The paper introduces the sweeping preconditioner, which is highly efficient for iterative solutions of the variable coefficient Helmholtz equation including very high frequency problems. The first central idea of this novel approach is to construct an approximate factorization of the discretized Helmholtz equation by sweeping the domain layer by layer, starting from an absorbing layer or boundary condition. Given this specific order of factorization, the second central idea of this approach is to represent the intermediate matrices in the hierarchical matrix framework. In two dimensions, both the construction and the application of the preconditioners are of linear complexity. The GMRES solver with the resulting preconditioner converges in an amazingly small number of iterations, which is essentially independent of the number of unknowns. This approach is also extended to the three dimensional case with some success. Numerical results are provided in both two and three dimensions to demonstrate the efficiency of this new approach.
This paper explores a family of generalized sweeping preconditionners for Helmholtz problems with non-overlapping checkerboard partition of the computational domain. The domain decomposition procedure relies on high-order transmission conditions and cross-point treatments, which cannot scale without an efficient preconditioning technique when the number of subdomains increases. With the proposed approach, existing sweeping preconditioners, such as the symmetric Gauss-Seidel and parallel double sweep preconditioners, can be applied to checkerboard partitions with different sweeping directions (…
Additive Sweeping Preconditioner for the Helmholtz Equation
Department of Mathematics, Stanford University（斯坦福大学数学系）；Institute for Computational and Mathematical Engineering, Stanford University（斯坦福大学计算与数学工程研究所）
In this paper we consider sweeping preconditioners for time harmonic wave propagation in stratified media, especially in the presence of reflections. In the most famous class of sweeping preconditioners Dirichlet-to-Neumann operators for half-space problems are approximated through absorbing boundary conditions. In the presence of reflections absorbing boundary conditions are not accurate resulting in an unsatisfactory performance of these sweeping preconditioners. We explore the potential of using more accurate Dirichlet-to-Neumann operators within the sweep. To this end, we make use of the s…
This paper is the first one of two serial articles, whose goal is to prove convergence of HX Preconditioner (proposed by Hiptmair and Xu 2007) for Maxwell's equations with jump coefficients. In this paper we establish various extensions of the regular Helmholtz decomposition for edge finite element functions defined in three dimensional domains. The functions defined by the regular Helmholtz decompositions can preserve the zero tangential complement on faces and edges of polyhedral domains and some non-Lipchitz domains, and possess stability estimates with only a $logarithm$ factor. These regu…
Comparing domain decomposition preconditioners for non-conforming Helmholtz discretizations
University of Vienna（维也纳大学）；Sorbonne Université, Université Paris Cité, CNRS, INRIA, Laboratoire Jacques-Louis Lions（索邦大学，巴黎西岱大学，法国国家科学研究中心，法国国家信息与自动化研究所，雅克-路易·利翁斯实验室）；Georg-August-Universität Göttingen（哥廷根大学）
By utilizing the perfectly matched layer (PML) and source transfer techniques, the diagonal sweeping domain decomposition method (DDM) was recently developed for solving the high-frequency Helmholtz equation in $\mathbb{R}^n$, which uses $2^n$ sweeps along respective diagonal directions with checkerboard domain decomposition. Although this diagonal sweeping DDM is essentially multiplicative, it is highly suitable for parallel computing of the Helmholtz problem with multiple right-hand sides when combined with the pipeline processing since the number of sequential steps in each sweep is much sm…
An Overlapping Domain Decomposition Preconditioner for the Helmholtz equation
State Key Laboratory of Scientific and Engineering Computing, Chinese Academy of Sciences（中国科学院计算数学与科学工程计算重点实验室）；Department of Mathematics, University of South Carolina（南卡罗来纳大学数学系）
In this paper, we propose and test a novel diagonal sweeping domain decomposition method (DDM) with source transfer for solving the high-frequency Helmholtz equation in $\mathbb{R}^n$. In the method the computational domain is partitioned into overlapping checkerboard subdomains for source transfer with the perfectly matched layer (PML) technique, then a set of diagonal sweeps over the subdomains are specially designed to solve the system efficiently. The method improves the additive overlapping DDM (W. Leng and L. Ju, 2019) and the L-sweeps method (M. Taus, et al., 2019) by employing a more e…
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 …
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…
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）
One of the main tools for solving linear systems arising from the discretization of the Helmholtz equation is the shifted Laplace preconditioner, which results from the discretization of a perturbed Helmholtz problem $-Δu - (k^2 + i \varepsilon )u = f$ where $0 \neq \varepsilon \in \mathbb{R}$ is an absorption parameter. In this work we revisit the idea of combining the shifted Laplace preconditioner with two-level deflation and apply it to Helmholtz problems discretized with linear finite elements. We use the convergence theory of GMRES based on the field of values to prove that GMRES applied…
