NUMERICAL LOCALIZATION OF ELECTROMAGNETIC IMPERFECTIONS FROM A PERTURBATION FORMULA IN THREE DIMENSIONS

This work deals with the numerical localization of small electromagnetic inhomogeneities. The underlying inverse problem considers, in a three-dimensional bounded domain, the time-harmonic Maxwell equations formulated in electric field. Typically, the domain contains a finite number of unknown inhomogeneities of small volume and the inverse problem attempts to localize these inhomogeneities from a finite number of boundary measurements. Our localization approach is based on a recent framework that uses an asymptotic expansion for the perturbations in the tangential boundary trace of the curl of the electric field. We present three numerical localization procedures resulting from the combination of this asymptotic expansion with each of the following inversion algorithms： the Current Projection method, the MUltiple Signal Classification （MUSIC） algorithm, and an Inverse Fourier method. We perform a numerical study of the asymptotic expansion and compare the numerical results obtained from the three localization procedures in different settings.     

ON THE ERROR ESTIMATES OF A NEW OPERATE SPLITTING METHOD FOR THE NAVIER-STOKES EQUATIONS

In this paper, a new operator splitting scheme is introduced for the numerical solution of the incompressible Navier-Stokes equations. Under some mild regularity assumptions on the PDE solution, the stability of the scheme is presented, and error estimates for the velocity and the pressure of the proposed operator splitting scheme are given.     

A NOTE ON PRESSURE APPROXIMATION OF FIRST AND HIGHER ORDER PROJECTION SCHEMES FOR THE NONSTATIONARY INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

Projection methods are efficient operator-splitting schemes to approximate solutions of the incompressible Navier-Stokes equations. As a major drawback, they introduce spurious layers, both in space and time. In this work, we survey convergence results for higher order projection methods, in the presence of only strong solutions of the limiting problem; in particular, we highlight concomitant difficulties in the construction process of accurate higher order schemes, such as limited regularities of the limiting solution, and a lack of accurate initial data for the pressure. Computational experiments are included to compare the presented schemes, and illustrate the difficulties mentioned.     

UNIFORM SUPERCONVERGENCE OF A FINITE ELEMENT METHOD WITH EDGE STABILIZATION FOR CONVECTION-DIFFUSION PROBLEMS

In the present paper the edge stabilization technique is applied to a convection-diffusion problem with exponential boundary layers on the unit square, using a Shishkin mesh with bilinear finite elements in the layer regions and linear elements on the coarse part of the mesh. An error bound is proved for ‖πu-u^h‖Е, where πu is some interpolant of the solution u and uh the discrete solution. This supercloseness result implies an optimal error estimate with respect to the L2 norm and opens the door to the application of postprocessing for improving the discrete solution.     

A NUMERICAL STUDY OF UNIFORM SUPERCONVERGENCE OF LDG METHOD FOR SOLVING SINGULARLY PERTURBED PROBLEMS

In this paper, we consider the local discontinuous Galerkin method （LDG） for solving singularly perturbed convection-diffusion problems in one- and two-dimensional settings. The existence and uniqueness of the LDG solutions are verified. Numerical experiments demonstrate that it seems impossible to obtain uniform superconvergence for numerical fluxes under uniform meshes. Thanks to the implementation of two-type different anisotropic meshes, i.e., the Shishkin and an improved grade meshes, the uniform 2p ＋ i-order superconvergence is observed numerically for both one-dimensional and twodimensional cases.     

NUMERICAL RECONSTRUCTION OF SMALL PERTURBATIONS IN THE ELECTROMAGNETIC COEFFICIENTS OF A DIELECTRIC MATERIAL

The aim of this paper is to solve numerically the inverse problem of reconstructing small amplitude perturbations in the magnetic permeability of a dielectric material from partial or total dynamic boundary measurements. Our numerical algorithm is based on the resolution of the time-dependent Maxwell equations, an exact controllability method and Fourier inversion for localizing the perturbations. Two-dimensional numerical experiments illustrate the performance of the reconstruction method for different configurations even in the case of limited-view data.     

VARIATIONAL DISCRETIZATION FOR OPTIMAL CONTROL GOVERNED BY CONVECTION DOMINATED DIFFUSION EQUATIONS

In this paper, we study variational discretization for the constrained optimal control problem governed by convection dominated diffusion equations, where the state equation is approximated by the edge stabilization Galerkin method. A priori error estimates are derived for the state, the adjoint state and the control. Moreover, residual type a posteriori error estimates in the L^2-norm are obtained. Finally, two numerical experiments are presented to illustrate the theoretical results.     

