. Linear Algebra Appl., 18 （2011）, 87-103].
文摘Iterative regularization multigrid methods have been successful applied to signal/image deblurring problems.When zero-Dirichlet boundary conditions are imposed the deblurring matrix has a Toeplitz structure and it is potentially full.A crucial task of a multilevel strategy is to preserve the Toeplitz structure at the coarse levels which can be exploited to obtain fast computations.The smoother has to be an iterative regularization method.The grid transfer operator should preserve the regularization property of the smoother.This paper improves the iterative multigrid method proposed in[11]introducing a wavelet soft-thresholding denoising post-smoother.Such postsmoother avoids the noise amplification that is the cause of the semi-convergence of iterative regularization methods and reduces ringing effects.The resulting iterative multigrid regularization method stabilizes the iterations so that and imprecise(over)estimate of the stopping iteration does not have a deleterious effect on the computed solution.Numerical examples of signal and image deblurring problems confirm the effectiveness of the proposed method.
基金the National Nuclear Security Administration of the U.S.Department of Energy at Los Alamos National Laboratory under Contract No.DE-AC52-06NA25396the DOE Office of Science Advanced Scientific Computing Research(ASCR)Program in Applied Mathematics Research.The first author has been supported in part by the Czech Ministry of Education projects MSM 6840770022 and LC06052(Necas Center for Mathematical Modeling).
文摘The maximum principle is a basic qualitative property of the solution of second-order elliptic boundary value problems.The preservation of the qualitative characteristics,such as the maximum principle,in discrete model is one of the key requirements.It is well known that standard linear finite element solution does not satisfy maximum principle on general triangular meshes in 2D.In this paper we consider how to enforce discrete maximum principle for linear finite element solutions for the linear second-order self-adjoint elliptic equation.First approach is based on repair technique,which is a posteriori correction of the discrete solution.Second method is based on constrained optimization.Numerical tests that include anisotropic cases demonstrate how our method works for problems for which the standard finite element methods produce numerical solutions that violate the discrete maximum principle.
基金This work is supported by NSFC(Grant Nos.11771035,11771162,11571128,61473126,91430216,91530204,11372354 and U1530401),a grant from the RGC of HK 11300517,China(Project No.CityU 11302915),China Postdoctoral Science Foundation under grant No.2016M602273,a grant DRA2015518 from 333 High-level Personal Training Project of Jiangsu Province,and the USA National Science Foundation grant DMS-1315259the USA Air Force Office of Scientific Research grant FA9550-15-1-0001.Jiwei Zhang also thanks the hospitality of Hong Kong City University during the period of his visiting.
文摘This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of L1-Galerkin finite element methods.The analysis of L1 methods for time-fractional nonlinear problems is limited mainly due to the lack of a fundamental Gronwall type inequality.In this paper,we establish such a fundamental inequality for the L1 approximation to the Caputo fractional derivative.In terms of the Gronwall type inequality,we provide optimal error estimates of several fully discrete linearized Galerkin finite element methods for nonlinear problems.The theoretical results are illustrated by applying our proposed methods to the time fractional nonlinear Huxley equation and time fractional Fisher equation.
文摘New conditions are derived for the l2-stability of time-varying linear and nonlinear discrete-time multiple-input multipleoutput （MIMO） systems, having a linear time time-invariant block with the transfer function F（z）, in negative feedback with a matrix of periodic/aperiodic gains A（k）, k = 0,1, 2,... and a vector of certain classes of non-monotone/monotone nonlinearities φp（-）, without restrictions on their slopes and also not requiring path-independence of their line integrals. The stability conditions, which are derived in the frequency domain, have the following features： i） They involve the positive definiteness of the real part （as evaluated on ｜z｜ = 1） of the product of Г （z） and a matrix multiplier function of z. ii） For periodic A（k）, one class of multiplier functions can be chosen so as to impose no constraint on the rate of variations A（k）, but for aperiodic A（k）, which allows a more general multiplier function, constraints are imposed on certain global averages of the generalized eigenvalues of （A（k ＋ 1）,A（k））, k = 1, 2 iii） They are distinct from and less restrictive than recent results in the literature.
文摘Unbounded operators can transform arbitrarily small vectors into arbitrarily large vectors—a phenomenon known as instability. Stabilization methods strive to approximate a value of an unbounded operator by applying a family of bounded operators to rough approximate data that do not necessarily lie within the domain of unbounded operator. In this paper we shall be concerned with the stable method of computing values of unbounded operators having perturbations and the stability is established for this method.
