This manuscript presents a fast direct solution technique for solving two dimensional wave scattering problems from quasi-periodic multilayered structures. The fast solver is built from the linear system that results from the discretization of a boundary integral formulation that is robust at Wood's anomalies. When the interface geometries are complex, the linear system is too large to be handled via dense linear algebra. The key building block of the proposed solver is a fast direct direct solver for the large sparse block system that corresponds to the discretization of boundary integral equations. The solver makes use of hierarchical matrix inversion techniques, has a cost that scales linearly with respect to the number of unknowns on the interfaces and the precomputation can be used for all choices of boundary data. By partitioning the remainder of the precomputation into parts based on their dependence on incident angle, the proposed direct solver is efficient for problems involving many incident angles like those that arise in applications. For example for a problem on an eleven layer geometry where the solution is desired for 287 incident angles, the proposed solution technique is 87 times faster than building a new fast direct solver for each new incident angle. An additional feature of the proposed solution technique is that solving a problem where an interface or layer property is changed require an update in the precomputation that cost linearly with respect to the number points on the affected interfaces with a small constant prefactor. The efficiency for modified geometries and multiple solves make the solution technique well suited for optimal design and inverse scattering applications.
Direct discretization of ) is possible via finite difference or finite element methods @cite_13 but it faces two challenges, (i) meshing to the interfaces to maintain accuracy and (ii) enforcing the radiation condition. Meshing can be effectively be handled using mesh generation software such as GMESH @cite_40 . Techniques such as perfectly matched layers @cite_14 can artificially enforce radiation conditions but lose accuracy if the points per wavelength remains fixed (the so-called effect) @cite_20 . Another alternative method for directly discretizing ) is the rigorous-coupled wave analysis (RCWA) or Fourier Modal Method. It is designed for multilayer gratings @cite_33 which is dependent on an iterative solve. While a Fourier factorization method @cite_29 @cite_36 can be used to accelerate convergence of an iterative solver, this solution approach is not ideal for problems with many right hand sides that arise in applications. An additional challenge of RCWA is that it is difficult to apply to arbitrary shaped interfaces.
@cite_14 @cite_33 @cite_36 @cite_29 @cite_40 @cite_13 @cite_20
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We consider the joint optimization of beamformers and linear receivers in a MIMO interference network. Each transmitter transmits a single beam corresponding to a rank-one precoder. When the number of users K is greater than the number of antennas at each terminal N, the maximum degrees of freedom is achieved via spatial interference alignment. Interference alignment is feasible for up to K = 2N−1 users, in which case there is a finite number of solutions to the alignment conditions. This number of solutions increases rapidly with N, and the solutions depend only on the cross-channel coefficients (i. e., they are independent of the direct channels). To maximize the achievable sum rate at high SNRs we therefore wish to select an aligned solution which is best matched to the direct channels. We evaluate the performance of this scheme for large K and N, assuming that the solution is the best out of a random subset of aligned solutions. We then compare numerically this performance with the performance of previously proposed numerical (e. g., forward-backward) techniques for optimizing beams, and a new technique which tracks the local optimum as the SNR is incrementally increased, similar to a homotopy method for improving convergence properties. We observe that the incremental technique typically achieves better performance than the previously proposed methods. SUMMARY In this paper, we address the solution of three-dimensional heterogeneous Helmholtz problems discretized with second-order finite difference methods with application to acoustic waveform inversion in geophysics. In this setting, the numerical simulation of wave propagation phenomena requires the approximate solution of possibly very large indefinite linear systems of equations. For that purpose, we propose and analyse an iterative two-grid method acting on the Helmholtz operator where the coarse grid problem is solved inaccurately. A cycle of a multigrid method applied to a complex shifted Laplacian operator is used as a preconditioner for the approximate solution of this coarse problem. A single cycle of the new method is then used as a variable preconditioner of a flexible Krylov subspace method. We analyse the properties of the resulting preconditioned operator by Fourier analysis. Numerical results demonstrate the effectiveness of the algorithm on three-dimensional applications. The proposed numerical method allows us to solve three-dimensional wave propagation problems even at high frequencies on a reasonable number of cores of a distributed memory computer. Copyright © 2012 John Wiley & Sons, Ltd. We are concerned with a finite element approximation for time-harmonic wave propagation governed by the Helmholtz equation. The usually oscillatory behavior of solutions, along with numerical dispersion, render standard finite element methods grossly inefficient already in medium-frequency regimes. As an alternative, methods that incorporate information about the solution in the form of plane waves have been proposed. We focus on a class of Trefftz-type discontinuous Galerkin methods that employs trial and test spaces spanned by local plane waves. In this paper we give a priori convergence estimates for the h -version of these plane wave discontinuous Galerkin methods in two dimensions. To that end, we develop new inverse and approximation estimates for plane waves and use these in the context of duality techniques. Asymptotic optimality of the method in a mesh dependent norm can be established. However, the estimates require a minimal resolution of the mesh beyond what it takes to resolve the wavelength. We give numerical evidence that this requirement cannot be dispensed with. It reflects the presence of numerical dispersion. In this paper we introduce a novel linear precoding technique. The approach used for the design of the precoding matrix is general and the resulting algorithm can address several optimization criteria with an arbitrary number of antennas at the user terminals. We have achieved this by designing the precoding matrices in two steps. In the first step we minimize the overlap of the row spaces spanned by the effective channel matrices of different users using a new cost function. In the next step, we optimize the system performance with respect to specific optimization criteria assuming a set of parallel single- user MIMO channels. By combining the closed form solution with Tomlinson-Harashima precoding we reach the maximum sum-rate capacity when the total number of antennas at the user terminals is less or equal to the number of antennas at the base station. By iterating the closed form solution with appropriate power loading we are able to extract the full diversity in the system and reach the maximum sum-rate capacity in case of high multi-user interference. Joint processing over a group of multi-user MIMO channels in different frequency and time slots yields maximum diversity regardless of the level of multi-user interference.