In this article, we consider a backward problem in time of the diffusion equation with local and nonlocal operators. This inverse problem is ill-posed because the solution does not depend continuously on the measured ...In this article, we consider a backward problem in time of the diffusion equation with local and nonlocal operators. This inverse problem is ill-posed because the solution does not depend continuously on the measured data. Inspired by the classical Landweber iterative method and Fourier truncation technique, we develops a modified Landweber iterative regularization method to restore the continuous dependence of solution on the measurement data. Under the a-priori and a-posteriori choice rules for the regularized parameter, the convergence estimates for the regularization method are derived. Some results of numerical simulation are provided to verify the stability and feasibility of our method in dealing with the considered problem.展开更多
A novel nonlocal operator theory based on the variational principle is proposed for the solution of partial differential equations.Common differential operators as well as the variational forms are defined within the ...A novel nonlocal operator theory based on the variational principle is proposed for the solution of partial differential equations.Common differential operators as well as the variational forms are defined within the context of nonlocal operators.The present nonlocal formulation allows the assembling of the tangent stiffness matrix with ease and simplicity,which is necessary for the eigenvalue analysis such as the waveguide problem.The present formulation is applied to solve the differential electromagnetic vector wave equations based on electric fields.The governing equations are converted into nonlocal integral form.An hourglass energy functional is introduced for the elimination of zeroenergy modes.Finally,the proposed method is validated by testing three classical benchmark problems.展开更多
We study the convergence and asymptotic compatibility of higher order collocation methods for nonlocal operators inspired by peridynamics,a nonlocal formulation of continuum mechanics.We prove that the methods are opt...We study the convergence and asymptotic compatibility of higher order collocation methods for nonlocal operators inspired by peridynamics,a nonlocal formulation of continuum mechanics.We prove that the methods are optimally convergent with respect to the polynomial degree of the approximation.A numerical method is said to be asymptotically compatible if the sequence of approximate solutions of the nonlocal problem converges to the solution of the corresponding local problem as the horizon and the grid sizes simultaneously approach zero.We carry out a calibration process via Taylor series expansions and a scaling of the nonlocal operator via a strain energy density argument to ensure that the resulting collocation methods are asymptotically compatible.We fnd that,for polynomial degrees greater than or equal to two,there exists a calibration constant independent of the horizon size and the grid size such that the resulting collocation methods for the nonlocal difusion are asymptotically compatible.We verify these fndings through extensive numerical experiments.展开更多
We discuss how to create more entanglement with nonlocal operations acting on two-particle states. For a given nonlocal operation, we find that some input states cannot produce entanglement and some produce the maxima...We discuss how to create more entanglement with nonlocal operations acting on two-particle states. For a given nonlocal operation, we find that some input states cannot produce entanglement and some produce the maximal entanglement, and find that any initial entangled states can produce more entanglement than initial product states.展开更多
We propose a method to probabilistically implement a nonlocal operation, exp[iζUAUB], between two distant qutrits A and B, where ζ∈ C [0,2π] and UA, UB are local unitary and Hermitian operations for qutrits A and ...We propose a method to probabilistically implement a nonlocal operation, exp[iζUAUB], between two distant qutrits A and B, where ζ∈ C [0,2π] and UA, UB are local unitary and Hermitian operations for qutrits A and B respectively. The consumptions of resource for one performance of the method are a single non-maximally entangled qutrit state and 1-trit classical communication. For a given ζ, the successful probability of the method depends on the forms of both entanglement resource and Bob＇s partial-measurement basis. We systematically discuss the optimal successful probabilities and their corresponding conditions for three cases： adjustable entanglement resource, adjustable partial-measurement basis, adjustable entanglement resource and partial-measurement basis. It is straightforward to generalize the method for producing nonlocal unitary operations between any two N-level systems.展开更多
In this paper, we provide an asymptotic analysis of a nonlocal reaction-diffusion equation and with a non-local stable operator of order α∈(0,1). Firstly, we prove the existence and uniqueness of positive and bounde...In this paper, we provide an asymptotic analysis of a nonlocal reaction-diffusion equation and with a non-local stable operator of order α∈(0,1). Firstly, we prove the existence and uniqueness of positive and bounded solutions for the stationary equation. Finally, we perform a long time-long range scaling in order to prove that the stable state invades the unstable state with a speed which is exponential in time.展开更多
We propose a method to implement a kind of non-local operation between spatially separated two systems with high dimensions by using only a low-dimensional qubit quantum channel and 2-bit classical communication. For ...We propose a method to implement a kind of non-local operation between spatially separated two systems with high dimensions by using only a low-dimensional qubit quantum channel and 2-bit classical communication. For qutrit systems, we further show the creation of non-local maximally entangled state and the construction of the non-local quantum XOR gate in terms of the obtained non-local operations as well as some single qutrit local gates.展开更多
In the paper,by exploring Stampacchia truncation method,some comparison techniques and variational approaches,we study the existence and regularity of positive solutions for a boundary value problem involving the frac...In the paper,by exploring Stampacchia truncation method,some comparison techniques and variational approaches,we study the existence and regularity of positive solutions for a boundary value problem involving the fractional p-Laplacian,where the nonlinear term satisfies some growth conditions but without the Ambrosetti and Rabinowitz condition.展开更多
We study a class of nonlocal-diffusion equations with drifts,and derive a priori-Hölder estimate for the solutions by using a purely probabilistic argument,whereis an intrinsic scaling function for the equation.
