In this paper, we propose a tailored-finite-point method for the numerical simulation of the Helmholtz equation with high wave numbers in heterogeneous medium. Our finite point method has been tailored to some particu...In this paper, we propose a tailored-finite-point method for the numerical simulation of the Helmholtz equation with high wave numbers in heterogeneous medium. Our finite point method has been tailored to some particular properties of the problem, which allows us to obtain approximate solutions with the same behaviors as that of the exact solution very naturally. Especially, when the coefficients are piecewise constant, we can get the exact solution with only one point in each subdomain. Our finite-point method has uniformly convergent rate with respect to wave number k in L^2-norm.展开更多
In this paper, we propose a tailored finite cell method for the computation of two- dimensional Helmholtz equation in layered heterogeneous medium. The idea underlying the method is to construct a numerical scheme bas...In this paper, we propose a tailored finite cell method for the computation of two- dimensional Helmholtz equation in layered heterogeneous medium. The idea underlying the method is to construct a numerical scheme based on a local approximation of the solution to Helmholtz equation. This provides a computational tool of achieving high accuracy with coarse mesh even for large wave number （high frequency）. The stability analysis and error estimates of this method are also proved. We present several numerical results to show its efficiency and accuracy.展开更多
In scientific applications from plasma to chemical kinetics, a wide range of temporal scales can present in a system of differential equations. A major difficulty is encountered due to the stiffness of the system and ...In scientific applications from plasma to chemical kinetics, a wide range of temporal scales can present in a system of differential equations. A major difficulty is encountered due to the stiffness of the system and it is required to develop fast numerical schemes that are able to access previously unattainable parameter regimes. In this work, we consider an initial-final value problem for a multi-scale singularly perturbed system of linear ordi- nary differential equations with discontinuous coefficients. We construct a tailored finite point method, which yields approximate solutions that converge in the maximum norm, uniformly with respect to the singular perturbation parameters, to the exact solution. A parameter-uniform error estimate in the maximum norm is also proved. The results of numerical experiments, that support the theoretical results, are reported.展开更多
In this paper,we propose using the tailored finite point method(TFPM)to solve the resulting parabolic or elliptic equations when minimizing the Huber regularization based image super-resolution model using the augment...In this paper,we propose using the tailored finite point method(TFPM)to solve the resulting parabolic or elliptic equations when minimizing the Huber regularization based image super-resolution model using the augmented Lagrangian method(ALM).The Hu-ber regularization based image super-resolution model can ameliorate the staircase for restored images.TFPM employs the method of weighted residuals with collocation tech-nique,which helps get more accurate approximate solutions to the equations and reserve more details in restored images.We compare the new schemes with the Marquina-Osher model,the image super-resolution convolutional neural network(SRCNN)and the classical interpolation methods:bilinear interpolation,nearest-neighbor interpolation and bicubic interpolation.Numerical experiments are presented to demonstrate that with the new schemes the quality of the super-resolution images has been improved.Besides these,the existence of the minimizer of the Huber regularization based image super-resolution model and the convergence of the proposed algorithm are also established in this paper.展开更多
In this paper we propose a development of the finite difference method,called the tailored finite point method,for solving steady magnetohydrodynamic(MHD)duct flow problems with a high Hartmann number.When the Hartman...In this paper we propose a development of the finite difference method,called the tailored finite point method,for solving steady magnetohydrodynamic(MHD)duct flow problems with a high Hartmann number.When the Hartmann number is large,the MHD duct flow is convection-dominated and thus its solution may exhibit localized phenomena such as the boundary layer.Most conventional numerical methods can not efficiently solve the layer problem because they are lacking in either stability or accuracy.However,the proposed tailored finite point method is capable of resolving high gradients near the layer regions without refining the mesh.Firstly,we devise the tailored finite point method for the scalar inhomogeneous convectiondiffusion problem,and then extend it to the MHD duct flow which consists of a coupled system of convection-diffusion equations.For each interior grid point of a given rectangular mesh,we construct a finite-point difference operator at that point with some nearby grid points,where the coefficients of the difference operator are tailored to some particular properties of the problem.Numerical examples are provided to show the high performance of the proposed method.展开更多
