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42
Sweeping preconditioner for the Helmholtz equation: moving perfectly matched layers
, 2010
"... The paper introduces the sweeping preconditioner, which is highly efficient for iterative solutions of the variablecoefficient Helmholtz equation including veryhighfrequency problems. The first central idea of this novel approach is to construct an approximate factorization of the discretized Hel ..."
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Cited by 46 (6 self)
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The paper introduces the sweeping preconditioner, which is highly efficient for iterative solutions of the variablecoefficient Helmholtz equation including veryhighfrequency 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 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 generalized minimal residual method (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 threedimensional case with some success. Numerical results are provided in both two and three dimensions to demonstrate the efficiency of this new approach. © 2011 Wiley Periodicals, Inc. 1
Discrete symbol calculus
, 2008
"... This paper deals with efficient numerical representation and manipulation of differential and integral operators as symbols in phasespace, i.e., functions of space x and frequency ξ. The symbol smoothness conditions obeyed by many operators in connection to smooth linear partial differential equati ..."
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Cited by 15 (9 self)
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This paper deals with efficient numerical representation and manipulation of differential and integral operators as symbols in phasespace, i.e., functions of space x and frequency ξ. The symbol smoothness conditions obeyed by many operators in connection to smooth linear partial differential equations allow to write fastconverging, nonasymptotic expansions in adequate systems of rational Chebyshev functions or hierarchical splines. The classical results of closedness of such symbol classes under multiplication, inversion and taking the square root translate into practical iterative algorithms for realizing these operations directly in the proposed expansions. Because symbolbased numerical methods handle operators and not functions, their complexity depends on the desired resolution N very weakly, typically only through log N factors. We present three applications to computational problems related to wave propagation: 1) preconditioning the Helmholtz equation, 2) decomposing wavefields into oneway components and 3) depthstepping in reflection seismology.
A scalable Helmholtz solver combining the shifted Laplace preconditioner with multigrid deflation, internal report
 Delft University of Technology, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft Institute of
"... A Helmholtz solver whose convergence is parameter independent can be obtained by combining the shifted Laplace preconditioner with multigrid deflation. To proof this claim, we develop a Fourier analysis of a twolevel variant of the algorithm proposed in [1]. In this algorithm those eigenvalues that ..."
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Cited by 5 (2 self)
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A Helmholtz solver whose convergence is parameter independent can be obtained by combining the shifted Laplace preconditioner with multigrid deflation. To proof this claim, we develop a Fourier analysis of a twolevel variant of the algorithm proposed in [1]. In this algorithm those eigenvalues that prevent the shifted Laplace preconditioner from being scalable are removed by deflation using multigrid vectors. Our analysis shows that the spectrum of the twogrid operator consists of a cluster surrounded by a few outliers, yielding a number of outer Krylov subspace iterations that remains constant as the wave number increases. Our analysis furthermore shows that the imaginary part of the shift in the twogrid operator can be made arbitrarily large without affecting the convergence. This opens promising perspectives on obtaining a very good preconditioner at very low cost. Numerical tests for problems with constant and nonconstant wave number illustrate our convergence theory. 3 4
A SOURCE TRANSFER DOMAIN DECOMPOSITION METHOD FOR HELMHOLTZ EQUATIONS IN UNBOUNDED DOMAIN
"... Abstract. We propose and study a domain decomposition method for solving the truncated perfectly matched layer (PML) approximation in bounded domain of Helmholtz scattering problems. The method is based on the decomposition of the domain into nonoverlapping layers and the idea of source transfer wh ..."
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Cited by 4 (0 self)
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Abstract. We propose and study a domain decomposition method for solving the truncated perfectly matched layer (PML) approximation in bounded domain of Helmholtz scattering problems. The method is based on the decomposition of the domain into nonoverlapping layers and the idea of source transfer which transfers the sources equivalently layer by layer so that the solution in the final layer can be solved using a PML method defined locally outside the last two layers. The convergence of the method is proved forthe case of constant wave number based on the analysis of the fundamental solution of the PML equation. The method can be used as an efficient preconditioner in the preconditioned GMRES method for solving discrete Helmholtz equations with constant and heterogeneous wave numbers. Numerical examples are included. Key words. Helmholtz equation, high frequency waves, PML, source transfer. 1. Introduction. We
A PARALLEL SWEEPING PRECONDITIONER FOR HETEROGENEOUS 3D HELMHOLTZ EQUATIONS∗
"... Abstract. A parallelization of a sweeping preconditioner for 3D Helmholtz equations without internal resonance is introduced and benchmarked for several challenging velocity models. The setup and application costs of the sequential preconditioner are shown to be O(γ2N4/3) and O(γN logN), where γ(ω) ..."