In this paper, we study the restoration of images simultaneously corrupted by blur and impulse noise via variational approach with a box constraint on the pixel values of an image. In the literature, the TV-l^1 variat...In this paper, we study the restoration of images simultaneously corrupted by blur and impulse noise via variational approach with a box constraint on the pixel values of an image. In the literature, the TV-l^1 variational model which contains a total variation （TV） regularization term and an l^1 data-fidelity term, has been proposed and developed. Several numerical methods have been studied and experimental results have shown that these methods lead to very promising results. However, these numerical methods are designed based on approximation or penalty approaches, and do not consider the box constraint. The addition of the box constraint makes the problem more difficult to handle. The main contribution of this paper is to develop numerical algorithms based on the derivation of exact total variation and the use of proximal operators. Both one-phase and two-phase methods are considered, and both TV and nonlocal TV versions are designed. The box constraint [0, 1] on the pixel values of an image can be efficiently handled by the proposed algorithms. The numerical experiments demonstrate that the proposed methods are efficient in computational time and effective in restoring images with impulse noise.展开更多
基金supported by the NSF of Ningxia(2022AAC03234)the NSF of China(11761004),the Construction Project of First-Class Disciplines in Ningxia Higher Education(NXYLXK2017B09)the Postgraduate Innovation Project of North Minzu University(YCX23074).
文摘In this article, we consider a backward problem in time of the diffusion equation with local and nonlocal operators. This inverse problem is ill-posed because the solution does not depend continuously on the measured data. Inspired by the classical Landweber iterative method and Fourier truncation technique, we develops a modified Landweber iterative regularization method to restore the continuous dependence of solution on the measurement data. Under the a-priori and a-posteriori choice rules for the regularized parameter, the convergence estimates for the regularization method are derived. Some results of numerical simulation are provided to verify the stability and feasibility of our method in dealing with the considered problem.
文摘A novel nonlocal operator theory based on the variational principle is proposed for the solution of partial differential equations.Common differential operators as well as the variational forms are defined within the context of nonlocal operators.The present nonlocal formulation allows the assembling of the tangent stiffness matrix with ease and simplicity,which is necessary for the eigenvalue analysis such as the waveguide problem.The present formulation is applied to solve the differential electromagnetic vector wave equations based on electric fields.The governing equations are converted into nonlocal integral form.An hourglass energy functional is introduced for the elimination of zeroenergy modes.Finally,the proposed method is validated by testing three classical benchmark problems.
文摘We study the convergence and asymptotic compatibility of higher order collocation methods for nonlocal operators inspired by peridynamics,a nonlocal formulation of continuum mechanics.We prove that the methods are optimally convergent with respect to the polynomial degree of the approximation.A numerical method is said to be asymptotically compatible if the sequence of approximate solutions of the nonlocal problem converges to the solution of the corresponding local problem as the horizon and the grid sizes simultaneously approach zero.We carry out a calibration process via Taylor series expansions and a scaling of the nonlocal operator via a strain energy density argument to ensure that the resulting collocation methods are asymptotically compatible.We fnd that,for polynomial degrees greater than or equal to two,there exists a calibration constant independent of the horizon size and the grid size such that the resulting collocation methods for the nonlocal difusion are asymptotically compatible.We verify these fndings through extensive numerical experiments.
