A Nonlocal Operator Method for Partial Differential Equations with Application to Electromagnetic Waveguide Problem

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.

Higher Order Collocation Methods for Nonlocal Problems and Their Asymptotic Compatibility

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.

Probabilistically implementing nonlocal operations between two distant qutrits

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.

Two-Phase Image Segmentation by the Allen-Cahn Equation and a Nonlocal Edge Detection Operator

Based on a nonlocal Laplacian operator,a novel edge detection method of the grayscale image is proposed in this paper.This operator utilizes the information of neighbor pixels for a given pixel to obtain effective and delicate edge detection.The nonlocal edge detection method is used as an initialization for solving the Allen-Cahn equation to achieve two-phase segmentation of the grayscale image.Efficient exponential time differencing(ETD)solvers are employed in the time integration,and finite difference method is adopted in space discretization.The maximum bound principle and energy stability of the proposed numerical schemes are proved.The capability of our segmentation method has been verified in numerical experiments for different types of grayscale images.

A Nonlocal Stokes Systemwith Volume Constraints

In this paper,we introduce a nonlocal model for linear steady Stokes system with physical no-slip boundary condition.We use the idea of volume constraint to enforce the no-slip boundary condition and prove that the nonlocal model is wellposed.We also show that and the solution of the nonlocal system converges to the solution of the original Stokes system as the nonlocality vanishes.

Hölder Estimates for Nonlocal-Diffusion Equations with Drifts

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.

EFFICIENT BOX-CONSTRAINED TV-TYPE-l^1 ALGORITHMS FOR RESTORING IMAGES WITH IMPULSE NOISE

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.