### Additive Sweeping Preconditioner for the Helmholtz Equation

, 2015

"... We introduce a new additive sweeping preconditioner for the Helmholtz equation based on the perfect matched layer (PML). This method divides the domain of interest into thin layers and proposes a new transmission condition between the subdomains where the emphasis is on the boundary values of the in ..."

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We introduce a new additive sweeping preconditioner for the Helmholtz equation based on the perfect matched layer (PML). This method divides the domain of interest into thin layers and proposes a new transmission condition between the subdomains where the emphasis is on the boundary values of the intermediate waves. This approach can be viewed as an effective approximation of an additive decomposition of the solution operator. When combined with the standard GMRES solver, the iteration number is essentially independent of the frequency. Several numerical examples are tested to show the efficiency of this new approach.

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"... Abstract. The Lippmann–Schwinger equation is an integral equation formulation for acous-tic and electromagnetic scattering from an inhomogeneous medium and quantum scattering from a localized potential. We present the sparsifying preconditioner for accelerating the iterative so-lution of the Lippman ..."

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Abstract. The Lippmann–Schwinger equation is an integral equation formulation for acous-tic and electromagnetic scattering from an inhomogeneous medium and quantum scattering from a localized potential. We present the sparsifying preconditioner for accelerating the iterative so-lution of the Lippmann–Schwinger equation. This new preconditioner transforms the discretized Lippmann–Schwinger equation into sparse form and leverages the efficient sparse linear algebra algo-rithms for computing an approximate inverse. This preconditioner is efficient and easy to implement. When combined with standard iterative methods, it results in almost frequency-independent iteration counts. We provide two- and three-dimensional numerical results to demonstrate the effectiveness of this new preconditioner. Key words. Lippmann–Schwinger equation, acoustic and electromagnetic scattering, quantum scattering, preconditioner, sparse linear algebra

### Preconditioning the Helmholtz Equation via Row-

"... 3D frequency-domain full waveform inversion relies on being able to efficiently solve the 3D Helmholtz equation. Iterative methods require sophisticated preconditioners because the Helmholtz matrix is typically indefinite. We review a preconditioning technique that is based on row-projections. Notab ..."

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3D frequency-domain full waveform inversion relies on being able to efficiently solve the 3D Helmholtz equation. Iterative methods require sophisticated preconditioners because the Helmholtz matrix is typically indefinite. We review a preconditioning technique that is based on row-projections. Notable advantages of this preconditioner over existing ones are that it has low algorithmic complexity, is easily parallelizable and extendable to time-harmonic vector equations. 3D frequency-domain full waveform inversion relies on being able to efficiently solve the 3D Helmholtz equation. A direct factorization of the matrix –as is commonly done for 2D waveform inversion – is infeasible because of the huge memory requirements, although some progress has been recently made in this area (Wang et al., 2011). Iterative methods have a small memory imprint, but require sophisticated preconditioners because the matrix is indefinite. Over the past years, several preconditioners have been proposed in the literature, and we briefly discuss a few. A good overview of classical preconditioning techniques, such as ILU and Schwarz is given by Osei-Kuffuor and Saad (2010). Sourbier et al. (2011) propose a hybrid method based on these techniques. Preconditioners based on the shifted Laplacian, first proposed by Bayliss et al. (1983), were further developed by Erlangga et al. (2004) who use a multi-grid method in conjunction with a complex shift; Riyanti et al. (2007) discuss a parallelization. Stolk

### Summary

"... The performance and design of a deterministic interleaver for short frame turbo codes is considered in this paper. The main characteristic of this class of deterministic interleaver is that their algebraic design selects the best permutation generator such that the points in smaller subsets of the i ..."

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The performance and design of a deterministic interleaver for short frame turbo codes is considered in this paper. The main characteristic of this class of deterministic interleaver is that their algebraic design selects the best permutation generator such that the points in smaller subsets of the interleaved output are uniformly spread over the entire range of the information data frame. It is observed that the interleaver designed in this manner improves the minimum distance of first few spectral lines of minimum distance spectrum. Finally we introduce a circular shift in the permutation function to reduce the correlation between the parity bits corresponding to the original and interleaved data frames to improve the decoding capability of MAP decoder. The design is focused on combining good permutations with de-correlation property. Our solution to design a deterministic interleaver outperforms the semi-random interleavers and the deterministic interleavers reported in the literature. (c) 2007

### unknown title

"... A sweeping preconditioner for Yee’s finite difference approximation of time-harmonic Maxwell’s equations ..."

