We consider the numerical solution of time-harmonic acoustic scattering by obstacles with uncertain geometries for Dirichlet, Neumann, impedance and transmission boundary conditions. In particular, we aim to quantify diffracted fields originated by small stochastic perturbations of a given relatively smooth nominal shape. Using first-order shape Taylor expansions, we derive tensor deterministic first kind boundary integral equations for the statistical moments of the scattering problems considered. These are then approximated by sparse tensor Galerkin discretizations via the combination technique (Griebel et al.). We supply extensive numerical experiments confirming the predicted error convergence rates with poly-logarithmic growth in the number of degrees of freedom and accuracy in approximation of the moments. Moreover, we discuss implementation details such as preconditioning to finally point out further research avenues.
The simulation of high-dimensional problems with manageable computational resource represents a long-standing challenge. In a series of our recent work [25, 17, 18, 24], a class of sparse grid DG methods has been formulated for solving various types of partial differential equations in high dimensions. By making use of the multiwavelet tensor-product bases on sparse grids in conjunction with the standard DG weak formulation, such a novel method is able to significantly reduce the computation and storage cost compared with full grid DG counterpart, while not compromising accuracy much for suffi…
In this paper a mathematical model is given for the scattering of an incident wave from a surface covered with microscopic small Helmholtz resonators, which are cavities with small openings. More precisely, the surface is built upon a finite number of Helmholtz resonators in a unit cell and that unit cell is repeated periodically. To solve the scattering problem, the mathematical framework elaborated in [Ammari et al., Asympt. Anal., 2019] is used. The main result is an approximate formula for the scattered wave in terms of the lengths of the openings. Our framework provides analytic expressio…
We present a fast spectral Galerkin scheme for the discretization of boundary integral equations arising from two-dimensional Helmholtz transmission problems in multi-layered periodic structures or gratings. Employing suitably parametrized Fourier basis and excluding Rayleigh-Wood anomalies, we rigorously establish the well-posedness of both continuous and discrete problems, and prove super-algebraic error convergence rates for the proposed scheme. Through several numerical examples, we confirm our findings and show performances competitive to those attained via Nyström methods.
Stability estimates of Nyström discretizations of Helmholtz decomposition boundary integral equation formulations for the solution of Navier scattering problems in two dimensions with Dirichlet boundary conditions
Over the last ten years, results from [Melenk-Sauter, 2010], [Melenk-Sauter, 2011], [Esterhazy-Melenk, 2012], and [Melenk-Parsania-Sauter, 2013] decomposing high-frequency Helmholtz solutions into "low"- and "high"-frequency components have had a large impact in the numerical analysis of the Helmholtz equation. These results have been proved for the constant-coefficient Helmholtz equation in either the exterior of a Dirichlet obstacle or an interior domain with an impedance boundary condition. Using the Helffer-Sjöstrand functional calculus, this paper proves analogous decompositions for scatt…
We investigate the asymptotic relation between the inverse problems relying on the Helmholtz equation and the radiative transfer equation (RTE) as physical models, in the high-frequency limit. In particular, we evaluate the asymptotic convergence of a generalized version of inverse scattering problem based on the Helmholtz equation, to the inverse scattering problem of the Liouville equation (a simplified version of RTE). The two inverse problems are connected through the Wigner transform that translates the wave-type description on the physical space to the kinetic-type description on the pha…
Finite Elements for Helmholtz Scattering with Infinity as a Computational Boundary
Institute of Analysis and Scientific Computing, Technische Universität Wien（维也纳工业大学分析与科学计算研究所）；Institute for Physical Science and Technology, University of Maryland（马里兰大学物理科学与技术研究所）
The Helmholtz wave scattering problem by screens in 2D can be recast into first-kind integral equations which lead to ill-conditioned linear systems after discretization. We introduce two new preconditioners, in the form of square-roots of local operators respectively for the corresponding problems with Dirichlet and Neumann conditions on the arc. They generalize the so-called "analytical" preconditioners available for Lipschitz scatterers. We introduce a functional setting adapted to the singularity of the problem and enabling the analysis of those preconditioners. The efficiency of the metho…
We introduce a novel multi-resolution Localized Orthogonal Decomposition (LOD) for time-harmonic acoustic scattering problems that can be modeled by the Helmholtz equation. The method merges the concepts of LOD and operator-adapted wavelets (gamblets) and proves its applicability for a class of complex-valued, non-hermitian and indefinite problems. It computes hierarchical bases that block-diagonalize the Helmholtz operator and thereby decouples the discretization scales. Sparsity is preserved by a novel localization strategy that improves stability properties even in the elliptic case. We pre…
In this paper, we consider the scattering of a plane wave by a rigid obstacle embedded in a homogeneous and isotropic elastic medium in three dimensions. Based on the Helmholtz decomposition, the elastic scattering problem is reduced to a coupled boundary value problem for the Helmholtz and Maxwell equations. A novel system of boundary integral equations is formulated and a spectral boundary integral method is developed for the coupled boundary value problem. Numerical experiments are presented to demonstrate the superior performance of the proposed method.
