Siegfried Cools

Local Fourier analysis of the complex shifted Laplacian preconditioner for Helmholtz problems

SUMMARY In this paper, we solve the Helmholtz equation with multigrid preconditioned Krylov subspace methods. The class of shifted Laplacian preconditioners is known to significantly speed up Krylov convergence. However, these preconditioners have a parameter β∈R, a measure of the complex shift. Because of contradictory requirements for the multigrid and Krylov convergence, the choice of this shift parameter can be a bottleneck in applying the method. In this paper, we propose a wavenumber-dependent minimal complex shift parameter, which is predicted by a rigorous k-grid local Fourier analysis (LFA) of the multigrid scheme. We claim that, given any (regionally constant) wavenumber, this minimal complex shift parameter provides the reader with a parameter choice that leads to efficient Krylov convergence. Numerical experiments in one and two spatial dimensions validate the theoretical results. It appears that the proposed complex shift is both the minimal requirement for a multigrid V-cycle to converge and being near optimal in terms of Krylov iteration count. Copyright

An Efficient Multigrid Calculation of the Far Field Map for Helmholtz and Schrödinger Equations

In this paper we present a new highly efficient calculation method for the far field amplitude pattern that arises from scattering problems governed by the

On rounding error resilience, maximal attainable accuracy and parallel performance of the pipelined Conjugate Gradients method for large-scale linear systems in PETSc

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A new level‐dependent coarse grid correction scheme for indefinite Helmholtz problems

-dimensional Helmholtz equation and, by extension, Schrodingers equation. The new technique is based upon a reformulation of the classical real-valued Greens function integral for the far field amplitude to an equivalent integral over a complex domain. It is shown that the scattered wave, which is essential for the calculation of the far field integral, can be computed very efficiently along this complex contour (or manifold, in multiple dimensions). Using the iterative multigrid method as a solver for the discretized damped scattered wave system, the proposed approach results in a fast and scalable calculation method for the far field map. The complex contour method is successfully validated on Helmholtz and Schrodinger model problems in two and three spatial dimensions, and multigrid convergence results are provided to substantiate the wavenumbe...

Hard Faults and Soft-Errors: Possible Numerical Remedies in Linear Algebra Solvers

Pipelined Krylov solvers typically display better strong scaling compared to standard Krylov methods for large linear systems. The synchronization bottleneck is mitigated by overlapping time-consuming global communications with computations. To achieve this hiding of communication, pipelined methods feature additional recurrence relations on auxiliary variables. This paper analyzes why rounding error effects have a significantly larger impact on the accuracy of pipelined algorithms. An algebraic model for the accumulation of rounding errors in the (pipelined) CG algorithm is derived. Furthermore, an automated residual replacement strategy is proposed to reduce the effect of rounding errors on the final solution. MPI parallel performance tests implemented in PETSc on an Intel Xeon X5660 cluster show that the pipelined CG method with automated residual replacement is more resilient to rounding errors while maintaining the efficient parallel performance obtained by pipelining.

A fast and robust computational method for the ionization cross sections of the driven Schrödinger equation using an O ( N ) multigrid-based scheme

SUMMARY In this paper, we construct and analyze a level-dependent coarse grid correction scheme for indefinite Helmholtz problems. This adapted multigrid (MG) method is capable of solving the Helmholtz equation on the finest grid using a series of MG cycles with a grid-dependent complex shift, leading to a stable correction scheme on all levels. It is rigorously shown that the adaptation of the complex shift throughout the MG cycle maintains the functionality of the two-grid correction scheme, as no smooth modes are amplified in or added to the error. In addition, a sufficiently smoothing relaxation scheme should be applied to ensure damping of the oscillatory error components. Numerical experiments on various benchmark problems show the method to be competitive with or even outperform the current state-of-the-art MG-preconditioned Krylov methods, for example, complex shifted Laplacian preconditioned flexible GMRES. Copyright

Analyzing the Effect of Local Rounding Error Propagation on the Maximal Attainable Accuracy of the Pipelined Conjugate Gradient Method

