Recurrence Relations for the Coefficients in Ultraspherical Series Solutions of Ordinary Differential Equations

A method is presented for obtaining recurrence relations for the coefficients in ultraspherical series of linear differential equations. This method applies Doha's method (1985) to generate polynomial approximations in terms of ultraspherical polynomials of $y(zx), -1\leq x\leq 1,z\in C,|z|\leq 1$, where y is a solution of a linear differential equation. In particular, rational approximations of $y(z)$ result if $x$ is set equal to unity. Two numerical examples are given to illustrate the application of the method to first and second order differential equations. In general, the rational approximations obtained by this method are better than the corresponding polynomial approximations, and compare favourably with Pade approximants.     

Pseudo-differential operators for embedding formulae

A new method is proposed for deriving embedding formulae in 2D diffraction problems. In contrast to the approach developed in Craster and Shanin (2005) [7], which is based on a differential operator, here a pseudo-differential, i.e., a non-local operator is applied to the wave field. Using this non-local operator a new embedding formula is derived for scattering by a single wedge. The formula has uniform structure for all opening angles, including angles irrational with respect to π; the earlier theory, Craster and Shanin (2005) [7], was valid only for rational angles.     

The inverse scattering problem for a partially coated cavity with interior measurements

We consider the interior inverse scattering problem of recovering the shape and the surface impedance of an impenetrable partially coated cavity from a knowledge of measured scatter waves due to point sources located on a closed curve inside the cavity. First, we prove uniqueness of the inverse problem, namely, we show that both the shape of the cavity and the impedance function on the coated part are uniquely determined from exact data. Then, based on the linear sampling method, we propose an inversion scheme for determining both the shape and the boundary impedance. Finally, we present some numerical examples showing the validity of our method.     

利用远场模式的完全与不完全数据反演声波阻尼区域

提出一种方法,利用远场模式的完全数据与不完全数据反演声波阻尼区域,证明了方法的收敛性,并给出若干数值例子.     

Numerical treatment of classic scattering from bounded objects

Classic scattering from objects of arbitrary shape must generally be treated by numerical methods. It has proven very difficult to describe scattering from general bounded objects without resorting to frequency-limiting approximations. The starting point of many numerical methods is the Helmholtz integral representation of a given wavefield. From that point several departures are possible for constructing computationally feasible approximate schemes. To date, attempts at direct solutions have been rare.One method (originated by P. Waterman) that attacks the exact numerical solution for a very broad class of problems begins with the Helmholtz integral representations for a point exterior and interior to the target in a partial wave expansion. After truncating the partial wave space, one arrives at a set of matrix equations useful in describing the field. This method is often referred to as the T-matrix method, null-field, or extended integral equation method. It leads to a unique solution of the exterior boundary integral equation by incorporating the interior solution (extinction theorem) as a constraint. In principle, there are no theoretical limitations on frequency, although numerical complications can arise and must be appropriately dealt with for the method to be computationally reliable.For submerged objects the formalism will be outlined for acoustical scattering from targets that are rigid; sound-soft and penetrable; elastic solids; elastic shells; and layered elastic objects. Finally, illustrations of several numerical examples for the above will be presented to emphasize specific response features peculiar to a variety of targets.     

Analysis of scattering properties of embedded particles by applying the discrete sources method

A numerical scheme based on the discrete sources method is constructed for the mathematical simulation of the scattering properties of nanoparticles embedded in a substrate. Both differential and integral scattering characteristics of particles embedded to various degrees are analyzed. It is shown that embedded particles can be distinguished from those lying on the substrate by using P-polarized external excitation waves incident at two different angles.     

Optimization via Chebyshev polynomials

This paper presents for the first time a robust exact line-search method based on a full pseudospectral (PS) numerical scheme employing orthogonal polynomials. The proposed method takes on an adaptive search procedure and combines the superior accuracy of Chebyshev PS approximations with the high-order approximations obtained through Chebyshev PS differentiation matrices. In addition, the method exhibits quadratic convergence rate by enforcing an adaptive Newton search iterative scheme. A rigorous error analysis of the proposed method is presented along with a detailed set of pseudocodes for the established computational algorithms. Several numerical experiments are conducted on one- and multi-dimensional optimization test problems to illustrate the advantages of the proposed strategy.     