The discretisation of boundary integral equations for the scalar Helmholtz equation leads to large dense linear systems. Efficient boundary element methods (BEM), such as the fast multipole method (FMM) and $\Hmat$ based methods, focus on structured low-rank approximations of subblocks in these systems. It is known that the ranks of these subblocks increase linearly with the wavenumber. We explore a data-sparse representation of BEM-matrices valid for a range of frequencies, based on extracting the known phase of the Green's function. Algebraically, this leads to a Hadamard product of a freque…
Injectivity of boundary integral operator in direct-indirect mixed Burton-Miller equation for wave scattering problems with transmissive circular inclusion
For the $h$-finite-element method ($h$-FEM) applied to the Helmholtz equation, the question of how quickly the meshwidth $h$ must decrease with the frequency $k$ to maintain accuracy as $k$ increases has been studied since the mid 80's. Nevertheless, there still do not exist in the literature any $k$-explicit bounds on the relative error of the FEM solution (the measure of the FEM error most often used in practical applications), apart from in one dimension. The main result of this paper is the sharp result that, for the lowest fixed-order conforming FEM (with polynomial degree, $p$, equal to …
Arbitrary high order numerical methods for time-harmonic acoustic scattering problems originally defined on unbounded domains are constructed. This is done by coupling recently developed high order local absorbing boundary conditions (ABCs) with finite difference methods for the Helmholtz equation. These ABCs are based on exact representations of the outgoing waves by means of farfield expansions. The finite difference methods, which are constructed from a deferred-correction (DC) technique, approximate the Helmholtz equation and the ABCs, with the appropriate number of terms, to any desired o…
This paper is concerned with the cavity scattering problem in an infinite thin plate, where the out-of-plane displacement is governed by the two-dimensional biharmonic wave equation. Based on an operator splitting, the scattering problem is recast into a coupled boundary value problem for the Helmholtz and modified Helmholtz equations. A novel boundary integral formulation is proposed for the coupled problem. By introducing an appropriate regularizer, the well-posedness is established for the system of boundary integral equations. Moreover, the convergence analysis is carried out for the semi-…
Hardy space infinite elements for Helmholtz-type problems with unbounded inhomogeneities
Institut für Numerische und Angewandte Mathematik, Georg-August Universität Göttingen（哥廷根大学数值与应用数学研究所）；Mathematisches Institut, Heinrich-Heine Universität Düsseldorf（杜塞尔多夫海因里希·海涅大学数学研究所）
An FFT-based algorithm for efficient computation of Green's functions for the Helmholtz and Maxwell's equations in periodic domains
LSEC and Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences（中国科学院数学与系统科学研究院应用数学研究所）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；Center for Industrial Mathematics, University of Bremen（不来梅大学工业数学中心）
We present three high-order Nystrom discretization strategies of various boundary integral equation formulations of the impenetrable time-harmonic Navier equations in two dimensions. One class of such formulations is based on the four classical Boundary Integral Operators (BIOs) associated with the Green's function of the Navier operator. We consider two types of Nystrom discretizations of these operators, one that relies on Kussmaul-Martensen logarithmic splittings and the other on Alpert quadratures. In addition, we consider an alternative formulation of Navier scattering problems based on H…
Fast and high-order accurate algorithms for three dimensional elastic scattering are of great importance when modeling physical phenomena in mechanics, seismic imaging, and many other fields of applied science. In this paper, we develop a novel boundary integral formulation for the three dimensional elastic scattering based on the Helmholtz decomposition of elastic fields, which converts the Navier equation to a coupled system consisted of Helmholtz and Maxwell equations. An FFT-accelerated separation of variables solver is proposed to efficiently invert boundary integral formulations of the c…
This paper is concerned with an inverse random source problem for the one-dimensional stochastic Helmholtz equation with attenuation. The source is assumed to be a microlocally isotropic Gaussian random field with its covariance operator being a classical pseudo-differential operator. The random sources under consideration are equivalent to the generalized fractional Gaussian random fields which include rough fields and can be even rougher than the white noise, and hence should be interpreted as distributions. The well-posedness of the direct source scattering problem is established in the dis…
We analyze the well posedness of certain field-only boundary integral equations (BIE) for frequency domain electromagnetic scattering from perfectly conducting spheres. Starting from the observations that (1) the three components of the scattered electric field $\mathbf{E}^s(\mathbf{x})$ and (2) scalar quantity $\mathbf{E}^s(\mathbf{x})\cdot\mathbf{x}$ are radiative solutions of the Helmholtz equation, novel boundary integral equation formulations of electromagnetic scattering from perfectly conducting obstacles can be derived using Green's identities applied to the aforementioned quantities a…
We introduce several spatially adaptive model order reduction approaches tailored to non-coercive elliptic boundary value problems, specifically, parametric-in-frequency Helmholtz problems. The offline information is computed by means of adaptive finite elements, so that each snapshot lives in a different discrete space that resolves the local singularities of the analytical solution and is adjusted to the considered frequency value. A rational surrogate is then assembled adopting either a least-squares or an interpolatory approach, yielding function-valued version of the standard rational int…
