In this paper, the Tikhonov regularization method was used to solve the nondegenerate compact hnear operator equation, which is a well-known ill-posed problem. Apart from the usual error level, the noise data were sup...In this paper, the Tikhonov regularization method was used to solve the nondegenerate compact hnear operator equation, which is a well-known ill-posed problem. Apart from the usual error level, the noise data were supposed to satisfy some additional monotonic condition. Moreover, with the assumption that the singular values of operator have power form, the improved convergence rates of the regularized solution were worked out.展开更多
In this paper,we consider a Cauchy problem of the time fractional diffusion equation(TFDE)in x∈[0,L].This problem is ubiquitous in science and engineering applications.The illposedness of the Cauchy problem is explai...In this paper,we consider a Cauchy problem of the time fractional diffusion equation(TFDE)in x∈[0,L].This problem is ubiquitous in science and engineering applications.The illposedness of the Cauchy problem is explained by its solution in frequency domain.Furthermore,the problem is formulated into a minimization problem with a modified Tikhonov regularization method.The gradient of the regularization functional based on an adjoint problem is deduced and the standard conjugate gradient method is presented for solving the minimization problem.The error estimates for the regularized solutions are obtained under Hp norm priori bound assumptions.Finally,numerical examples illustrate the effectiveness of the proposed method.展开更多
Interaction between mesoscale perturbations of sea surface temperature(SSTmeso)and wind stress(WSmeso)has great influences on the ocean upwelling system and turbulent mixing in the atmospheric boundary layer.Using dai...Interaction between mesoscale perturbations of sea surface temperature(SSTmeso)and wind stress(WSmeso)has great influences on the ocean upwelling system and turbulent mixing in the atmospheric boundary layer.Using daily Quik-SCAT wind speed data and AMSR-E SST data,SSTmeso and WSmeso fields in the western coast of South America are extracted by using a locally weighted regression method(LOESS).The spatial patterns of SSTmeso and WSmeso indicate strong mesoscale SST-wind stress coupling in the region.The coupling coefficient between SSTmeso and WSmeso is about 0.0095 N/(m^2·℃)in winter and 0.0082 N/(m^2·℃)in summer.Based on mesoscale coupling relationships,the mesoscale perturbations of wind stress divergence(Div(WSmeso))and curl(Curl(WSmeso))can be obtained from the SST gradient perturbations,which can be further used to derive wind stress vector perturbations using the Tikhonov regularization method.The computational examples are presented in the western coast of South America and the patterns of the reconstructed WS meso are highly consistent with SSTmeso,but the amplitude can be underestimated significantly.By matching the spatially averaged maximum standard deviations of reconstructed WSmeso magnitude and observations,a reasonable magnitude of WSmeso can be obtained when a rescaling factor of 2.2 is used.As current ocean models forced by prescribed wind cannot adequately capture the mesoscale wind stress response,the empirical wind stress perturbation model developed in this study can be used to take into account the feedback effects of the mesoscale wind stress-SST coupling in ocean modeling.Further applications are discussed for taking into account the feedback effects of the mesoscale coupling in largescale climate models and the uncoupled ocean models.展开更多
Newton type methods are one kind of the efficient methods to solve nonlinear ill-posed problems, which have attracted extensive attention. However, computational cost of Newton type methods is high because practical p...Newton type methods are one kind of the efficient methods to solve nonlinear ill-posed problems, which have attracted extensive attention. However, computational cost of Newton type methods is high because practical problems are complicated. We propose a mixed Newton-Tikhonov method, i.e., one step Newton-Tikhonov method with several other steps of simplified Newton-Tikhonov method. Convergence and stability of this method are proved under some conditions. Numerical experiments show that the proposed method has obvious advantages over the classical Newton method in terms of computational costs.展开更多
A Cauchy problem for the elliptic equation with variable coefficients is considered. This problem is severely ill-posed. Then, we need use the regularization techniques to overcome its ill-posedness and get a stable n...A Cauchy problem for the elliptic equation with variable coefficients is considered. This problem is severely ill-posed. Then, we need use the regularization techniques to overcome its ill-posedness and get a stable numerical solution. In this paper, we use a modified Tikhonov regularization method to treat it. Under the a-priori bound assumptions for the exact solution, the convergence estimates of this method are established. Numerical results show that our method works well.展开更多
文摘In this paper, the Tikhonov regularization method was used to solve the nondegenerate compact hnear operator equation, which is a well-known ill-posed problem. Apart from the usual error level, the noise data were supposed to satisfy some additional monotonic condition. Moreover, with the assumption that the singular values of operator have power form, the improved convergence rates of the regularized solution were worked out.
