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43
Preasymptotic error analysis of CIPFEM and FEM for the Helmholtz equation with high wave number. Part I: linear version
, 2014
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Efficient scalable algorithms for hierarchically semiseparable matrices
 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 ..."
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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 rankrevealing 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 finegrained level. We have appplied our new parallel HSSembedded 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 largescale realworld problems. Our parallel strategies can be easily adapted to the parallelization of the other rank structured methods. Key words. HSS matrix, parallel HSS algorithm, lowrank property, compression, HSS construction, direct solver AMS subject classifications. 15A23, 65F05, 65F30, 65F50
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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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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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.
SWEEPING PRECONDITIONERS FOR ELASTIC WAVE PROPAGATION WITH SPECTRAL ELEMENT METHODS
, 2013
"... Abstract.We present a parallel preconditioning method for the iterative solution of the timeharmonic elastic wave equation which makes use of higherorder 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 timeharmonic elastic wave equation which makes use of higherorder 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 largescale problems from exploration geophysics are presented.
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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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
NEAROPTIMAL PERFECTLY MATCHED LAYERS FOR INDEFINITE HELMHOLTZ PROBLEMS
, 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 nearbest 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 nearbest 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 threeterm 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, NeumanntoDirichlet 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 wavelike phenomena.
50 Years of Time Parallel Time Integration
"... 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, spacetime multigrid methods
Recursive Sweeping Preconditioner for the 3D Helmholtz Equation
, 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 ..."
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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 quasi2D auxiliary problems introduced in the 3D sweeping preconditioner. Compared to the nonrecursive 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 compared with the nonrecursive sweeping preconditioner to demonstrate the efficiency of the new approach.
POLLUTIONFREE FINITE DIFFERENCE SCHEMES FOR NONHOMOGENEOUS HELMHOLTZ EQUATION
"... Abstract. In this paper, we develop pollutionfree finite difference schemes for solving the nonhomogeneous Helmholtz equation in one dimension. A family of highorder algorithms is derived by applying the Taylor expansion and imposing the conditions that the resulting finite difference schemes sat ..."
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Abstract. In this paper, we develop pollutionfree finite difference schemes for solving the nonhomogeneous Helmholtz equation in one dimension. A family of highorder algorithms is derived by applying the Taylor expansion and imposing the conditions that the resulting finite difference schemes satisfied the original equation and the boundary conditions to certain degrees. The most attractive features of the proposed schemes are: first, the new difference schemes have a 2n−order of rate of convergence and are pollutionfree. Hence, the error is bounded even for the equation at high wave numbers. Secondly, the resulting difference scheme is simple, namely it has the same structure as the standard threepoint central differencing regardless the order of accuracy. Convergence analysis is presented, and numerical simulations are reported for the nonhomogeneous Helmholtz equation with both constant and varying wave numbers. The computational results clearly confirm the superior performance of the proposed schemes.