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Sweeping Preconditioner for the Helmholtz Equation: Moving Perfectly Matched Layers,” Multiscale Model. (2011)

by B Engquist, L Ying
Venue:Simul.
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Pre-asymptotic error analysis of CIP-FEM and FEM for the Helmholtz equation with high wave number. Part I: linear version

by Haijun Wu , 2014
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Abstract - Cited by 11 (3 self) - Add to MetaCart
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Efficient scalable algorithms for hierarchically semiseparable matrices

by Shen Wang, Xiaoye S. Li, Jianlin Xia, Yingchong Situ, Maarten V. De Hoop - Submitted SIAM Journal on Scientific Computing , 2012
"... Abstract. Hierarchically semiseparable (HSS) matrix algorithms are emerging techniques in constructing the superfast direct solvers for both dense and sparse linear systems. Here, we develope a set of novel parallel algorithms for the key HSS operations that are used for solving large linear systems ..."
Abstract - Cited by 5 (0 self) - Add to MetaCart
Abstract. Hierarchically semiseparable (HSS) matrix algorithms are emerging techniques in constructing the superfast direct solvers for both dense and sparse linear systems. Here, we develope a set of novel parallel algorithms for the key HSS operations that are used for solving large linear systems. These include the parallel rank-revealing QR factorization, the HSS constructions with hierarchical compression, the ULV HSS factorization, and the HSS solutions. The HSS tree based parallelism is fully exploited at the coarse level. The BLACS and ScaLAPACK libraries are used to facilitate the parallel dense kernel operations at the fine-grained level. We have appplied our new parallel HSS-embedded multifrontal solver to the anisotropic Helmholtz equations for seismic imaging, and were able to solve a linear system with 6.4 billion unknowns using 4096 processors, in about 20 minutes. The classical multifrontal solver simply failed due to high demand of memory. To our knowledge, this is the first successful demonstration of employing the HSS algorithms in solving the truly large-scale real-world problems. Our parallel strategies can be easily adapted to the parallelization of the other rank structured methods. Key words. HSS matrix, parallel HSS algorithm, low-rank property, compression, HSS construction, direct solver AMS subject classifications. 15A23, 65F05, 65F30, 65F50
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...en leads to very large sparse matrices A which are highly indefinite and ill-conditioned. It has been observed that, in the direct factorization of A the dense intermediate matrix may be compressible =-=[11, 25]-=-. We implemented a parallel multifrontal factorization method for such a matrix A based on the multifrontal method [9] with the nested dissection reordering [12]. In nested dissection, an n × n mesh i...

A SOURCE TRANSFER DOMAIN DECOMPOSITION METHOD FOR HELMHOLTZ EQUATIONS IN UNBOUNDED DOMAIN

by Zhiming Chen, Xueshuang Xiang
"... 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 non-overlapping layers and the idea of source transfer wh ..."
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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 non-overlapping 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
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...rid methods. We refer to the review articles Erlangga [15], Osei-Kuffuor and Saad [22] and the references therein for further information. Our work is inspired by the recent work of Engquist and Ying =-=[14]-=- in which a sweeping preconditioner is constructed by an approximate LDL t factorization which eliminates the unknowns layer by layer. The Schur complement matrix of the factorization is approximated ...

SWEEPING PRECONDITIONERS FOR ELASTIC WAVE PROPAGATION WITH SPECTRAL ELEMENT METHODS

by Paul Tsuji, Jack Poulson, Lexing Ying , 2013
"... Abstract.We present a parallel preconditioning method for the iterative solution of the time-harmonic elastic wave equation which makes use of higher-order spectral elements to reduce pollution error. In particular, the method leverages perfectly matched layer boundary conditions to efficiently appr ..."
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Abstract.We present a parallel preconditioning method for the iterative solution of the time-harmonic elastic wave equation which makes use of higher-order spectral elements to reduce pollution error. In particular, the method leverages perfectly matched layer boundary conditions to efficiently approximate the Schur complement matrices of a block LDLT factorization. Both sequential and parallel versions of the algorithm are discussed and results for large-scale problems from exploration geophysics are presented.
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...4305, USA. lexing@math.stanford.edu Article published by EDP Sciences c© EDP Sciences, SMAI 2014 434 P. TSUJI ET AL. degrade as the frequency is increased [9], and so several sweeping preconditioners =-=[6,7]-=- were recently introduced in order to alleviate this problem in the context of Helmholtz equations. Both of the original sweeping preconditioner approaches approximate a block LDLT factorization organ...