基金Supported by the National Natural Science Foundation of China(Grant No.11471253 and No.11571311)
文摘In this paper,we consider a Cauchy problem of the time fractional diffusion equation(TFDE)in x∈[0,L].This problem is ubiquitous in science and engineering applications.The illposedness of the Cauchy problem is explained by its solution in frequency domain.Furthermore,the problem is formulated into a minimization problem with a modified Tikhonov regularization method.The gradient of the regularization functional based on an adjoint problem is deduced and the standard conjugate gradient method is presented for solving the minimization problem.The error estimates for the regularized solutions are obtained under Hp norm priori bound assumptions.Finally,numerical examples illustrate the effectiveness of the proposed method.
基金Supported by the National Key Research and Development Program of China(No.2017YFC1404102(2017YFC1404100))the National Program on Global Change and Air-sea Interaction(No.GASI-IPOVAI-06)+3 种基金the National Natural Science Foundation of China(Nos.41490644(41490640),41690122(41690120))the Chinese Academy of Sciences Strategic Priority Project(No.XDA19060102)the NSFC Shandong Joint Fund for Marine Science Research Centers(No.U1406402)the Taishan Scholarship and the Recruitment Program of Global Experts。
文摘Interaction between mesoscale perturbations of sea surface temperature(SSTmeso)and wind stress(WSmeso)has great influences on the ocean upwelling system and turbulent mixing in the atmospheric boundary layer.Using daily Quik-SCAT wind speed data and AMSR-E SST data,SSTmeso and WSmeso fields in the western coast of South America are extracted by using a locally weighted regression method(LOESS).The spatial patterns of SSTmeso and WSmeso indicate strong mesoscale SST-wind stress coupling in the region.The coupling coefficient between SSTmeso and WSmeso is about 0.0095 N/(m^2·℃)in winter and 0.0082 N/(m^2·℃)in summer.Based on mesoscale coupling relationships,the mesoscale perturbations of wind stress divergence(Div(WSmeso))and curl(Curl(WSmeso))can be obtained from the SST gradient perturbations,which can be further used to derive wind stress vector perturbations using the Tikhonov regularization method.The computational examples are presented in the western coast of South America and the patterns of the reconstructed WS meso are highly consistent with SSTmeso,but the amplitude can be underestimated significantly.By matching the spatially averaged maximum standard deviations of reconstructed WSmeso magnitude and observations,a reasonable magnitude of WSmeso can be obtained when a rescaling factor of 2.2 is used.As current ocean models forced by prescribed wind cannot adequately capture the mesoscale wind stress response,the empirical wind stress perturbation model developed in this study can be used to take into account the feedback effects of the mesoscale wind stress-SST coupling in ocean modeling.Further applications are discussed for taking into account the feedback effects of the mesoscale coupling in largescale climate models and the uncoupled ocean models.
基金supported by the Key Disciplines of Shanghai Municipality (Operations Research & Cybernetics, No. S30104)Shanghai Leading Academic Discipline Project (No. J50101)
文摘Newton type methods are one kind of the efficient methods to solve nonlinear ill-posed problems, which have attracted extensive attention. However, computational cost of Newton type methods is high because practical problems are complicated. We propose a mixed Newton-Tikhonov method, i.e., one step Newton-Tikhonov method with several other steps of simplified Newton-Tikhonov method. Convergence and stability of this method are proved under some conditions. Numerical experiments show that the proposed method has obvious advantages over the classical Newton method in terms of computational costs.
文摘A Cauchy problem for the elliptic equation with variable coefficients is considered. This problem is severely ill-posed. Then, we need use the regularization techniques to overcome its ill-posedness and get a stable numerical solution. In this paper, we use a modified Tikhonov regularization method to treat it. Under the a-priori bound assumptions for the exact solution, the convergence estimates of this method are established. Numerical results show that our method works well.