New Lower Bounds for the Least Common Multiples of Arithmetic Progressions

Abstract For relatively prime positive integers u0 and r, and for 0 〈 k ≤ n, define uk ：= u0 ＋ kr. Let Ln ：= 1cm（u0,u1,... ,un） and let a,l≥2 be any integers. In this paper, the authors show that, for integers α≥ a, r ≥max（a,l - 1） and n ≥lατ, the following inequality holds Ln≥u0r^（l-1）α＋a-l（r＋1）^n.Particularly, letting l = 3 yields an improvement on the best previous lower bound on Ln obtained by Hong and Kominers in 2010.     

ON THE QUASI-RANDOM CHOICE METHOD FOR THE LIOUVILLE EQUATION OF GEOMETRICAL OPTICS WITH DISCONTINUOUS WAVE SPEED

We study the quasi-random choice method （QRCM） for the Liouville equation of ge- ometrical optics with discontinuous locM wave speed. This equation arises in the phase space computation of high frequency waves through interfaces, where waves undergo partial transmissions and reflections. The numerical challenges include interface, contact discon- tinuities, and measure-valued solutions. The so-called QRCM is a random choice method based on quasi-random sampling （a deterministic alternative to random sampling）. The method not only is viscosity-free but also provides faster convergence rate. Therefore, it is appealing for the prob!em under study which is indeed a Hamiltonian flow. Our analy- sis and computational results show that the QRCM 1） is almost first-order accurate even with the aforementioned discontinuities; 2） gives sharp resolutions for all discontinuities encountered in the problem; and 3） for measure-valued solutions, does not need the level set decomposition for finite difference/volume methods with numerical viscosities.     

A COARSENING ALGORITHM ON ADAPTIVE GRIDS BY NEWEST VERTEX BISECTION AND ITS APPLICATIONS

In this paper, an efficient and easy-to-implement coarsening algorithm is proposed for adaptive grids obtained using the newest vertex bisection method in two dimemsions. The new coarsening algorithm does not require storing the binary refinement tree explicitly. Instead, the structure is implicitly contained in a special ordering of triangular elements. Numerical experiments demonstrate that the proposed coarsening algorithm is efficient when applied for multilevel preconditioners and mesh adaptivity for time-dependent problems.     

SIMPLE PROOFS OF THE CAUCHY-SCHWARTZ INEQUALITY AND THE NEGATIVE DISCRIMINANTPROPERTY IN ARCHIMEDEAN ALMOST f-ALGEBRAS

The paper presents simple proofs of the Cauchy-Schwartz inequality and the negative discriminant property in archimedean almost f-algebras^[5], based on a sequence approximation.     

Edge coloring by total labelings of outerplanar graphs

An edge coloring totalk-labeling is a labeling of the vertices and the edges of a graph G with labels{1,2,...,k}such that the weights of the edges defne a proper edge coloring of G.Here the weight of an edge is the sum of its label and the labels of its two end vertices.This concept was introduce by Brandt et al.They defnedχt(G)to be the smallest integer k for which G has an edge coloring total k-labeling and proposed a question:Is there a constant K withχt(G)≤Δ(G)+12+K for all graphs G of maximum degreeΔ(G)?In this paper,we give a positive answer for outerplanar graphs by showing thatχt(G)≤Δ(G)+12+1 for each outerplanar graph G with maximum degreeΔ(G).     

A NUMERICAL STUDY FOR THE PERFORMANCE OF THE WENO SCHEMES BASED ON DIFFERENT NUMERICAL FLUXES FOR THE SHALLOW WATER EQUATIONS

In this paper we investigate the performance of the weighted essential non-oscillatory （WENO） methods based on different numerical fluxes, with the objective of obtaining better performance for the shallow water equations by choosing suitable numerical fluxes. We consider six numerical fluxes, i.e., Lax-Friedrichs, local Lax-Friedrichs, Engquist-Osher, Harten-Lax-van Leer, HLLC and the first-order centered fluxes, with the WENO finite volume method and TVD Runge-Kutta time discretization for the shallow water equations. The detailed numerical study is performed for both one-dimensional and two-dimensional shallow water equations by addressing the property, and resolution of discontinuities. issues of CPU cost, accuracy, non-oscillatory     