This paper presents two uniformly convergent numerical schemes for the two dimensional steady state discrete ordinates transport equation in the diffusive regime,which is valid up to the boundary and interface layers....This paper presents two uniformly convergent numerical schemes for the two dimensional steady state discrete ordinates transport equation in the diffusive regime,which is valid up to the boundary and interface layers.A five-point nodecentered and a four-point cell-centered tailored finite point schemes(TFPS)are introduced.The schemes first approximate the scattering coefficients and sources by piecewise constant functions and then use special solutions to the constant coefficient equation as local basis functions to formulate a discrete linear system.Numerically,both methods can not only capture the diffusion limit,but also exhibit uniform convergence in the diffusive regime,even with boundary layers.Numerical results show that the five-point scheme has first-order accuracy and the four-point scheme has second-order accuracy,uniformly with respect to the mean free path.Therefore a relatively coarse grid can be used to capture the two dimensional boundary and interface layers.展开更多
It is of great interest to solve the inverse problem of stationary radiative transport equation(RTE)in optical tomography.The standard way is to formulate the inverse problem into an optimization problem,but the bottl...It is of great interest to solve the inverse problem of stationary radiative transport equation(RTE)in optical tomography.The standard way is to formulate the inverse problem into an optimization problem,but the bottleneck is that one has to solve the forward problem repeatedly,which is time-consuming.Due to the optical property of biological tissue,in real applications,optical thin and thick regions coexist and are adjacent to each other,and the geometry can be complex.To use coarse meshes and save the computational cost,the forward solver has to be asymptotic preserving across the interface(APAL).In this paper,we propose an offline/online solver for RTE.The cost at the offline stage is comparable to classical methods,while the cost at the online stage is much lower.Two cases are considered.One is to solve the RTE with fixed scattering and absorption cross sections while the boundary conditions vary;the other is when cross sections vary in a small domain and the boundary conditions change many times.The solver can be decomposed into offline/online stages in these two cases.One only needs to calculate the offline stage once and update the online stage when the parameters vary.Our proposed solver is much cheaper when one needs to solve RTE with multiple right-hand sides or when the cross sections vary in a small domain,thus can accelerate the speed of solving inverse RTE problems.We illustrate the online/offline decomposition based on the Tailored Finite Point Method(TFPM),which is APAL on general quadrilateral meshes.展开更多
基金the NSFC Projects No.10471073No.10676017the National Basic Research Program of China under the grant 2005CB321701
文摘In this paper, we propose a tailored-finite-point method for the numerical simulation of the Helmholtz equation with high wave numbers in heterogeneous medium. Our finite point method has been tailored to some particular properties of the problem, which allows us to obtain approximate solutions with the same behaviors as that of the exact solution very naturally. Especially, when the coefficients are piecewise constant, we can get the exact solution with only one point in each subdomain. Our finite-point method has uniformly convergent rate with respect to wave number k in L^2-norm.
文摘In this paper, we propose a tailored finite cell method for the computation of two- dimensional Helmholtz equation in layered heterogeneous medium. The idea underlying the method is to construct a numerical scheme based on a local approximation of the solution to Helmholtz equation. This provides a computational tool of achieving high accuracy with coarse mesh even for large wave number （high frequency）. The stability analysis and error estimates of this method are also proved. We present several numerical results to show its efficiency and accuracy.
基金Acknowledgments. H. Han was supported by the NSFC Project No. 10971116. M. Tang is supported by Natural Science Foundation of Shanghai under Grant No. 12ZR1445400.
文摘In scientific applications from plasma to chemical kinetics, a wide range of temporal scales can present in a system of differential equations. A major difficulty is encountered due to the stiffness of the system and it is required to develop fast numerical schemes that are able to access previously unattainable parameter regimes. In this work, we consider an initial-final value problem for a multi-scale singularly perturbed system of linear ordi- nary differential equations with discontinuous coefficients. We construct a tailored finite point method, which yields approximate solutions that converge in the maximum norm, uniformly with respect to the singular perturbation parameters, to the exact solution. A parameter-uniform error estimate in the maximum norm is also proved. The results of numerical experiments, that support the theoretical results, are reported.
基金partially supported by the NSFC Project Nos.12001529,12025104,11871298,81930119.
