This paper presents a numerical compression strategy for the boundary integral equation of acoustic scattering in two dimensions. These equations have oscillatory kernels that we represent in a basis of wave atoms, and compress by thresholding the small coefficients to zero. This phenomenon was perhaps first observed in 1993 by Bradie, Coifman, and Grossman, in the context of local Fourier bases BCG . Their results have since then been extended in various ways. The purpose of this paper is to bridge a theoretical gap and prove that a well-chosen fixed expansion, the nonstandard wave atom form, provides a compression of the acoustic single and double layer potentials with wave number @math as @math -by- @math matrices with @math nonnegligible entries, with a constant that depends on the relative @math accuracy @math in an acceptable way. The argument assumes smooth, separated, and not necessarily convex scatterers in two dimensions. The essential features of wave atoms that enable to write this result as a theorem is a sharp time-frequency localization that wavelet packets do not obey, and a parabolic scaling wavelength @math (essential diameter) @math . Numerical experiments support the estimate and show that this wave atom representation may be of interest for applications where the same scattering problem needs to be solved for many boundary conditions, for example, the computation of radar cross sections.
Most of the approaches on sparsifying in well-chosen bases require the construction of the full integral operator. Since this step itself takes @math operations, it poses computational difficulty for large @math values. In @cite_19 , proposed a solution to the related problem of sparsifying the boundary integral operator of the Laplace equation. They successfully avoided the construction of the full integral operator by predicting the location of the large coefficients and applying a special one-point quadrature rule to compute the coefficients. The corresponding solution for the integral operator of the Helmholtz equation is still missing.
@cite_19
2078691815 1517670975 2083690159 2734896008
Abstract We revisit the efficient approximation of functions by sums of exponentials or Gaussians in Beylkin and Monzon (2005) [16] to discuss several new results and applications of these approximations. By using the Poisson summation to discretize integral representations of e.g., power functions r − β , β > 0 , we obtain approximations with uniform relative error on the whole real line. Our approach is applicable to a class of functions and, in particular, yields a separated representation for the function e − x y . As a result, we obtain sharper error estimates and a simpler method to derive trapezoidal-type quadratures valid on finite intervals. We also introduce a new reduction algorithm for the case where our representation has an excessive number of terms with small exponents. As an application of these new estimates, we simplify and improve previous results on separated representations of operators with radial kernels. For any finite but arbitrary accuracy, we obtain new separated representations of solutions of Laplace's equation satisfying boundary conditions on the half-space or the sphere. These representations inherit a multiresolution structure from the Gaussian approximation leading to fast algorithms for the evaluation of the solutions. In the case of the sphere, our approach provides a foundation for a new multiresolution approach to evaluating and estimating models of gravitational potentials used for satellite orbit computations. Preface to the Classics Edition Preface Symbols 1. The Riesz-Fredholm theory for compact operators 2. Regularity properties of surface potentials 3. Boundary-value problems for the scalar Helmholtz equation 4. Boundary-value problems for the time-harmonic Maxwell equations and the vector Helmholtz equation 5. Low frequency behavior of solutions to boundary-value problems in scattering theory 6. The inverse scattering problem: exact data 7. Improperly posed problems and compact families 8. The determination of the shape of an obstacle from inexact far-field data 9. Optimal control problems in radiation and scattering theory References Index. We present a novel boundary integral formulation of the Helmholtz transmission problem for bounded composite scatterers (that is, piecewise constant material parameters in "subdomains") that directly lends itself to operator preconditioning via Calderon projectors. The method relies on local traces on subdomains and weak enforcement of transmission conditions. The variational formulation is set in Cartesian products of standard Dirichlet and special Neumann trace spaces for which restriction and extension by zero are well defined. In particular, the Neumann trace spaces over each subdomain boundary are built as piecewise @math -distributions over each associated interface. Through the use of interior Calderon projectors, the problem is cast in variational Galerkin form with an operator matrix whose diagonal is composed of block boundary integral operators associated with the subdomains. We show existence and uniqueness of solutions based on an extension of Lions' projection lemma for non-closed subspaces. We also investigate asymptotic quasi-optimality of conforming boundary element Galerkin discretization. Numerical experiments in 2-D confirm the efficacy of the method and a performance matching that of another widely used boundary element discretization. They also demonstrate its amenability to different types of preconditioning. We propose a new nonconforming finite element algorithm to approximate the solution to the elliptic problem involving the fractional Laplacian. We first derive an integral representation of the bilinear form corresponding to the variational problem. The numerical approximation of the action of the corresponding stiffness matrix consists of three steps: (i) apply a sinc quadrature scheme to approximate the integral representation by a finite sum where each term involves the solution of an elliptic partial differential equation defined on the entire space, (ii) truncate each elliptic problem to a bounded domain, (iii) use the finite element method for the space approximation on each truncated domain. The consistency error analysis for the three steps is discussed together with the numerical implementation of the entire algorithm. The results of computations are given illustrating the error behavior in terms of the mesh size of the physical domain, the domain truncation parameter and the quadrature spacing parameter.