BIOLUMINESCENCE TOMOGRAPHY： BIOMEDICAL BACKGROUND,MATHEMATICAL THEORY,AND NUMERICAL APPROXIMATION

Over the last couple of years molecular imaging has been rapidly developed to study physiological and pathological processes in vivo at the cellular and molecular levels. Among molecular imaging modalities, optical imaging stands out for its unique advantages, especially performance and cost-effectiveness. Bioluminescence tomography （BLT） is an emerging optical imaging mode with promising biomedical advantages. In this survey paper, we explain the biomedical significance of BLT, summarize theoretical results on the analysis and numerical solution of a diffusion based BLT model, and comment on a few extensions for the study of BLT.     

EFFICIENT NUMERICAL ALGORITHMS FOR THREE-DIMENSIONAL FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS

This paper detailedly discusses the locally one-dimensional numerical methods for ef- ficiently solving the three-dimensional fractional partial differential equations, including fractional advection diffusion equation and Riesz fractional diffusion equation. The second order finite difference scheme is used to discretize the space fractional derivative and the Crank-Nicolson procedure to the time derivative. We theoretically prove and numerically verify that the presented numerical methods are unconditionally stable and second order convergent in both space and time directions. In particular, for the Riesz fractional dif- fusion equation, the idea of reducing the splitting error is used to further improve the algorithm, and the unconditional stability and convergency are also strictly proved and numerically verified for the improved scheme.     

SHOCK AND BOUNDARY STRUCTURE FORMATION BY SPECTRAL-LAGRANGIAN METHODS FOR THE INHOMOGENEOUS BOLTZMANN TRANSPORT EQUATION

The numerical approximation of the Spectral-Lagrangian scheme developed by the authors in [30] for a wide range of homogeneous non-linear Boltzmann type equations is extended to the space inhomogeneous case and several shock problems are benchmark. Recognizing that the Boltzmann equation is an important tool in the analysis of formation of shock and boundary layer structures, we present the computational algorithm in Section 3.3 and perform a numerical study case in shock tube geometries well modeled in for ID in x times 3D in v in Section 4. The classic Riemann problem is numerically analyzed for Knudsen numbers close to continuum. The shock tube problem of Aoki et al [2], where the wall temperature is suddenly increased or decreased, is also studied. We consider the problem of heat transfer between two parallel plates with diffusive boundary conditions for a range of Knudsen numbers from close to continuum to a highly rarefied state. Finally, the classical infinite shock tube problem that generates a non-moving shock wave is studied. The point worth noting in this example is that the flow in the final case turns from a supersonic flow to a subsonic flow across the shock.     

ON A PROBLEM OF SUMS OF MIXED POWERS （II）

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FIXED-POINT CONTINUATION APPLIED TO COMPRESSED SENSING： IMPLEMENTATION AND NUMERICAL EXPERIMENTS

Fixed-point continuation （FPC） is an approach, based on operator-splitting and continuation, for solving minimization problems with l1-regularization：min ｜｜x｜｜1＋uf（x）.We investigate the application of this algorithm to compressed sensing signal recovery, in which f（x） = 1/2｜｜Ax-b｜｜2M,A∈m×n and m≤n. In particular, we extend the original algorithm to obtain better practical results, derive appropriate choices for M and u under a given measurement model, and present numerical results for a variety of compressed sensing problems. The numerical results show that the performance of our algorithm compares favorably with that of several recently proposed algorithms.     

UNIFORM ESTIMATES OF SOLUTIONS OF δ^--EQUATION ON STEIN MANIFOLDS

By means of the Hermitian metric and Chern connection, Qiu [4] obtained the Koppelman-Leray-Norguet formula for （p, q） differential forms on an open set with C^1 piecewise smooth boundary on a Stein manifold, and under suitable conditions gave the solutions of δ^--equation on a Stein manifold. In this article, using the method of Range and Siu [5], under suitable conditions, the authors complicatedly calculate to give the uniform estimates of solutions of δ^--equation for （p, q） differential forms on a Stein manifold.     

NORM CALCULATIONS OF A CLASS OF INTEGRAL OPERATORS

In this paper we calculate the norm of a special class of integral operators acting on LP （C^n, dvs）, where dvs is the Gaussian measure on C^n.