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A sweeping preconditioner for Yee’s finite difference approximation of time-harmonic Maxwell’s equations

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"... difference grid, is generalized to an unstructured mesh with finite elements. The method dramatically reduces the number of GMRES iterations necessary for convergence, resulting in an almost linear complexity solver. Numerical examples including electromagnetic cloaking simulations are presented to ..."

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difference grid, is generalized to an unstructured mesh with finite elements. The method dramatically reduces the number of GMRES iterations necessary for convergence, resulting in an almost linear complexity solver. Numerical examples including electromagnetic cloaking simulations are presented to demonstrate the efficiency of the proposed method. 2012 Elsevier Inc. All rights reserved. the number of elements per wavelength to be increased if the same level of accuracy is desired at a higher frequency. If the domain is K wavelengths wide in each dimension, the number of degrees of freedom N is at least of order O(Kd), where the dimension is d = 2, 3; for high frequencies, this number becomes extremely large. Second, the oscillatory nature of the dyadic Green’s function for Maxwell’s equations causes the stiffness matrix to be highly indefinite and ill-conditioned.

### SWEEPINGPRECONDITIONERFORTHEHELMHOLTZ EQUATION: MOVING PERFECTLY MATCHED LAYERS*

"... Abstract. This paper introduces a new sweeping preconditioner for the iterative solution of the variable coefficient Helmholtz equation in two and three dimensions. The algorithms follow the general structure of constructing an approximate LDLt factorization by eliminating the unknowns layer by laye ..."

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Abstract. This paper introduces a new sweeping preconditioner for the iterative solution of the variable coefficient Helmholtz equation in two and three dimensions. The algorithms follow the general structure of constructing an approximate LDLt factorization by eliminating the unknowns layer by layer starting from an absorbing layer or boundary condition. The central idea of this paper is to approximate the Schur complement matrices of the factorization using moving perfectly matched layers (PMLs) introduced in the interior of the domain. Applying each Schur complement matrix is equivalent to solving a quasi-1D problem with a banded LU factorization in the 2D case and to solving a quasi-2D problem with a multifrontal method in the 3D case. The resulting preconditioner has linear application cost, and the preconditioned iterative solver converges in a number of iterations that is essentially independent of the number of unknowns or the frequency. Numerical results are presented in both two and three dimensions to demonstrate the efficiency of this new preconditioner.

### unknown title

"... Noname manuscript No. (will be inserted by the editor) Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: What is the largest shift for which wavenumber-independent convergence is guaranteed? ..."

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Noname manuscript No. (will be inserted by the editor) Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: What is the largest shift for which wavenumber-independent convergence is guaranteed?

### OPTIMIZED SCHWARZ METHODS FOR CIRCULAR DOMAIN DECOMPOSITIONS WITH OVERLAP

"... Abstract. Optimized Schwarz methods are based on transmission conditions between subdo-mains which are optimized for the problem class that is being solved. Such optimizations have been performed for many different types of partial differential equations, but almost exclusively based on the assumpti ..."

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Abstract. Optimized Schwarz methods are based on transmission conditions between subdo-mains which are optimized for the problem class that is being solved. Such optimizations have been performed for many different types of partial differential equations, but almost exclusively based on the assumption of straight interfaces. We study in this paper the influence of curvature on the opti-mization, and we obtain four interesting new results: first, we show that the curvature does indeed enter the optimized parameters and the contraction factor estimates. Second, we had to develop an asymptotically accurate approximation technique, based on Turán type inequalities in our case, to solve the much harder optimization problem on the curved interface, and this approximation tech-nique will also be applicable to currently too complex best approximation problems in the area of optimized Schwarz methods. Third, we show that one can obtain transmission conditions from a simple circular model decomposition which have also been found using microlocal analysis, but that these are not the best choices for the performance of the optimized Schwarz method. And finally, we find that in the case of curved interfaces, optimized Schwarz methods are not necessarily convergent for all admissible parameters. Our optimization leads however to parameter choices that give the same good performance for a circular decomposition as for a straight interface decomposition. We illustrate our analysis with numerical experiments.