文摘We discuss how to create more entanglement with nonlocal operations acting on two-particle states. For a given nonlocal operation, we find that some input states cannot produce entanglement and some produce the maximal entanglement, and find that any initial entangled states can produce more entanglement than initial product states.
基金Project supported by the National Major Fundamental Research Project, China （Grant No 2001CB309310）, the National Natural Science Foundation of China （Grant Nos 10347128, 10325523 and 90203018）, the Natural Science Foundation of Hunan Province （Grant No 04JJ3017）, the Science Foundation for Post Doctorate of China （Grant No 2005037695）, and the Scientific Research Fund of Hunan Provincial Education Bureau.
文摘We propose a method to probabilistically implement a nonlocal operation, exp[iζUAUB], between two distant qutrits A and B, where ζ∈ C [0,2π] and UA, UB are local unitary and Hermitian operations for qutrits A and B respectively. The consumptions of resource for one performance of the method are a single non-maximally entangled qutrit state and 1-trit classical communication. For a given ζ, the successful probability of the method depends on the forms of both entanglement resource and Bob＇s partial-measurement basis. We systematically discuss the optimal successful probabilities and their corresponding conditions for three cases： adjustable entanglement resource, adjustable partial-measurement basis, adjustable entanglement resource and partial-measurement basis. It is straightforward to generalize the method for producing nonlocal unitary operations between any two N-level systems.
文摘In this paper, we provide an asymptotic analysis of a nonlocal reaction-diffusion equation and with a non-local stable operator of order α∈(0,1). Firstly, we prove the existence and uniqueness of positive and bounded solutions for the stationary equation. Finally, we perform a long time-long range scaling in order to prove that the stable state invades the unstable state with a speed which is exponential in time.
基金The project supported by the Scientific Research Fund of the Education Bureau of Hunan Province under Grant No. 05B041, the Key Project of the Ministry of Education under Grant No. 206103, the China Postdoctoral Science Foundation under Grant No. 2005037695, National Natural Science Foundation of China under Grant Nos. 2001CB309310, 10325523, and 90203018, and the Natural Science Foundation of Hunan Province under Grant Nos. 04JJ3017 and 05JJ30012
文摘We propose a method to implement a kind of non-local operation between spatially separated two systems with high dimensions by using only a low-dimensional qubit quantum channel and 2-bit classical communication. For qutrit systems, we further show the creation of non-local maximally entangled state and the construction of the non-local quantum XOR gate in terms of the obtained non-local operations as well as some single qutrit local gates.
基金supported by the National Natural Science Foundation of China(No.12171470)。
文摘In the paper,by exploring Stampacchia truncation method,some comparison techniques and variational approaches,we study the existence and regularity of positive solutions for a boundary value problem involving the fractional p-Laplacian,where the nonlinear term satisfies some growth conditions but without the Ambrosetti and Rabinowitz condition.
基金The research of ZC was partially supported by NSF Grant DMS-1206276.The research of XZ was partially supported by NSFC Grant of China(Nos.11271294,11325105).
文摘We study a class of nonlocal-diffusion equations with drifts,and derive a priori-Hölder estimate for the solutions by using a purely probabilistic argument,whereis an intrinsic scaling function for the equation.
文摘In this paper, we study the restoration of images simultaneously corrupted by blur and impulse noise via variational approach with a box constraint on the pixel values of an image. In the literature, the TV-l^1 variational model which contains a total variation （TV） regularization term and an l^1 data-fidelity term, has been proposed and developed. Several numerical methods have been studied and experimental results have shown that these methods lead to very promising results. However, these numerical methods are designed based on approximation or penalty approaches, and do not consider the box constraint. The addition of the box constraint makes the problem more difficult to handle. The main contribution of this paper is to develop numerical algorithms based on the derivation of exact total variation and the use of proximal operators. Both one-phase and two-phase methods are considered, and both TV and nonlocal TV versions are designed. The box constraint [0, 1] on the pixel values of an image can be efficiently handled by the proposed algorithms. The numerical experiments demonstrate that the proposed methods are efficient in computational time and effective in restoring images with impulse noise.