文摘In this paper,we propose using the tailored finite point method(TFPM)to solve the resulting parabolic or elliptic equations when minimizing the Huber regularization based image super-resolution model using the augmented Lagrangian method(ALM).The Hu-ber regularization based image super-resolution model can ameliorate the staircase for restored images.TFPM employs the method of weighted residuals with collocation tech-nique,which helps get more accurate approximate solutions to the equations and reserve more details in restored images.We compare the new schemes with the Marquina-Osher model,the image super-resolution convolutional neural network(SRCNN)and the classical interpolation methods:bilinear interpolation,nearest-neighbor interpolation and bicubic interpolation.Numerical experiments are presented to demonstrate that with the new schemes the quality of the super-resolution images has been improved.Besides these,the existence of the minimizer of the Huber regularization based image super-resolution model and the convergence of the proposed algorithm are also established in this paper.
基金supported by the National Science Council of Taiwan under the Grant NSC 97-2115-M-008-015-MY2.
文摘In this paper we propose a development of the finite difference method,called the tailored finite point method,for solving steady magnetohydrodynamic(MHD)duct flow problems with a high Hartmann number.When the Hartmann number is large,the MHD duct flow is convection-dominated and thus its solution may exhibit localized phenomena such as the boundary layer.Most conventional numerical methods can not efficiently solve the layer problem because they are lacking in either stability or accuracy.However,the proposed tailored finite point method is capable of resolving high gradients near the layer regions without refining the mesh.Firstly,we devise the tailored finite point method for the scalar inhomogeneous convectiondiffusion problem,and then extend it to the MHD duct flow which consists of a coupled system of convection-diffusion equations.For each interior grid point of a given rectangular mesh,we construct a finite-point difference operator at that point with some nearby grid points,where the coefficients of the difference operator are tailored to some particular properties of the problem.Numerical examples are provided to show the high performance of the proposed method.
基金supported by the NSFC Project No.10971116.M.Tang is supported by Natural Science Foundation of Shanghai under Grant No.12ZR1445400Shanghai Pujiang Program 13PJ1404700+1 种基金supported in part by the National Natural Science Foundation of China under Grant DMS-11101278the Young Thousand Talents Program of China.
文摘This paper presents two uniformly convergent numerical schemes for the two dimensional steady state discrete ordinates transport equation in the diffusive regime,which is valid up to the boundary and interface layers.A five-point nodecentered and a four-point cell-centered tailored finite point schemes(TFPS)are introduced.The schemes first approximate the scattering coefficients and sources by piecewise constant functions and then use special solutions to the constant coefficient equation as local basis functions to formulate a discrete linear system.Numerically,both methods can not only capture the diffusion limit,but also exhibit uniform convergence in the diffusive regime,even with boundary layers.Numerical results show that the five-point scheme has first-order accuracy and the four-point scheme has second-order accuracy,uniformly with respect to the mean free path.Therefore a relatively coarse grid can be used to capture the two dimensional boundary and interface layers.
文摘It is of great interest to solve the inverse problem of stationary radiative transport equation(RTE)in optical tomography.The standard way is to formulate the inverse problem into an optimization problem,but the bottleneck is that one has to solve the forward problem repeatedly,which is time-consuming.Due to the optical property of biological tissue,in real applications,optical thin and thick regions coexist and are adjacent to each other,and the geometry can be complex.To use coarse meshes and save the computational cost,the forward solver has to be asymptotic preserving across the interface(APAL).In this paper,we propose an offline/online solver for RTE.The cost at the offline stage is comparable to classical methods,while the cost at the online stage is much lower.Two cases are considered.One is to solve the RTE with fixed scattering and absorption cross sections while the boundary conditions vary;the other is when cross sections vary in a small domain and the boundary conditions change many times.The solver can be decomposed into offline/online stages in these two cases.One only needs to calculate the offline stage once and update the online stage when the parameters vary.Our proposed solver is much cheaper when one needs to solve RTE with multiple right-hand sides or when the cross sections vary in a small domain,thus can accelerate the speed of solving inverse RTE problems.We illustrate the online/offline decomposition based on the Tailored Finite Point Method(TFPM),which is APAL on general quadrilateral meshes.
