A fast method is proposed for solving the high frequency Helmholtz equation. The building block of the new fast method is an overlapping source transfer domain decomposition method for layered medium, which is an extension of the source transfer domain decomposition method proposed by Chen and Xiang \cite{Chen2013a,Chen2013b}. The new fast method contains a setup phase and a solving phase. In the setup phase, the computation domain is decomposed hierarchically into many subdomains of different levels, and the mapping from incident traces to field traces on all the subdomains are set up bottom-up. In the solving phase, first on the bottom level, the local problem on the subdomains with restricted source is solved, then the wave propagates on the boundaries of all the subdomains bottom-up, at last the local solutions on all the subdomains are summed up top-down. The total computation cost of the new fast method is $O(n^{\frac{3}{2}} \log n)$ for 2D problem. Numerical experiments shows that with the new fast method, Helmholtz equations with half billion unknowns could be solved efficiently on massively parallel machines.
We solve first-kind Fredholm boundary integral equations arising from Helmholtz and Laplace problems on bounded, smooth screens in three-dimensions with either Dirichlet or Neumann conditions. The proposed Galerkin-Bubnov method takes as discretization elements pushed-forward weighted azimuthal projections of standard spherical harmonics onto the canonical disk. We show that these bases allow for spectral convergence and provide fully discrete error analysis. Numerical experiments support our claims, with results comparable to Nyström-type and $hp$-methods.
We study wave propagation phenomena modelled in the frequency domain by the Helmholtz equation in heterogeneous media with focus on media with discontinuous, highly oscillating wave speed. We restrict to problems with spherical symmetry and will derive explicit representations of the Green's operator and stability estimates which are explicit in the frequency and the wave speed.
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.
Taylor expansion based fast Multipole Methods for 3-D Helmholtz equations in Layered Media
Hunan Normal University（湖南师范大学）；Southern Methodist University（南方卫理公会大学）；University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校）；Indiana University（印第安纳大学）；University of Massachusetts Lowell（马萨诸塞大学洛厄尔分校）
We propose and analyze a quasi-Monte Carlo (QMC) algorithm for efficient simulation of wave propagation modeled by the Helmholtz equation in a bounded region in which the refractive index is random and spatially heterogenous. Our focus is on the case in which the region can contain multiple wavelengths. We bypass the usual sign-indefiniteness of the Helmholtz problem by switching to an alternative sign-definite formulation recently developed by Ganesh and Morgenstern (Numerical Algorithms, 83, 1441-1487, 2020). The price to pay is that the regularity analysis required for QMC methods becomes m…
An overlapped continuous model framework, for the Helmholtz wave propagation problem in unbounded regions comprising bounded heterogeneous media, was recently introduced and analyzed by the authors ({\tt J. Comput. Phys., {\bf 403}, 109052, 2020}). The continuous Helmholtz system incorporates a radiation condition (RC) and our equivalent hybrid framework facilitates application of widely used finite element methods (FEM) and boundary element methods (BEM), and the resulting discrete systems retain the RC exactly. The FEM and BEM discretizations, respectively, applied to the designed interior h…
Fast multipole methods (FMM) were originally developed for accelerating $N$-body problems for particle-based methods. FMM is more than an $N$-body solver, however. Recent efforts to view the FMM as an elliptic Partial Differential Equation (PDE) solver have opened the possibility to use it as a preconditioner for a broader range of applications. FMM can solve Helmholtz problems with optimal $\mathcal{O}(N \log N)$ complexity, has compute-bound inner kernels, and highly asynchronous communication patterns. The combination of these features makes FMM an interesting candidate as a preconditioner …
In this paper, a fast multipole method (FMM) is proposed to compute long-range interactions of wave sources embedded in 3-D layered media. The layered media Green's function for the Helmholtz equation, which satisfies the transmission conditions at material interfaces, is decomposed into a free space component and four types of reaction field components arising from wave reflections and transmissions through the layered media. The proposed algorithm is a combination of the classic FMM for the free space component and FMMs specifically designed for the four types reaction components, made possi…
Full waveform inversion (FWI) is one of a family of methods that allows the reconstruction of earth subsurface parameters from measurements of waves at or near the surface. This is a numerical optimization problem that uses the whole waveform information of all arrivals to update the subsurface parameters that govern seismic wave propagation. We apply FWI in the multi-scale approach on two well-known benchmarks: Marmousi and 2004 BP velocity model. For the forward modeling, we use an RBF-FD solver on hexagonal grids and quasi-optimal shape parameters.
