A decomposition scheme for acoustic obstacle scattering in a multilayered medium

We consider the acoustic wave scattering by an impenetrable obstacle embedded in a multilayered background medium, which is modelled by a linear system constituted by the Helmholtz equations with different wave numbers and the transmission conditions across the interfaces. The aim of this article is to construct an efficient computing scheme for the scattered waves for this complex scattering process, with a rigorous mathematical analysis. First, we construct a set of functions by a series of coupled transmission problems, which are proven to be well-defined. Then, the solution to our complex scattering in each layer is decomposed as the summation in terms of these functions, which are essentially the contributions from two interfaces enclosing this layer. These contributions physically correspond to the scattered fields for simple scattering problems, which do not involve the multiple scattering and are coupled via the boundary conditions. Finally, we propose an iteration scheme to compute the wave field in each layer decoupling the multiple scattering effects, with the advantage that only the solvers for the well-known transmission problems and an obstacle scattering problem in a homogeneous background medium are applied. The convergence property of this iteration scheme is proven.     

An improved time domain linear sampling method for Robin and Neumann obstacles

We consider inverse obstacle scattering problems for the wave equation with Robin or Neumann boundary conditions. The problem of reconstructing the geometry of such obstacles from measurements of scattered waves in the time domain is tackled using a time domain linear sampling method. This imaging technique yields a picture of the scatterer by solving a linear operator equation involving the measured data for many right-hand sides given by singular solutions to the wave equation. We analyse this algorithm for causal and smooth impulse shapes, we discuss the effect of different choices of the singular solutions used in the algorithm, and finally we propose a fast FFT-based implementation.     

基于遗传算法重建多个散射体的组合Newton法

本文研究由单个入射声波或电磁波及其远场数据反演多个柔性散射体边界的逆散射问题.通过建立边界到边界总场的非线性算子及其n6chet导数,本文首先给出了基于单层位势的组合Newton法.将组合Newton法转化为泛响优化问题,从而获得了该方法重建单个散射体的收敛性分析.然后,基于遗传算法和正则化参数选取的模型函数方法,给出...     

求解二维多区域声波散射问题的边界元方法

对于多散射区域的声波散射问题的外Neumann边值问题,用单层位势来逼近每个散射域上的散射波,再利用位势理论的跳跃关系将问题转换为第二类边界积分方程组的求解问题,然后用Nystrom方法进行了求解.对多个随机散射区域的声波散射问题,数值例子体现了该求解方法的可行性和准确性.     

The numerical solution of an inverse periodic transmission problem

We consider the inverse problem of recovering a 2D periodic structure from scattered waves measured above and below the structure. We discuss convergence and implementation of an optimization method for solving the inverse TE transmission problem, following an approach first developed by Kirsch and Kress for acoustic obstacle scattering. The convergence analysis includes the case of Lipschitz grating profiles and relies on variational methods and solvability properties of periodic boundary integral equations. Numerical results for exact and noisy data demonstrate the practicability of the inversion algorithm. Copyright © 2004 John Wiley & Sons, Ltd.     

Fast iteration method in the problem of waves interacting with a set of thin screens

A new iteration method is proposed for the wave equation describing the scattering of a harmonic wave from an arbitrary configuration in the form of an array of thin straight barriers. The problem is reduced to a system of boundary integral equations, which are discretized by applying the Belotserkovskii-Lifanov method. In discrete form, a finite number of systems with Toeplitz matrices (the number of systems is equal to the number of barriers) are solved at each iteration step by applying special fast methods. The algorithm is tested on several geometries, and its convergence in these cases is analyzed.     

On the convergence of the multigrid method for a hypersingular integral equation of the first kind

Summary We present a multigrid method to solve linear systems arising from Galerkin schemes for a hypersingular boundary integral equation governing three dimensional Neumann problems for the Laplacian. Our algorithm uses damped Jacobi iteration, Gauss-Seidel iteration or SOR as preand post-smoothers. If the integral equation holds on a closed, Lipschitz surface we prove convergence ofV- andW-cycles with any number of smoothing steps. If the integral equation holds on an open, Lipschitz surface (covering crack problems) we show convergence of theW-cycle. Numerical experiments are given which underline the theoretical results.     

