Recent Results on Domain Decomposition Preconditioning for the High-Frequency Helmholtz Equation Using Absorption (original) (raw)
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Domain decomposition preconditioning for high-frequency Helmholtz problems with absorption
Mathematics of Computation, 2017
In this paper we give new results on domain decomposition preconditioners for GM-RES when computing piecewise-linear finite-element approximations of the Helmholtz equation −∆u − (k 2 + iε)u = f , with absorption parameter ε ∈ R. Multigrid approximations of this equation with ε = 0 are commonly used as preconditioners for the pure Helmholtz case (ε = 0). However a rigorous theory for such (so-called "shifted Laplace") preconditioners, either for the pure Helmholtz equation, or even the absorptive equation (ε = 0), is still missing. We present a new theory for the absorptive equation that provides rates of convergence for (left-or right-) preconditioned GMRES, via estimates of the norm and field of values of the preconditioned matrix. This theory uses a k-and ε-explicit coercivity result for the underlying sesquilinear form and shows, for example, that if |ε| ∼ k 2 , then classical overlapping additive Schwarz will perform optimally for the absorptive problem, provided the subdomain and coarse mesh diameters are carefully chosen. Extensive numerical experiments are given that support the theoretical results. While the theory applies to a certain weighted variant of GMRES, the experiments for both weighted and classical GMRES give comparable results. The theory for the absorptive case gives insight into how its domain decomposition approximations perform as preconditioners for the pure Helmholtz case ε = 0. At the end of the paper we propose a (scalable) multilevel preconditioner for the pure Helmholtz problem that has an empirical computation time complexity of about O(n 4/3) for solving finite element systems of size n = O(k 3), where we have chosen the mesh diameter h ∼ k −3/2 to avoid the pollution effect. Experiments on problems with h ∼ k −1 , i.e. a fixed number of grid points per wavelength, are also given.
L-Sweeps: A scalable, parallel preconditioner for the high-frequency Helmholtz equation
Journal of Computational Physics, 2020
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 • 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 propagate through the CDD. The new approach decouples the subdomains at each wave front, so that they can be processed in parallel, resulting in better parallel scalability than previously demonstrated in the literature. The method has an overall O (N/p) log ω empirical run-time for N = n d total degrees-of-freedom in a d-dimensional problem, frequency ω, and p = O(n) processors. We introduce the algorithm and provide a complexity analysis for our parallel implementation of the solver. We corroborate all claims in several two-and three-dimensional numerical examples involving constant, smooth, and discontinuous wave speeds.
Numerische Mathematik, 2000
We present a Lagrange multiplier based two-level domain decomposition method for solving iteratively large-scale systems of equations arising from the finite element discretization of high-frequency exterior Helmholtz problems. The proposed method is essentially an extension of the regularized FETI (Finite Element Tearing and Interconnecting) method to indefinite problems. Its two key ingredients are the regularization of each subdomain matrix by a complex interface lumped mass matrix, and the preconditioning of the interface problem by an auxiliary coarse problem. constructed to enforce at each iteration the orthogonality of the residual to a set of carefully chosen planar waves. We show numerically that the proposed method is scalable with respect to the mesh size, the subdomain size, and the wavenumber. We report performance results for a submarine application that highlight the efficiency of the proposed method for the solution of high frequency acoustic scattering problems discretized by finite elements.
Cornell University - arXiv, 2022
The convergence rate of domain decomposition methods (DDMs) strongly depends on the transmission condition at the interfaces between subdomains. Thus, an important aspect in improving the efficiency of such solvers is careful design of appropriate transmission conditions. In this work, we will develop an efficient solver for Helmholtz equations based on perfectly matched layers (PMLs) as transmission conditions at the interfaces within an optimised restricted additive Schwarz (ORAS) domain decomposition preconditioner, in both two and three dimensional domains. We perform a series of numerical simulations on a model problem and will assess the convergence rate and accuracy of our solutions compared to the situation where impedance boundary conditions are used.
Algebraic Multilevel Preconditioner for the Helmholtz Equation in Heterogeneous Media
SIAM Journal on Scientific Computing, 2009
An algebraic multilevel (ML) preconditioner is presented for the Helmholtz equation in heterogeneous media. It is based on a multilevel incomplete LDL T factorization and preserves the inherent (complex) symmetry of the Helmholtz equation. The ML preconditioner incorporates two key components for efficiency and numerical stability: symmetric maximum weight matchings and an inverse-based pivoting strategy. The former increases the block-diagonal dominance of the system, whereas the latter controls L −1 for numerical stability. When applied recursively, their combined effect yields an algebraic coarsening strategy, similar to algebraic multigrid methods, even for highly indefinite matrices. The ML preconditioner is combined with a Krylov subspace method and applied as a "black-box" solver to a series of challenging two-and three-dimensional test problems, mainly from geophysical seismic imaging. The numerical results demonstrate the robustness and efficiency of the ML preconditioner, even at higher frequency regimes.
An algebraic multigrid based shifted-Laplacian preconditioner for the Helmholtz equation
Journal of Computational Physics, 2007
A preconditioner defined by an algebraic multigrid cycle for a damped Helmholtz operator is proposed for the Helmholtz equation. This approach is well-suited for acoustic scattering problems in complicated computational domains and with varying material properties. The spectral properties of the preconditioned systems and the convergence of the GMRES method are studied with linear, quadratic, and cubic finite element discretizations. Numerical experiments are performed with two-dimensional problems describing acoustic scattering in a cross section of a car cabin and in a layered medium. Asymptotically the number of iterations grows linearly with respect to the frequency while for lower frequencies the growth is milder. The proposed preconditioner is particularly effective for low-frequency and mid-frequency problems.
A Non-Overlapping Domain Decomposition Method for the Exterior Helmholtz Problem
In this paper, we first show that the domain decomposition methods that are usually efficient for solving elliptic problems typically fail when applied to acoustics problems. Next, we present an alternative domain decomposition algorithm that is better suited for the exterior Helmholtz problem. We describe it in a formalism that can use either one or two Lagrange multiplier fields for solving the corresponding interface problem by a Krylov method. In order to improve convergence and ensure scalability with respect the number of subdomains, we propose two complementary preconditioning techniques. The first preconditioner is based on a spectral analysis of the resulting interface operator and targets the high frequency components of the error. The second preconditioner is based on a coarsening technique, employs plane waves, and addresses the low frequency components of the error. Finally, we show numerically that, using both preconditioners, the convergence rate of the proposed domain decomposition method is quasi independent of the number of elements in the mesh, the number of subdomains, and depends only weakly on the wave number, which makes this method uniquely suitable for solving large scale high frequency exterior acoustics problems.
A domain decomposition method for the Helmholtz equation in a multilayer domain
SIAM Journal on Scientific Computing, 1999
coupled system composed by the direct and adjoint Helmholtz equation and the optimality condition which varia-We present an iterative domain decomposition method to solve the Helmholtz equation and related optimal control problems. The tionally expresses that the control is optimal. This method proof of convergence of this method relies on energy techniques. actually solves at the same time the equations and the This method leads to efficient algorithms for the numerical resoluoptimization problem, whereas classical methods require tion of harmonic wave propagation problems in homogeneous and the iterated resolution of direct and adjoint problems in heterogeneous media. ᮊ 1997 Academic Press order to compute descent directions for a gradient-type method. The proposed method is easy to implement and naturally adapted to parallel computers, the use of which 1