A PARALLEL SWEEPING PRECONDITIONER FOR HETEROGENEOUS 3D HELMHOLTZ EQUATIONS∗

by Jack Poulson, Siwei Li, Lexing Ying
"... 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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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 frequency-dependent number of grid points per Perfectly Matched Layer. Several computational and memory improvements are introduced relative to using black-box sparse-direct solvers for the auxiliary problems, and competitive runtimes and iteration counts are reported for high-frequency problems distributed over thousands of cores. Two open-source packages are released along with this paper: Parallel Sweeping Preconditioner (PSP) and the underlying distributed multifrontal solver, Clique.
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...ω3), every linear solve required Ω(ω4) work with iterative techniques. Engquist and Ying recently introduced two classes of sweeping preconditioners for Helmholtz equations without internal resonance =-=[14, 15]-=-: Both approaches approximate a block LDLT factorization of the Helmholtz operator in block tridiagonal form in a manner which exploits a radiation boundary condition. The first approach performs a bl...

The method of polarized traces for the 2D Helmholtz equation. ArXiv e-prints

by Laurent Demanet , 2014
"... We present a solver for the 2D high-frequency 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 ..."
Abstract - Cited by 3 (0 self) - Add to MetaCart
We present a solver for the 2D high-frequency 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 con-ditions in boundary integral form to explicitly define “polarized traces”, i.e., up- and down-going 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 low-rank partitioning of the integral kernels is used to speed up their application to interface data. The method uses second-order finite differences. The complexity scalings are empirical but motivated by an analysis of ranks of off-diagonal 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
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...ach can be traced back to the AILU method of Gander and Nataf in 2001 [35, 37]. The first linear complexity claim was perhaps made in the work of Engquist and Ying on sweeping preconditioners in 2011 =-=[28, 29]-=- – a special kind of domain decomposition into grid-spacing-thin layers. In 2013, Stolk [65] restored the flexibilty to consider coarser layerings, with a domain decomposition method that realizes int...

NEAR-OPTIMAL PERFECTLY MATCHED LAYERS FOR INDEFINITE HELMHOLTZ PROBLEMS

by Vladimir Druskin, Leonid Knizhnerman, Mims Eprint, Vladimir Druskin, Leonid Knizhnerman , 2013
"... Abstract. A new construction of an absorbing boundary condition for indefinite Helmholtz problems on unbounded domains is presented. This construction is based on a near-best uniform rational interpolant of the inverse square root function on the union of a negative and positive real interval, desig ..."
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Abstract. A new construction of an absorbing boundary condition for indefinite Helmholtz problems on unbounded domains is presented. This construction is based on a near-best uniform rational interpolant of the inverse square root function on the union of a negative and positive real interval, designed with the help of a classical result by Zolotarev. Using Krein’s interpretation of a Stieltjes continued fraction, this interpolant can be converted into a three-term finite difference discretization of a perfectly matched layer (PML) which converges exponentially fast in the number of grid points. The convergence rate is asymptotically optimal for both propagative and evanescent wave modes. Several numerical experiments and illustrations are included. Key words. Helmholtz equation, Neumann-to-Dirichlet map, perfectly matched layer, rational approximation, Zolotarev problem, continued fraction AMS subject classifications. 35J05, 65N06, 65N55, 30E10, 65D25 1. Introduction. An important task in science and engineering is the numerical solution of a partial differential equation (PDE) on an unbounded spatial domain. Unbounded spatial domains need to be truncated for computational purposes and this turns out to be particularly difficult when the PDE models wave-like phenomena.
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...ments of certain PDE discretization matrices, and they became a crucial component of modern Helmholtz preconditioners, such as Schwarz domain decomposition [12, 23] and the sweeping preconditioner in =-=[22]-=-. Preliminary results have shown successful application to a multilevel domain decomposition preconditioner, and a related Schlumberger patent application is pending. 20 V. DRUSKIN, S. GÜTTEL, AND L....