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Cited by 4 (3 self)
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Abstract. A parallelization of a sweeping preconditioner for 3D Helmholtz equations without internal resonance is introduced and benchmarked for several challenging velocity models. The setup and application costs of the sequential preconditioner are shown to be O(γ2N4/3) and O(γN logN), where γ(ω) denotes the modestly frequencydependent number of grid points per Perfectly Matched Layer. Several computational and memory improvements are introduced relative to using blackbox sparsedirect solvers for the auxiliary problems, and competitive runtimes and iteration counts are reported for highfrequency problems distributed over thousands of cores. Two opensource packages are released along with this paper: Parallel Sweeping Preconditioner (PSP) and the underlying distributed multifrontal solver, Clique.
Parallel multistep methods for linear evolution problems
 IMA J. Numer. Anal
"... Parallel multistep methods for linear evolution problems ..."
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Cited by 3 (1 self)
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Parallel multistep methods for linear evolution problems
The method of polarized traces for the 2D Helmholtz equation. ArXiv eprints
, 2014
"... We present a solver for the 2D highfrequency Helmholtz equation in heterogeneous acoustic media, with online parallel complexity that scales optimally as O(NL), where N is the number of volume unknowns, and L is the number of processors, as long as L grows at most like a small fractional power of N ..."
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Cited by 3 (0 self)
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We present a solver for the 2D highfrequency Helmholtz equation in heterogeneous acoustic media, with online parallel complexity that scales optimally as O(NL), where N is the number of volume unknowns, and L is the number of processors, as long as L grows at most like a small fractional power of N. The solver decomposes the domain into layers, and uses transmission conditions in boundary integral form to explicitly define “polarized traces”, i.e., up and downgoing waves sampled at interfaces. Local direct solvers are used in each layer to precompute traces of local Green’s functions in an embarrassingly parallel way (the offline part), and incomplete Green’s formulas are used to propagate interface data in a sweeping fashion, as a preconditioner inside a GMRES loop (the online part). Adaptive lowrank partitioning of the integral kernels is used to speed up their application to interface data. The method uses secondorder finite differences. The complexity scalings are empirical but motivated by an analysis of ranks of offdiagonal blocks of oscillatory integrals. They continue to hold in the context of standard geophysical community models such as BP and Marmousi 2, where convergence occurs in 5 to 10 GMRES iterations. 1
On solving the illconditioned system Ax= b: generalpurpose conditioners obtained from the boundarycollocation solution of the Laplace equation, using Trefftz expansions with multiple length scales. CMES: Computer Modeling in Engineering
 Sciences
, 2009
"... Abstract: Here we develop a general purpose pre/post conditioner T, to solve an illposed system of linear equations, Ax = b. The conditioner T is obtained in the course of the solution of the Laplace equation, through a boundarycollocation Trefftz method, leading to: Ty = x, where y is the vector ..."
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Cited by 3 (2 self)
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Abstract: Here we develop a general purpose pre/post conditioner T, to solve an illposed system of linear equations, Ax = b. The conditioner T is obtained in the course of the solution of the Laplace equation, through a boundarycollocation Trefftz method, leading to: Ty = x, where y is the vector of coefficients in the Trefftz expansion, and x is the boundary data at the discrete points on a unit circle. We show that the quality of the conditioner T is greatly enhanced by using multiple characteristic lengths (Multiple Length Scales) in the Trefftz expansion. We further show that T can be multiplicatively decomposed into a dilation TD and a rotation TR. For an oddordered A, we develop four conditioners based on the solution of the Laplace equation for Dirichlet boundary conditions, while for an evenordered A we develop four conditioners employing the Neumann boundary conditions. All these conditioners are wellbehaved and easily invertible. Several examples involving illconditioned A, such as the Hilbert matrices, those arising from the Method of Fundamental Solutions, those arising from veryhigh order polynomial interpolations, and those resulting from the solution of the firstkind Fredholm integral equations, are presented. The results demonstrate that the presently proposed conditioners result in very high computational efficiency and accuracy, when Ax = b is highly illconditioned, and b is noisy.