The interior inverse scattering problem for cavities with an artificial obstacle

The interior inverse scattering by an impenetrable cavity is considered. Both the sources and the measurements are placed on a curve or surface inside the cavity. As a rule of thumb, both the direct and the inverse problems suffer from interior eigenvalues. The interior eigenvalues are removed by adding an artificial obstacle with impedance boundary condition to the underlying scattering system. For this new system, we prove a reciprocity relation for the scattered field and a uniqueness theorem for the inverse problem. Some new techniques are used in the arguments of the uniqueness proof because of the Lipschitz regularity of the boundary of the cavity. The linear sampling method is used for this new scattering system for reconstructing the shape of the cavity. Finally, some numerical experiments are presented to demonstrate the feasibility and effectiveness of the linear sampling method. In particular, the introduction of the artificial obstacle makes the linear sampling method robust to frequency.     

Reconstruction of cracks of different types from far‐field measurements

In this paper, we deal with the acoustic inverse scattering problem for reconstructing cracks of possibly different types from the far‐field map. The scattering problem models the diffraction of waves by thin two‐sided cylindrical screens. The cracks are characterized by their shapes, the type of boundary conditions and the boundary coefficients (surface impedance). We give explicit formulas of the indicator function of the probe method, which can be used to reconstruct the shape of the cracks, distinguish their types of boundary conditions, the two faces of each of them and reconstruct the possible material coefficients on them by using the far‐field map. To test the validity of these formulas, we present some numerical implementations for a single crack, which show the efficiency of the proposed method for suitably distributed surface impedances. The difficulties for numerically recovering the properties of the crack in the concave side as well as near the tips are presented and some explanations are given. Copyright © 2009 John Wiley & Sons, Ltd.     

Efficient nonlinear reanalysis for structural modifications

In this paper an approach for nonlinear reanalysis based on residual increment approximations (RIA) is presented. The method requires only the evaluation of residual vectors, which can be done very fast and efficient. Moreover, the results are improved by using a rational approximation method. The approach is general and can be applied to linear and nonlinear problems with different kind of design modifications. Furthermore, the proposed method is very efficient and easily to implement in existing finite element programs. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Low-energy vortex dynamics in the self-dual Chern–Simons–Higgs model

The relativistic Chern–Simons–Higgs theory finds application in anyonic superconductivity and contains topological vortices whose dynamics are poorly understood. The gauge fields are defined by a set of nonlinear constraint equations that can be accurately solved with effective Green’s functions, spectral methods, and a discretization scheme using lattice gauge techniques. Simulations show that low-energy two-vortex interactions are elastic with final scattering angles sensitive to vortex velocity; furthermore, vortex pairs form rotating breather states for certain impact parameters. In this study, a function that reproduces scattering angles in the adiabatic limit for nontangential collisions is presented. Simulation results are discussed in the context of analytical methods that extract vortex dynamics from low-energy effective Lagrangians, and a numerical method to calculate the effective Lagrangian is suggested. The numerical techniques used can be applied to the study of other Chern–Simon theories.     

A method of local convex majorants for solving variational-like inequalities

A numerical method based on convex approximations that locally majorize a gap function is proposed for solving a variational-like inequality. The algorithm is theoretically validated and the results of comparison of its numerical efficiency to that of the conventional methods are presented.     

Finite Element Method and Discontinuous Galerkin Method for Stochastic Scattering Problem of Helmholtz Type in ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript> (<Emphasis Type="Italic">d</Emphasis> = 2, 3)

In this paper, we address the finite element method and discontinuous Galerkin method for the stochastic scattering problem of Helmholtz type in ℝ ^d (d = 2, 3). Convergence analysis and error estimates are presented for the numerical solutions. The effects of the noises on the accuracy of the approximations are illustrated. Results of the numerical experiments are provided to examine our theoretical results. The first author is supported by NSF under grand number 0609918 and AFOSR under grant numbers FA9550-06-1-0234 and FA9550-07-1-0154, the second author is supported by NSFC(10671082, 10626026, 10471054), and the third author is supported by NNSF (No. 10701039 of China) and 985 program of Jilin University.     