Time-harmonic solutions to the wave equation can be computed in the frequency or in the time domain. In the frequency domain, one solves a discretized Helmholtz equation, while in the time domain, the periodic solutions to a discretized wave equation are sought, e.g. by simulating for a long time with a time-harmonic forcing term. Disadvantages of the time-domain method are that the solutions are affected by temporal discretization errors and that the spatial discretization cannot be freely chosen, since it is inherited from the time-domain scheme. In this work we address these issues. Given a…
Two-level hybrid Schwarz preconditioners with preasymptotic piecewise-polynomial coarse spaces for the high-frequency Helmholtz equation
University College London（伦敦大学学院）；University of Bath（巴斯大学）；Sorbonne Université, Université Paris Cité, CNRS, INRIA, Laboratoire Jacques-Louis Lions（索邦大学，巴黎西岱大学，法国国家科学研究中心，法国国家信息与自动化研究所，雅克-路易·利翁斯实验室）
Multilevel Preconditioner with Stable Coarse Grid Corrections for the Helmholtz Equation
Xiamen University（厦门大学）；Nanjing University（南京大学）；Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）
We consider one-level additive Schwarz preconditioners for a family of Helmholtz problems with absorption and increasing wavenumber $k$. These problems are discretized using the Galerkin method with nodal conforming finite elements of any (fixed) order on meshes with diameter $h = h(k)$, chosen to maintain accuracy as $k$ increases. The action of the preconditioner requires solution of independent (parallel) subproblems (with impedance boundary conditions) on overlapping subdomains of diameter $H$ and overlap $δ\leq H$. The solutions of these subproblems are linked together using prolongation/…
We consider one-level additive Schwarz domain decomposition preconditioners for the Helmholtz equation with variable coefficients (modelling wave propagation in heterogeneous media), subject to boundary conditions that include wave scattering problems. Absorption is included as a parameter in the problem. This problem is discretised using $H^1$-conforming nodal finite elements of fixed local degree $p$ on meshes with diameter $h = h(k)$, chosen so that the error remains bounded with increasing $k$. The action of the one-level preconditioner consists of the parallel solution of problems on subd…
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 is the second one of two serial articles, whose goal is to prove convergence of HX Preconditioner (proposed by Hiptmair and Xu, 2007) for Maxwell's equations with jump coefficients. In this paper, based on the auxiliary results developed in the first paper (Hu, 2017), we establish a new regular Helmholtz decomposition for edge finite element functions in three dimensions, which is nearly stable with respect to a weight function. By using this Helmholtz decomposition, we give an analysis of the convergence of the HX preconditioner for the case with strongly discontinuous coefficients…
Solving the indefinite Helmholtz equation is not only crucial for the understanding of many physical phenomena but also represents an outstandingly-difficult benchmark problem for the successful application of numerical methods. Here we introduce a new approach for evolving efficient preconditioned iterative solvers for Helmholtz problems with multi-objective grammar-guided genetic programming. Our approach is based on a novel context-free grammar, which enables the construction of multigrid preconditioners that employ a tailored sequence of operations on each discretization level. To find sol…
Due to its significance in terms of wave phenomena a considerable effort has been put into the design of preconditioners for the Helmholtz equation. One option to derive a preconditioner is to apply a multigrid method on a shifted operator. In such an approach, the wavenumber is shifted by some imaginary value. This step is motivated by the observation that the shifted problem can be more efficiently handled by iterative solvers when compared to the standard Helmholtz equation. However, up to now, it is not obvious what the best strategy for the choice of the shift parameter is. It is well kno…
Discrete Edge Multiscale Space based Hybrid Schwarz Preconditioner for High-Frequency Helmholtz Problems
Eastern Institute for Advanced Study, Eastern Institute of Technology（宁波东方理工大学高级研究院）；School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）理工学院）；Department of Mathematics, The University of Hong Kong（香港大学数学系）；Department of Mathematics, Purdue University（普渡大学数学系）
In this paper, we present a data-driven approach to iteratively solve the discrete heterogeneous Helmholtz equation at high wavenumbers. In our approach, we combine classical iterative solvers with convolutional neural networks (CNNs) to form a preconditioner which is applied within a Krylov solver. For the preconditioner, we use a CNN of type U-Net that operates in conjunction with multigrid ingredients. Two types of preconditioners are proposed 1) U-Net as a coarse grid solver, and 2) U-Net as a deflation operator with shifted Laplacian V-cycles. Following our training scheme and data-augmen…
A two-level domain-decomposition preconditioner for the time-harmonic Maxwell's equations
Université Côte d’Azur, CNRS, LJAD（蔚蓝海岸大学，法国国家科学研究中心，拉格朗日分析几何实验室）；University of Strathclyde（思克莱德大学）；University of Bath（巴斯大学）；UPMC Univ Paris 06, LJLL（巴黎第六大学皮埃尔和玛丽·居里大学，拉格朗日分析实验室）