A Priori Estimates for High Frequency Scattering by Obstacles of Arbitrary Shape

High frequency estimates for the Dirichlet-to-Neumann and Neumann-to-Dirichlet operators are obtained for the Helmholtz equation in the exterior of bounded obstacles. These a priori estimates are used to study the scattering of plane waves by an arbitrary bounded obstacle and to prove that the total cross section of the scattered wave does not exceed four geometrical cross sections of the obstacle in the limit as the wave number k → ∞. This bound of the total cross section is sharp.     

A NON-OVERLAPPING DOMAIN DECOMPOSITION ALGORITHM BASED ON THE NATURAL BOUNDARY REDUCTION FOR WAVE EQUATIONS IN AN UNBOUNDED DOMAIN

In this paper, a new domain decomposition method based on the natural boundary reduction, which solves wave problems over an unbounded domain, is suggestted. An circular artificial boundary is introduced. The original unbounded domain is divided into two subdomains, an internal bounded region and external unbounded region outside the artificial boundary. A Dirichlet-Neumann(D-N) alternating iteration algorithm is constructed. We prove that the algorithm is equavilent to preconditional Richardson iteration method. Numerical studies are performed by finite element method. The numerical results show that the convergence rate of the discrete D-N iteration is independent of the finite element mesh size.     

An interior–exterior Schwarz algorithm and its convergenceUn algorithme de Schwarz intérieur–extérieur et sa convergence

In this work we study the solution of Laplace's equation in a domain with holes by an iteration consisting of splitting the problem in an exterior one, around the holes, plus an interior problem in the unholed domain. We show the existence of a decomposition of the solution when the exterior problem is represented by means of a single-layer protential. Also, for the three-dimensional case and with some adjustments for the two-dimensional case, we prove convergence of the method by writing the iteration as a Jacobi iteration for an operator equation and studying the spectrum of the iteration operator. To cite this article: R. Celorrio et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 923–926.     

A hybrid method for sound-hard obstacle reconstruction

We are interested in solving the inverse problem of acoustic wave scattering to reconstruct the position and the shape of sound-hard obstacles from a given incident field and the corresponding far field pattern of the scattered field. The method we suggest is an extension of the hybrid method for the reconstruction of sound-soft cracks as presented in [R. Kress, P. Serranho, A hybrid method for two-dimensional crack reconstruction, Inverse Problems 21 (2005) 773–784] to the case of sound-hard obstacles. The designation of the method is justified by the fact that it can be interpreted as a hybrid between a regularized Newton method applied to a nonlinear operator equation with the operator that maps the unknown boundary onto the solution of the direct scattering problem and a decomposition method in the spirit of the potential method as described in [A. Kirsch, R. Kress, On an integral equation of the first kind in inverse acoustic scattering, in: Cannon, Hornung (Eds.), Inverse Problems, ISNM, vol. 77, 1986, pp. 93–102. Since the method does not require a forward solver for each Newton step its computational costs are reduced. By some numerical examples we illustrate the feasibility of the method.     

Mathematical basis of the linear sampling method for partially coated obstacles

We consider the inverse scattering problem of determining the shape of a partially coated obstacle D. To this end, we solve a scattering problem for the Helmholtz equation where the scattered field satisfies mixed Dirichlet–Neumann-impedance boundary conditions on the Lipschitz boundary of the scatterer D. Based on the analysis of the boundary integral system to the direct scattering problem, we propose how to reconstruct the shape of the obstacle D by using the linear sampling method.     

Convergence of Monte Carlo algorithms for reconstructing the scattering phase function with polarization

The problem of reconstructing the atmospheric scattering phase function from groundbased solar almucantar sky brightness observations is considered.A newiterative algorithm for solving this problem was developed as a combination of existing additive and multiplicative methods of refining the single-scattering contribution to the observed brightness with polarization of scattered radiation in the atmosphere. Also, some modifications of these methods were proposed. The objective of this paper is to numerically substantiate the convergence of these methods. For this purpose, an algorithm of Jacobi matrix calculation for the iteration operators of the methods was developed, and calculations were carried out for various parameters of the atmosphere.     

