In this paper, we consider the problem for determining an unknown source in the heat equation. The Tikhonov regularization method in Hilbert scales is presented to deal with ill-posedness of the problem and error esti...In this paper, we consider the problem for determining an unknown source in the heat equation. The Tikhonov regularization method in Hilbert scales is presented to deal with ill-posedness of the problem and error estimates are obtained with a posteriori choice rule to find the regularization parameter. The smoothness parameter and the a priori bound of exact solution are not needed for the choice rule. Numerical tests show that the proposed method is effective and stable.展开更多
The inverse heat conduction problem （IHCP） is a severely ill-posed problem in the sense that the solution （ if it exists） does not depend continuously on the data. But now the results on inverse heat conduction pr...The inverse heat conduction problem （IHCP） is a severely ill-posed problem in the sense that the solution （ if it exists） does not depend continuously on the data. But now the results on inverse heat conduction problem are mainly devoted to the standard inverse heat conduction problem. Some optimal error bounds in a Sobolev space of regularized approximation solutions for a sideways parabolic equation, i. e. , a non-standard inverse heat conduction problem with convection term which appears in some applied subject are given.展开更多
This paper deals with the inverse time problem for an axisymmetric heat equation. The problem is ill-posed. A modified Tikhonov regularization method is applied to formulate regularized solution which is stably conver...This paper deals with the inverse time problem for an axisymmetric heat equation. The problem is ill-posed. A modified Tikhonov regularization method is applied to formulate regularized solution which is stably convergent to the exact one. estimate between the approximate solution and exact technical inequality and improving a priori smoothness Meanwhile, a logarithmic-HSlder type error solution is obtained by introducing a rather assumption.展开更多
The regularization of ill-posed problems has become a useful tool in studying initial value problems that do not adhere to certain desired properties such as continuous dependence of solutions on initial data.Because ...The regularization of ill-posed problems has become a useful tool in studying initial value problems that do not adhere to certain desired properties such as continuous dependence of solutions on initial data.Because direct computation of the solution becomes difficult in this situation,many authors have alternatively approximated the solution by the solution of a closely defined well posed problem.In this paper,we demonstrate this process of regularization for nonautonomous ill-posed problems including the backward heat equation with a time-dependent diffusion coefficient.In the process,we provide two different approximate well posed models and numerically compare convergence rates of their solutions to a known solution of the original ill-posed problem.展开更多
For the backward diffusion equation,a stable discrete energy regularization algorithm is proposed.Existence and uniqueness of the numerical solution are given.Moreover,the error between the solution of the given backw...For the backward diffusion equation,a stable discrete energy regularization algorithm is proposed.Existence and uniqueness of the numerical solution are given.Moreover,the error between the solution of the given backward diffusion equation and the numerical solution via the regularization method can be estimated.Some numerical experiments illustrate the efficiency of the method,and its application in image deblurring.展开更多
In this paper,we consider an inverse time-dependent source problem of heat conduction equation.Firstly,the ill-posedness and conditional stability of this inverse source problem is analyzed.Then,a finite difference in...In this paper,we consider an inverse time-dependent source problem of heat conduction equation.Firstly,the ill-posedness and conditional stability of this inverse source problem is analyzed.Then,a finite difference inversion method is proposed for reconstructing the time-dependent source from a nonlocal measurement.The existence and uniqueness of the finite difference inverse solutions are rigorously analyzed,and the convergence is proved.Combined with the mollification method,the proposed finite difference inversion method can obtain more stable reconstructions from the nonlocal data with noise.Finally,numerical examples are given to illustrate the efficiency and convergence of the proposed finite difference inversion method.展开更多
文摘In this paper, we consider the problem for determining an unknown source in the heat equation. The Tikhonov regularization method in Hilbert scales is presented to deal with ill-posedness of the problem and error estimates are obtained with a posteriori choice rule to find the regularization parameter. The smoothness parameter and the a priori bound of exact solution are not needed for the choice rule. Numerical tests show that the proposed method is effective and stable.
文摘The inverse heat conduction problem （IHCP） is a severely ill-posed problem in the sense that the solution （ if it exists） does not depend continuously on the data. But now the results on inverse heat conduction problem are mainly devoted to the standard inverse heat conduction problem. Some optimal error bounds in a Sobolev space of regularized approximation solutions for a sideways parabolic equation, i. e. , a non-standard inverse heat conduction problem with convection term which appears in some applied subject are given.
基金Supported by National Natural Science Foundation of China (Grant No.10671085)Fundamental Research Fund for Natural Science of Education Department of He'nan Province of China (Grant No.2009Bl10007)Hight-level Personnel fund of He'nan University of Technology (Grant No.2007BS028)
文摘This paper deals with the inverse time problem for an axisymmetric heat equation. The problem is ill-posed. A modified Tikhonov regularization method is applied to formulate regularized solution which is stably convergent to the exact one. estimate between the approximate solution and exact technical inequality and improving a priori smoothness Meanwhile, a logarithmic-HSlder type error solution is obtained by introducing a rather assumption.
文摘The regularization of ill-posed problems has become a useful tool in studying initial value problems that do not adhere to certain desired properties such as continuous dependence of solutions on initial data.Because direct computation of the solution becomes difficult in this situation,many authors have alternatively approximated the solution by the solution of a closely defined well posed problem.In this paper,we demonstrate this process of regularization for nonautonomous ill-posed problems including the backward heat equation with a time-dependent diffusion coefficient.In the process,we provide two different approximate well posed models and numerically compare convergence rates of their solutions to a known solution of the original ill-posed problem.
基金National Natural Science Foundation of China(No.10471073)。
文摘For the backward diffusion equation,a stable discrete energy regularization algorithm is proposed.Existence and uniqueness of the numerical solution are given.Moreover,the error between the solution of the given backward diffusion equation and the numerical solution via the regularization method can be estimated.Some numerical experiments illustrate the efficiency of the method,and its application in image deblurring.
基金supported by National Natural Science Foundation of China(11561003,11661004,11761007)Natural Science Foundation of Jiangxi Province(20161BAB201034)Foundation of Academic and Technical Leaders Program for Major Subjects in Jiangxi Province(20172BCB22019)。
文摘In this paper,we consider an inverse time-dependent source problem of heat conduction equation.Firstly,the ill-posedness and conditional stability of this inverse source problem is analyzed.Then,a finite difference inversion method is proposed for reconstructing the time-dependent source from a nonlocal measurement.The existence and uniqueness of the finite difference inverse solutions are rigorously analyzed,and the convergence is proved.Combined with the mollification method,the proposed finite difference inversion method can obtain more stable reconstructions from the nonlocal data with noise.Finally,numerical examples are given to illustrate the efficiency and convergence of the proposed finite difference inversion method.