Asymptotic dispersion correction for the isotropic elastic Helmholtz equation discretized with a MAC scheme
Université de Pau et des Pays de l’Adour（波城及阿杜尔地区大学）；Normandie University, INSA de Rouen Normandie（诺曼底大学，鲁昂诺曼底国立应用科学学院）；Ben-Gurion University of the Negev（内盖夫本-古里安大学）
A Class of Iterative Solvers for the Helmholtz Equation: Factorizations, Sweeping Preconditioners, Source Transfer, Single Layer Potentials, Polarized Traces, and Optimized Schwarz Methods
University of Geneva（日内瓦大学）；Zhejiang Ocean University（浙江海洋大学）
A crucial part of successful wave propagation related inverse problems is an efficient and accurate numerical scheme for solving the seismic wave equations. In particular, the numerical solution to a multi-dimensional Helmholtz equation can be troublesome when the perfectly matched layer (PML) boundary condition is implemented. In this paper, we present a general approach for constructing fourth-order finite difference schemes for the Helmholtz equation with PML in the two-dimensional domain based on point-weighting strategy. Particularly, we develop two optimal fourth-order finite difference …
Asymptotic spectral properties and preconditioning of an approximated nonlocal Helmholtz equation with Caputo fractional Laplacian and variable coefficient wave number $μ$
We consider a fast, data-sparse directional method to realize matrix-vector products related to point evaluations of the Helmholtz kernel. The method is based on a hierarchical partitioning of the point sets and the matrix. The considered directional multi-level approximation of the Helmholtz kernel can be applied even on high-frequency levels efficiently. We provide a detailed analysis of the almost linear asymptotic complexity of the presented method. Our numerical experiments are in good agreement with the provided theory.
In this paper, we propose a fast multipole method (FMM) for 3-D linearized Poisson-Boltzmann (PB) equation in layered media. The main framework of the algorithm is analogous to the FMM for Helmholtz and Laplace equation in layered media [1,2], using an extension of the Funk-Hecke formula for pure imaginary wave number. Moreover, a recurrence formula is provided for the run-time computation of the Sommerfeld-type integrals used in the FMM algorithm. Due to the similarity between Helmholtz and linearized PB equation, the recurrence formula can also be used for the FMM of Helmholtz equation in la…
Numerical resolution of exterior Helmholtz problems requires some approach to domain truncation. As an alternative to approximate nonreflecting boundary conditions and invocation of the Dirichlet-to-Neumann map, we introduce a new, nonlocal boundary condition. This condition is exact and requires the evaluation of layer potentials involving the free space Green's function. However, it seems to work in general unstructured geometry, and Galerkin finite element discretization leads to convergence under the usual mesh constraints imposed by Gårding-type inequalities. The nonlocal boundary conditi…
Quasi-Helmholtz decompositions are fundamental tools in integral equation modeling of electromagnetic problems because of their ability of rescaling solenoidal and non-solenoidal components of solutions, operator matrices, and radiated fields. These tools are however incapable, per se, of modifying the refinement-dependent spectral behavior of the different operators and often need to be combined with other preconditioning strategies. This paper introduces the new concept of filtered quasi-Helmholtz decompositions proposing them in two incarnations: the filtered Loop-Star functions and the qua…
We present the first fast solver for the high-frequency Helmholtz equation that scales optimally in parallel, for a single right-hand side. The L-sweeps approach achieves this scalability by departing from the usual propagation pattern, in which information flows in a 180 degree cone from interfaces in a layered decomposition. Instead, with L-sweeps, information propagates in 90 degree cones induced by a checkerboard domain decomposition (CDD). We extend the notion of accurate transmission conditions to CDDs and introduce a new sweeping strategy to efficiently track the wave fronts as they pro…