Recovering the Dirichlet-to-Neumann map in inverse scattering problems using integral equation methods

Consider the reconstruction of Dirichlet-to-Neumann map(D-to-N map) from the far-field patterns of the scattered waves in inverse scattering problems, which is the first step in detecting the obstacle boundary by the probe method using far-field measurements corresponding to all incident plane waves. In principle, this problem can be reduced to solving an integral equation of the second kind with the kernels involving the derivatives of the scattered waves for point sources. Based on the mixed reciprocity principle, we propose two simple and feasible numerical schemes for reconstructing D-to-N map. Compared with the well-known obstacle boundary recovering schemes using the simulation of D-to-N map directly, the proposed schemes give the possible ways to realizing the probe methods using practical far-field data, with the advantage of no numerical differentiation for scattered wave in their implementations. We present some numerical examples for the D-to-N map, showing the validity and stability of our schemes.     

Plane Wave Discontinuous Galerkin Methods: Exponential Convergence of the $$$$-Version

We consider the two-dimensional Helmholtz equation with constant coefficients on a domain with piecewise analytic boundary, modelling the scattering of acoustic waves at a sound-soft obstacle. Our discretisation relies on the Trefftz-discontinuous Galerkin approach with plane wave basis functions on meshes with very general element shapes, geometrically graded towards domain corners. We prove exponential convergence of the discrete solution in terms of number of unknowns.     

局部扰动半平面上的散射问题

该文讨论半平面上有局部扰动情况下的散射问题．通过位势理论,应用边界积分方程的方法研究了该问题解的存在与唯一性．主要方法是运用对称反射,使该无界区域上的散射问题变成一个有界区域上的散射问题,只是这一有界区域的边界不光滑．通过仔细分析相应的边界积分算子,作者得到了其解的存在与唯一性．     

A Neural Network-Based Policy Iteration Algorithm with Global $$H^2$$ -Superlinear Convergence for Stochastic Games on Domains

In this work, we propose a class of numerical schemes for solving semilinear Hamilton–Jacobi–Bellman–Isaacs (HJBI) boundary value problems which arise naturally from exit time problems of diffusion processes with controlled drift. We exploit policy iteration to reduce the semilinear problem into a sequence of linear Dirichlet problems, which are subsequently approximated by a multilayer feedforward neural network ansatz. We establish that the numerical solutions converge globally in the \(H^2\)-norm and further demonstrate that this convergence is superlinear, by interpreting the algorithm as an inexact Newton iteration for the HJBI equation. Moreover, we construct the optimal feedback controls from the numerical value functions and deduce convergence. The numerical schemes and convergence results are then extended to oblique derivative boundary conditions. Numerical experiments on the stochastic Zermelo navigation problem are presented to illustrate the theoretical results and to demonstrate the effectiveness of the method.

     

Décomposition de domaine pour un calcul hybride de l'équation de helmholtz

We prove an estimate for the Dirichlet-Neumann operator, and for the H¹ local norm for solutions of Helmholtz equation outside an obstacle without trapping rays. We give an algorithm solving Helmholtz equation outside a union of such obstacles. Convergence follows from this estimate. At each step of the resolution, only one obstacle is considered for itself; this defines a decomposition domain technique fitting this equation. One can use different numerical schemes, one at each step, adapted to the considered component of the obstacle; therefore, this algorithm is a hybrid computation. The results are given for two obstacles, and the generalization is straightforward     

A convergence theorem for the fast multipole method for 2 dimensional scattering problems

The Fast Multipole Method (FMM) designed by V. Rokhlin rapidly computes the field scattered from an obstacle. This computation consists of solving an integral equation on the boundary of the obstacle. The main result of this paper shows the convergence of the FMM for the two dimensional Helmholtz equation. Before giving the theorem, we give an overview of the main ideas of the FMM. This is done following the papers of V. Rokhlin. Nevertheless, the way we present the FMM is slightly different. The FMM is finally applied to an acoustic problem with an impedance boundary condition. The moment method is used to discretize this continuous problem.