ANALYSIS OF FDTD TO UPML FOR MAXWELL EQUATIONS IN POLAR COORDINATES

An FDTD system associated with uniaxial perfectly matched layer（UPML） for an electromagnetic scattering problem in two-dimensional space in polar coordinates is considered.Particularly the FDTD system of an initial-boundary value problems of the transverse magnetic（TM） mode to Maxwell＇s equations is obtained by Yee＇s algorithm,and the open domain of the scattering problem is truncated by a circle with a UPML.Besides,an artificial boundary condition is imposed on the outer boundary of the UPML.Afterwards,stability of the FDTD system on the truncated domain is established through energy estimates by the Gronwall inequality.Numerical experiments are designed to approve the theoretical analysis.

New Reflection Principles for Maxwell's Equations and Their Applications

Some new reflection principles for Maxwell's equations are first established, which are then applied to derive two novel identifiability results in inverse electromagnetic obstacle scattering problems with polyhedral scatterers.

An Inverse Problem for Maxwell's Equations in Anisotropic Media

The authors consider Maxwell＇s equations for an isomagnetic anisotropic and inhomogeneous medium in two dimensions, and discuss an inverse problem of determining the permittivity tensor （ε1,ε2,ε2,ε3 ） and the permeability μ in the constitutive relations from a finite number of lateral boundary measurements. Applying a Carleman estimate, the authors prove an estimate of the Lipschitz type for stability, provided that ε1,ε2,ε3,μ satisfy some a priori conditions.

Maxwell方程反问题的小波-微分正则化算法

构造了一种求解Maxwell方程反问题的小波-微分正则化混合反演算法。利用小波将反问题分解到不同尺度上,在最大尺度上采用微分正则化方法求得次级尺度的初始解,在其它尺度上进行迭代修正以获得全局最小点。算法结合了小波多尺度反演和微分正则化方法的优点,数值模拟说明了其较强的全局搜索能力。

Some Universal Properties of the Green’s Functions Associated with the Wave Equation in Bounded Partially-Homogeneous Domains and Their Use in Acoustic Tomography

Direct and inverse scattering problems connected with the wave equation in non-homogeneous bounded domains constitute challenging actual subjects for both mathematicians and engineers. Among them one can mention, for example, inverse source problems in seismology, nondestructive archeological probing, mine prospecting, inverse initial-value problems in acoustic tomography, etc. In spite of its crucial importance, almost all of the available rigorous investigations concern the case of unbounded simple domains such as layered planar or cylindrical or spherical structures. The main reason for the lack of the works related to non-homogeneous bounded structures is the extreme complexity of the explicit expressions of the Green’s functions. The aim of the present work consists in discovering some universal properties of the Green’s functions in question, which reduce enormously the difficulties arising in various applications. The universality mentioned here means that the properties are not depend on the geometrical and physical properties of the configuration. To this end one considers first the case when the domain is partially-homogeneous. Then the results are generalized to the most general case. To show the importance of the universal properties in question, they are applied to an inverse initial-value problem connected with photo-acoustic tomography.

Determination of Time-Dependent Coefficients for a Weakly Degenerate Heat Equation

In this paper,we consider solving numerically for the first time inverse problems of determining the time-dependent thermal diffusivity coefficient for a weakly degenerate heat equation,which vanishes at the initial moment of time,and/or the convection coefficient along with the temperature for a one-dimensional parabolic equation,from some additional information about the process(the so-called over-determination conditions).Although uniquely solvable these inverse problems are still ill-posed since small changes in the input data can result in enormous changes in the output solution.The finite difference method with the Crank-Nicolson scheme combined with the nonlinear Tikhonov regularization are employed.The resulting minimization problem is computationally solved using the MATLAB toolbox routine lsqnonlin.For both exact and noisy input data,accurate and stable numerical results are obtained.