On future large-scale systems, the mean time between failures (MTBF) of the system is expected to decrease so that many faults could occur during the solution of large problems. Consequently, it becomes critical to design parallel numerical linear algebra kernels that can survive faults. In that framework, we investigate the relevance of approaches relying on numerical techniques, which might be combined with more classical techniques for real large-scale parallel implementations. Our main objective is to provide robust resilient schemes so that the solver may keep converging in the presence of the hard fault without restarting the calculation from scratch. For this purpose, we study interpolation-restart (IR) strategies. For a given numerical scheme, the IR strategies consist of extracting relevant information from available data after a fault. After data extraction, a well-selected part of the missing data is regenerated through interpolation strategies to constitute a meaningful input to restart the numerical algorithm. In this paper, we revisit a few state-of-the-art methods in numerical linear algebra in the light of our IR strategies. Through a few numerical experiments, we illustrate the respective robustness of the resulting resilient schemes with respect to the MTBF via qualitative illustrations.

A multi-level preconditioned Krylov method for the efficient solution of algebraic tomographic reconstruction problems

This paper improves the convergence and robustness of a multigrid-based solver for the cross sections of the driven Schrodinger equation. Adding a Coupled Channel Correction Step (CCCS) after each multigrid (MG) V-cycle efficiently removes the errors that remain after the V-cycle sweep. The combined iterative solution scheme (MG-CCCS) is shown to feature significantly improved convergence rates over the classical MG method at energies where bound states dominate the solution, resulting in a fast and scalable solution method for the complex-valued Schrodinger break-up problem for any energy regime. The proposed solver displays optimal scaling; a solution is found in a time that is linear in the number of unknowns. The method is validated on a 2D Temkin-Poet model problem, and convergence results both as a solver and preconditioner are provided to support the O ( N ) scalability of the method. This paper extends the applicability of the complex contour approach for far field map computation (Cools et al. (2014) 10).

On the Optimality of Shifted Laplacian in a Class of Polynomial Preconditioners for the Helmholtz Equation

Pipelined Krylov subspace methods typically offer improved strong scaling on parallel HPC hardware compared to standard Krylov subspace methods for large and sparse linear systems. In pipelined methods the traditional synchronization bottleneck is mitigated by overlapping time-consuming global communications with useful computations. However, to achieve this communication hiding strategy, pipelined methods introduce additional recurrence relations for a number of auxiliary variables that are required to update the approximate solution. This paper aims at studying the influence of local rounding errors that are introduced by the additional recurrences in the pipelined Conjugate Gradient method. Specifically, we analyze the impact of local round-off effects on the attainable accuracy of the pipelined CG algorithm and compare to the traditional CG method. Furthermore, we estimate the gap between the true residual and the recursively computed residual used in the algorithm. Based on this estimate we suggest an automated residual replacement strategy to reduce the loss of attainable accuracy on the final iterative solution. The resulting pipelined CG method with residual replacement improves the maximal attainable accuracy of pipelined CG, while maintaining the efficient parallel performance of the pipelined method. This conclusion is substantiated by numerical results for a variety of benchmark problems.

The communication-hiding pipelined BiCGstab method for the parallel solution of large unsymmetric linear systems

Classical iterative methods for tomographic reconstruction include the class of Algebraic Reconstruction Techniques (ART). Convergence of these stationary linear iterative methods is however notably slow. In this paper we propose the use of Krylov solvers for tomographic linear inversion problems. These advanced iterative methods feature fast convergence at the expense of a higher computational cost per iteration, causing them to be generally uncompetitive without the inclusion of a suitable preconditioner. Combining elements from standard multigrid (MG) solvers and the theory of wavelets, a novel wavelet-based multi-level (WMG) preconditioner is introduced, which is shown to significantly speed-up Krylov convergence. The performance of the WMG-preconditioned Krylov method is analyzed through a spectral analysis, and the approach is compared to existing methods like the classical Simultaneous Iterative Reconstruction Technique (SIRT) and unpreconditioned Krylov methods on a 2D tomographic benchmark problem. Numerical experiments are promising, showing the method to be competitive with the classical Algebraic Reconstruction Techniques in terms of convergence speed and overall performance (CPU time) as well as precision of the reconstruction.