50 Years of Time Parallel Time Integration

by Martin J. Gander
"... Time parallel time integration methods have received renewed interest over the last decade because of the advent of massively parallel computers, which is mainly due to the clock speed limit reached on today’s processors. When solving time dependent partial differential equations, the time directi ..."
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Time parallel time integration methods have received renewed interest over the last decade because of the advent of massively parallel computers, which is mainly due to the clock speed limit reached on today’s processors. When solving time dependent partial differential equations, the time direction is usually not used for parallelization. But when parallelization in space saturates, the time direction offers itself as a further direction for parallelization. The time direction is however special, and for evolution problems there is a causality principle: the solution later in time is affected (it is even determined) by the solution earlier in time, but not the other way round. Algorithms trying to use the time direction for parallelization must therefore be special, and take this very different property of the time dimension into account. We show in this chapter how time domain decomposition methods were invented, and give an overview of the existing techniques. Time parallel methods can be classified into four different groups: methods based on multiple shooting, methods based on domain decomposition and waveform relaxation, space-time multigrid methods
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...ized to N iterations with N subdomains, or one iteration when using an alternating sweep, and is the underlying mechanism for the good convergence of the sweeping preconditioner recently presented in =-=[16]-=-. Since the DtN operators are in general expensive, so-called optimized Schwarz waveform relaxation methods use local approximations; for a complete treatment of advection reaction diffusion equations...

2010] On the generalization of wavelet diagonal preconditioning to the Helmholtz equation. Arxiv preprint arXiv:1010.4764

by Christiaan C. Stolk
"... We present a preconditioning method for the multi-dimensional Helmholtz equation with smoothly varying coefficient. The method is based on a frame of functions, that approximately separates components associated with different singular values of the op-erator. For the small singular values, correspo ..."
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We present a preconditioning method for the multi-dimensional Helmholtz equation with smoothly varying coefficient. The method is based on a frame of functions, that approximately separates components associated with different singular values of the op-erator. For the small singular values, corresponding to propagating waves, the frame functions are constructed using ray-theory. A series of 2-D numerical experiments demon-strates that the number of iterations required for convergence is small and independent of the frequency. In this sense the method is optimal. Acknowledgement This research was partly funded by the Netherlands Organisation for
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...α(x)ω Lc(x) , (1) where c(x) and α(x) are positive, smoothly varying coefficents, ω is the frequency parameter, and L is a typical length of the domain, which is inserted to make α dimensionless. See =-=[3, 11, 14, 23, 27, 12, 13]-=- for examples of preconditioners from the literature. For elliptic equations there are good preconditioning methods, such as multigrid and wavelet diagonal preconditioning [6, 8]. Their effect can be ...

Recursive Sweeping Preconditioner for the 3D Helmholtz Equation

by Fei Liu, Lexing Ying , 2015
"... This paper introduces the recursive sweeping preconditioner for the numerical solution of the Helmholtz equation in 3D. This is based on the earlier work of the sweeping preconditioner with the moving perfectly matched layers (PMLs). The key idea is to apply the sweeping preconditioner recursively t ..."
Abstract - Cited by 1 (1 self) - Add to MetaCart
This paper introduces the recursive sweeping preconditioner for the numerical solution of the Helmholtz equation in 3D. This is based on the earlier work of the sweeping preconditioner with the moving perfectly matched layers (PMLs). The key idea is to apply the sweeping preconditioner recursively to the quasi-2D auxiliary problems introduced in the 3D sweeping preconditioner. Compared to the non-recursive 3D sweeping preconditioner, the setup cost of this new approach drops from O(N4/3) to O(N), the application cost per iteration drops from O(N logN) to O(N), and the iteration count only increases mildly when combined with the standard GMRES solver. Several numerical examples are tested and the results are com-pared with the non-recursive sweeping preconditioner to demonstrate the efficiency of the new approach.
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...itioner, the method uses the absorbing boundary condition (ABC), which is less effective compared to the PML, and hence the iteration count grows much more rapidly. Since the sweeping preconditioners =-=[6, 7]-=- were proposed, there have been a number of exciting developments for the numerical solutions of the high frequency Helmholtz equation, including but not limited to [15, 14, 18, 16, 17, 19, 2, 3, 20]....

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