This paper proposes a plane wave activation based neural network (PWNN) for solving Helmholtz equation, the basic partial differential equation to represent wave propagation, e.g. acoustic wave, electromagnetic wave, and seismic wave. Unlike using traditional activation based neural network (TANN) or $sin$ activation based neural network (SIREN) for solving general partial differential equations, we instead introduce a complex activation function $e^{\mathbf{i}{x}}$, the plane wave which is the basic component of the solution of Helmholtz equation. By a simple derivation, we further find that …
Solving time-harmonic wave propagation problems in the frequency domain and within heterogeneous media brings many mathematical and computational challenges, especially in the high frequency regime. We will focus here on computational challenges and try to identify the best algorithm and numerical strategy for a few well-known benchmark cases arising in applications. The aim is to cover, through numerical experimentation and consideration of the best implementation strategies, the main two-level domain decomposition methods developed in recent years for the Helmholtz equation. The theory for t…
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…
Neural network-driven domain decomposition for efficient solutions to the Helmholtz equation
Department of Mathematics and Computer Science, Eindhoven University of Technology（数学与计算机科学系，埃因霍温理工大学）；Atlantis project-team, Inria Research Center at Université Côte d'Azur（Atlantis项目组，法国国家信息与自动化技术研究院（Inria）在阿尔卑斯大学（Université Côte d'Azur）的研究中心）；MACARON project-team, Université de Strasbourg, CNRS, Inria, IRMA（MACARON项目组，斯特拉斯堡大学，法国国家科学研究中心（CNRS），法国国家信息与自动化技术研究院（Inria），IRMA）
We present a butterfly-compressed representation of the Hadamard-Babich (HB) ansatz for the Green's function of the high-frequency Helmholtz equation in smooth inhomogeneous media. For a computational domain discretized with $N_v$ discretization cells, the proposed algorithm first solves and tabulates the phase and HB coefficients via eikonal and transport equations with observation points and point sources located at the Chebyshev nodes using a set of much coarser computation grids, and then butterfly compresses the resulting HB interactions from all $N_v$ cell centers to each other. The over…
An unconditionally convergent discontinuous Galerkin neural network method activated by plane waves for Helmholtz equation and Maxwell's equations
College of Mathematics and Systems Science, Shandong University of Science and Technology（山东科技大学数学与系统科学学院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
A new scheme is presented for imposing periodic boundary conditions on unit cells with arbitrary source distributions. We restrict our attention here to the Poisson, modified Helmholtz, Stokes and modified Stokes equations. The approach extends to the oscillatory equations of mathematical physics, including the Helmholtz and Maxwell equations, but we will address these in a companion paper, since the nature of the problem is somewhat different and includes the consideration of quasiperiodic boundary conditions and resonances. Unlike lattice sum-based methods, the scheme is insensitive to the u…
In $d$ dimensions, accurately approximating an arbitrary function oscillating with frequency $\lesssim k$ requires $\sim k^d$ degrees of freedom. A numerical method for solving the Helmholtz equation (with wavenumber $k$ and in $d$ dimensions) suffers from the pollution effect if, as $k\to\infty$, the total number of degrees of freedom needed to maintain accuracy grows faster than this natural threshold (i.e., faster than $k^d$ for domain-based formulations, such as finite element methods, and $k^{d-1}$ for boundary-based formulations, such as boundary element methods). It is well known that t…
