Nested Domain Decomposition with Polarized Traces for the 2D Helmholtz Equation

Zepeda-Núñez, Leonardo; Demanet, Laurent

doi:10.1137/15m104582x

Cited by 16 publications

(30 citation statements)

References 89 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[29,30,60,110,137,155,167,177,178]. A recursive version that solves the 2-D sub-problems in a 3-D domain recursively by decomposing them into 1-D lines and sweeping can be found in [119], see also [176] for a similar idea. A recursive sweeping algorithm with low-order ABCs was already proposed earlier, see [1].…”

Section: Direct and Iterative Solvers After Discretization Of Equationmentioning

confidence: 99%

A Class of Iterative Solvers for the Helmholtz Equation: Factorizations, Sweeping Preconditioners, Source Transfer, Single Layer Potentials, Polarized Traces, and Optimized Schwarz Methods

Gander¹,

Zhang²

2019

SIAM Rev.

141

114

View full text Add to dashboard Cite

Solving time-harmonic wave propagation problems by iterative methods is a difficult task, and over the last two decades, an important research effort has gone into developing preconditioners for the simplest representative of such wave propagation problems, the Helmholtz equation. A specific class of these new preconditioners are considered here. They were developed by researchers with various backgrounds using formulations and notations that are very different, and all are among the most promising preconditioners for the Helmholtz equation.The goal of the present manuscript is to show that this class of preconditioners are based on a common mathematical principle, and they can all be formulated in the context of domain decomposition methods called optimized Schwarz methods. This common formulation allows us to explain in detail how and why all these methods work. The domain decomposition formulation also allows us to avoid technicalities in the implementation description we give of these recent methods.The equivalence of these methods with optimized Schwarz methods translates at the discrete level into equivalence with approximate block LU decomposition preconditioners, and we give in each case the algebraic version, including a detailed description of the approximations used. While we chose to use the Helmholtz equation for which these methods were developed, our notation is completely general and the algorithms we give are written for an arbitrary second order elliptic operator. The algebraic versions are even more general, assuming only a connectivity pattern in the discretization matrix.All these new methods studied here are based on sequential decompositions of the problem in space into a sequence of subproblems, and they have in their optimal form the property to lead to nilpotent iterations, like an exact block LU factorization. Using our domain decomposition formulation, we finally present an algorithm for two dimensional decompositions, i.e. decompositions that contain cross points, which is still nilpotent in its optimal form. Its approximation is currently an active area of research, and it would have been difficult to discover such an algorithm without the domain decomposition framework. 1 We use this simplest form of the Helmholtz equation only here at the beginning, and treat in the main part the more complete formulation given in Equation (11). 2 We assume here homogeneous Dirichlet boundary conditions and well-posedness for simplicity at the beginning, see Section 4 for more information.The overlap has no influence on the two step convergence property of the optimal parallel Schwarz method 5 . With J subdomains, as indicated in Figure 1 on the right, 4 The right hand side on the interface is in fact an exact or transparent boundary condition for the neighboring subdomain.5 This will be different if one uses approximations of the DtN operators, as we will see. * 10 we use Matlab notation for concatenating column vectors vertically to avoid having to use the transpose symbol T .

show abstract

Section: Direct and Iterative Solvers After Discretization Of Equationmentioning

confidence: 99%

A Class of Iterative Solvers for the Helmholtz Equation: Factorizations, Sweeping Preconditioners, Source Transfer, Single Layer Potentials, Polarized Traces, and Optimized Schwarz Methods

Gander¹,

Zhang²

2019

SIAM Rev.