振荡介质Maxwell方程透射特征值的下界估计

内部透射特征值问题在研究非均匀介质的逆散射问题中起着至关重要的作用,它源于入射波关于非均匀介质的散射问题。主要研究带有小周期振荡系数的Maxwell方程组的透射特征值问题,此模型在物理上对应于电磁波在周期微结构介质中的传播。给出了该类数学问题对应的电磁场散射的物理背景,明确了透射特征值实际上是一类特殊频率,使得该频率的入射波在给定的介质中不产生散射波和透射波。对此重要的应用问题,给出了正透射特征值的下界估计。

Stability analysis of FDTD to UPML for time dependent Maxwell equations

We study an finite-difference time-domain (FDTD) system of uniaxial perfectly matched layer (UPML) method for electromagnetic scattering problems. Particularly we analyze the discrete initial-boundary value problems of the transverse magnetic mode (TM) to Maxwell's equations with Yee's algorithm. An exterior domain in two spacial dimension is truncated by a square with a perfectly matched layer filled by a certain artificial medium. Besides, an artificial boundary condition is imposed on the outer boundary of the UPML. Using energy method, we obtain the stability of this FDTD system on the truncated domain. Numerical experiments are designed to approve the theoretical analysis.

Uniqueness and stability in an inverse problem for a Poisson's equation

Consider the Poisson’s equation Ψ″ (x) = ?e v?Ψ + e Ψ?v ? N(x) with the Dirichlet boundary data, and we mainly investigate the inverse problem of determining the unknown function N(x) from a parameter function family. Some uniqueness and stability results in the inverse problem are obtained.

Adaptive hp-Finite Element Computations for Time-Harmonic Maxwell’s Equations

In this paper,hp-adaptive finite element methods are studied for timeharmonic Maxwell’s equations.We propose the parallel hp-adaptive algorithms on conforming unstructured tetrahedral meshes based on residual-based a posteriori error estimates.Extensive numerical experiments are reported to investigate the efficiency of the hp-adaptive methods for point singularities,edge singularities,and an engineering benchmark problem of Maxwell’s equations.The hp-adaptive methods show much better performance than the h-adaptive method.

Uniqueness and local stability for the inverse scattering problem of determining the cavity

Considering a time-harmonic electromagnetic plane wave incident on an arbitrarily shaped open cavity embedded in infinite ground plane, the physical process is modelled by Maxwell's equations. We investigate the inverse problem of determining the shape of the open cavity from the information of the measured scattered field. Results on the uniqueness and the local stability of the inverse problem in the 2-dimensional TM (transverse magnetic) polarization are proved in this paper.

Inverse problem for Chaplygin’s nonholonomic systems

Chaplygin’s nonholonomic systems are familiar mechanical systems subject to unintegrable linear constraints, which can be reduced into holonomic nonconservative systems in a subspace of the original state space. The inverse problem of the calculus of variations or Lagrangian inverse problem for such systems is analyzed by making use of a reduction of the systems into new ones with time reparametrization symmetry and a genotopic transformation related with a conformal transformation. It is evident that the Lagrangian inverse problem does not have a direct universality. By meaning of a reduction of Chaplygin’s nonholonomic systems into holonomic, regular, analytic, nonconservative, first-order systems, the systems admit a Birkhoffian representation in a star-shaped neighborhood of a regular point of their variables, which is universal due to the Cauchy-Kovalevski theorem and the converse of the Poincaré lemma.

ANALYSIS OF TWO-DIMENSIONAL ELECTROMAGNETIC SCATTERING BY A PERFECTLY CONDUCTING OBSTACLE IN A HOMOGENEOUS CHIRAL ENVIRONMENT

The time-harmonic electromagnetic plane waves incident on a perfectly conducting obstacle in a homogeneous chiral environment are considered. A two-dimensional direct scattering model is established and the existence and uniqueness of solutions to the problem are discussed by an integral equation approach. The inverse scattering problem to find the shape of scatterer with the given far-field data is formulated. Result on the uniqueness of the inverse problem is proved.

AMBARZUMYAN’S THEOREM WITH EIGENPARAMETER IN THE BOUNDARY CONDITIONS

In this paper, the classical Ambarzumyan’s theorem for the regular SturmLiouville problem is extended to the case in which the boundary conditions are eigenparameter dependent. Specifically, we show that if the spectrum of the operator D 2 ＋q with eigenparameter dependent boundary conditions is the same as the spectrum belonging to the zero potential, then the potential function q is actually zero.