Integral equation methods for scattering from an impedance crack

For the scattering problem for time-harmonic waves from an impedance crack in two dimensions, we give a uniqueness and existence analysis via a combined single- and double-layer potential approach in a Hölder space setting leading to a system of integral equations that contains a hypersingular operator. For its numerical solution we describe a fully discrete collocation method based on trigonometric interpolation and interpolatory quadrature rules including a convergence analysis and numerical examples.     

A step size control algorithm for the weak approximation of stochastic differential equations

A vriable step size control algorithm for the weak approximation of stochastic differential equations is introduced. The algorithm is based on embedded Runge–Kutta methods which yield two approximations of different orders with a negligible additional computational effort. The difference of these two approximations is used as an estimator for the local error of the less precise approximation. Some numerical results are presented to illustrate the effectiveness of the introduced step size control method.      

A finite element method for computing the bifurcation function for semilinear elliptic BVPs

The method of alternative problems can be used to show that a semilinear elliptic boundary value problem (Lu + g(x, u) = 0 with g_u(x, u) bounded below) is equivalent to a finite-dimensional problem ( , ), in the sense that their solution sets, which are not necessarily singletons, are in a one-to-one correspondence. This correspondence is based on a map σ from low-frequency to high-frequency Fourier components of solutions. A numerical method is presented for approximating σ and hence also solutions of the BVP. The method uses finite element approximations and avoids the use of eigenfunction expansions. Existence, uniqueness, and error estimates for the approximations of σ and solutions u are derived.     

Scattering of the electromagnetic field on a finite impedance section of an interface

The scattering of a three-dimensional electromagnetic field on a local impedance section of a wavy surface is considered. The boundary-value problem for the system of Maxwell equations is reduced to solving a system of hypersingular integral equations. A numerical algorithm is developed and a program is designed for computing the electrodynamic characteristics of the given scattering problem. The accuracy of the simulation results is investigated. __________ Translated from Prikladnaya Matematika i Informatika, No. 25, pp. 56–69, 2007.     

Coherent Reflection of Electromagnetic Plane Waves from Infinite Rough Slabs

We adopt the prospect of an observer interested to optimise the signal-to-noise ratio in the reception of the backward radiation coming from a surface illuminated by an electromagnetic wave with a wavelength chosen to minimize the diffuse scattering so that he has just to point his receiver in the direction of the coherent reflection. Then, to analyse the coherent reflection for harmonic plane waves impinging on a dielectric infinite film deposited on a metallic substrate we develop a formalism generalizing the customary angular spectrum representation used to tackle this kind of problem. This new approach whose efficiency is proved in the easier situation of a dielectric film endowed with an impedance, is used to get the coherent reflection from a structured 1D-dielectric film illuminated by TE and TM electromagnetic plane waves when the rough amplitude h is small enough to justify 0(h ²) approximations. The Idemen technique is used to get the boundary conditions needed to tackle these scattering problems.     

On parallel algorithms for semidiscretized parabolic partial differential equations based on subdiagonal Padé approximations

It is well known that methods for solving semidiscretized parabolic partial differential equations based on the second-order diagonal [1/1] Padé approximation (the Crank–Nicolson or trapezoidal method) can produce poor numerical results when a time discretization is imposed with steps that are “too large” relative to the spatial discretization. A monotonicity property is established for all diagonal Padé approximants from which it is shown that corresponding higher-order methods suffer a similar time step restriction as the [1/1] Padé. Next, various high-order methods based on subdiagonal Padé approximations are presented which, through a partial fraction expansion, are no more complicated to implement than the first-order implicit Euler method based on the [0/1] Padé approximation; moreover, the resulting algorithms are free of a time step restriction intrinsic to those based on diagonal Padé approximations. Numerical results confirm this when various test problems from the literature are implemented on a Multiple Instruction Multiple Data (MIMD) machine such as an Alliant FX/8. © 1993 John Wiley & Sons, Inc.     

Spectral Method for Nonlinear Stochastic Partial Differential Equations of Elliptic Type

This paper is concerned with the numerical approximations of semi-linear stochastic partial differential equations of elliptic type in multi-dimensions. Convergence analysis and error estimates are presented for the numerical solutions based on the spectral method. Numerical results demonstrate the good performance of the spectral method.