141

114

View full text Add to dashboard Cite

show abstract

“…2, left) of size 2(L − 1)n 2 × 2(L − 1)n 2 . Theorem 1 of Zepeda-Núñez and Demanet (2016) gives that the solution of Eq. 14 is exactly the restriction of the solution of Eq.…”

Section: Reduction To a Surface Integral Equationmentioning

confidence: 99%

“…This reduction in storage, when combined with the reduced storage due to the relatively small number of GMRES iterations required for convergence, yields a smaller memory footprint for the outer GMRES iteration than methods requiring to update the full volume. It is possible to further reduce the memory footprint by using Bi-CGSTAB instead of GMRES, keeping the computational cost almost constant (Zepeda-Núñez and Demanet, 2015).…”

Section: Pipeliningmentioning

confidence: 99%

“…• Sweeping-like preconditioners (e.g., Gander and Nataf (2005); Engquist and Ying (2011a,b); Chen and Xiang (2013a,b); Stolk (2013); Vion and Geuzaine (2014); Liu and Ying (2015); Zepeda-Núñez and Demanet (2016)), which are a relatively recent domain decomposition based approach that has been shown to achieve linear or nearly-linear asymptotic complexity.…”

Section: Introductionmentioning

confidence: 99%

“…While most sweeping algorithms require visiting each subdomain in sequential fashion, Stolk (2015b) introduced a modified sweeping pattern, which changes the data dependencies during the sweeps to improve parallelism. Finally, Zepeda-Núñez and Demanet (2016) introduced the method of polarized traces, which reduces the solver's runtime by leveraging parallelism and fast summation methods. This paper builds on top of the general framework of the method of polarized traces, which we elaborate on in the sequel.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The method of polarized traces for the 3D Helmholtz equation

et al. 2019

Self Cite

View full text Add to dashboard Cite

We present a fast solver for the 3D high-frequency Helmholtz equation in heterogeneous, constant density, acoustic media. The solver is based on the method of polarized traces, coupled with distributed linear algebra libraries and pipelining to obtain an empirical online runtime O(max(1, R/n)N log N ) where N = n 3 is the total number of degrees of freedom and R is the number of right-hand sides. Such a favorable scaling is a prerequisite for large-scale implementations of full waveform inversion (FWI) in frequency domain.Suppose that L, the number of layers in the domain decomposition, scales as L ∼ n, i.e., each layer has a constant thickness in number of grid points * . Each layer is * This hypothesis is critical for obtaining a quasi-linear complexity algorithm and is related to the complexity of solving a quasi-

show abstract

Recent Results on Domain Decomposition Preconditioning for the High-Frequency Helmholtz Equation Using Absorption

Graham

Spence

Vainikko

2017

Modern Solvers for Helmholtz Problems

View full text Add to dashboard Cite

In this paper we present an overview of recent progress on the development and analysis of domain decomposition preconditioners for discretised Helmholtz problems, where the preconditioner is constructed from the corresponding problem with added absorption. Our preconditioners incorporate local subproblems that can have various boundary conditions, and include the possibility of a global coarse mesh. While the rigorous analysis describes preconditioners for the Helmholtz problem with added absorption, this theory also informs the development of efficient multilevel solvers for the "pure" Helmholtz problem without absorption. For this case, 2D experiments for problems containing up to about 50 wavelengths are presented. The experiments show iteration counts of order about O(n 0.2 ) and times (on a serial machine) of order about O(n α ), with α ∈ [1.3, 1.4] for solving systems of dimension n. This holds both in the pollution-free case corresponding to meshes with grid size O(k −3/2 ) (as the wavenumber k increases), and also for discretisations with a fixed number of grid points per wavelength, commonly used in applications. Parallelisation of the algorithms is also briefly discussed.

show abstract

Nested Domain Decomposition with Polarized Traces for the 2D Helmholtz Equation

Cited by 16 publications

References 89 publications

A Class of Iterative Solvers for the Helmholtz Equation: Factorizations, Sweeping Preconditioners, Source Transfer, Single Layer Potentials, Polarized Traces, and Optimized Schwarz Methods

A Class of Iterative Solvers for the Helmholtz Equation: Factorizations, Sweeping Preconditioners, Source Transfer, Single Layer Potentials, Polarized Traces, and Optimized Schwarz Methods

The method of polarized traces for the 3D Helmholtz equation

Recent Results on Domain Decomposition Preconditioning for the High-Frequency Helmholtz Equation Using Absorption

Contact Info

Product

Resources

About