Well-posedness of the time-varying linear electromagnetic initial-boundary value proble

The well-posedness of the initial-boundary value problem of the time-varying linear electromagnetic field in a multi-medium region is investigated. Function spaces are defined, with Faraday＇s law of electromagnetic induction and the initial-boundary conditions considered as constraints. Gauss＇s formula applied to a multi-medium region is used to derive the energy-estimating inequality. After converting the initial-boundary conditions into homogeneous ones and analysing the characteristics of an operator introduced according to the total current law, the existence, uniqueness and stability of the weak solution to the initial-boundary value problem of the time-varying linear electromagnetic field are proved.

Dirichlet-to-Neumann Map for a Hyperbolic Equation

In this paper, we provide an explicit expression for the full Dirichlet-to-Neumann map corresponding to a radial potential for a hyperbolic differential equation in 3-dimensional. We show that the Dirichlet-Neumann operators corresponding to a potential radial have the same properties for hyperbolic differential equations as for elliptic differential equations. We numerically implement the coefficients of the explicit formulas. Moreover, a Lipschitz type stability is established near the edge of the domain by an estimation constant. That is necessary for the reconstruction of the potential from Dirichlet-to-Neumann map in the inverse problem for a hyperbolic differential equation.

Heuristic Solution to the Conundrum of the Zitterbewegung

Assuming the Dirac wavefunction describes the state of a single particle. We propose that the relation derived by Schrödinger, which contains the Zitterbewegung term, is a position equation for an amplitude modulated wave. Namely, the elementary constituents are amplitude modulated waves. Indeed, we surmise that a second wave is associated with the particle, which corresponds to a signal. At the same time, we interpret that Broglie’s wave corresponds to a carrier. Furthermore, the quantum object is a recording medium and, like in a hologram, information encoded on its surface. We suggest a description and the cause of the Zitterbewegung heretofore never considered regarding the previous assertions. Hereunder, we shall also apply the quantum amplitude modulation interpretation to the single-photon wave function by Bialynicki-Birula. The predictions are testable, thence providing evidence for the proposed hypothesis.

A SURVEY OF OPEN CAVITY SCATTERING PROBLEMS

This paper gives a brief survey of recent developments on mathematical modeling and analysis of the open cavity scattering problems, which arise in diverse scientific areas and have significant industrial and military applications. The scattering problems are studied for the two-dimensional Helmholtz equation corresponding to the transverse magnetic or electric polarization, and the three-dimensional time-harmonic and time-domain Maxwell equations. Since these problems are imposed in open domains, a key step of the analysis is to develop transparent boundary conditions and reformulate them equivalently into boundary value problems in bounded domains. The well-posedness of weak solutions are shown for the associated variational problems by using either the Lax-Milgram theorem or the Fredholm alternative.

用地质雷达数据资料反演二维地下介质的方法

从二维麦克斯韦方程组出发推导出反演介电常数和电导率等二维介质物性参数的反演公式.反演的步骤是:建立初始猜测模型,利用电磁波时间域有限差分法模拟正演数据,用正演数据与观测数据之间的数据残差建立目标函数,通过引入一个由麦克斯韦方程计算的伴随场,将目标函数对介质参数的导数表示成显式形式,应用最优化理论得出对初始猜测模型的修改,用共轭梯度法迭代,最终得到反演结果.用合成数据反演具有粗糙地表的非导电介质的介电常数,用实验数据同时反演介电常数和电导率,并比较了麦克斯韦方程反演结果与声波方程反演结果、波动方程偏移剖面的差异.

探地雷达反问题的小波-同伦混合反演算法

构造一种求解探地雷达反问题的小波-同伦混合反演算法.利用小波将反问题分解到不同尺度上,在最大尺度上采用同伦反演方法,求得次级尺度的初始近似解,在其余尺度上进行多次迭代修正,以获得全局最小点.算法结合小波反演和同伦反演的优点,数值结果表明其具有较强的全局搜索能力.