Bram Reps

Analyzing the wave number dependency of the convergence rate of a multigrid preconditioned Krylov method for the Helmholtz equation with an absorbing layer

This paper analyzes the Krylov convergence rate of a Helmholtz problem preconditioned with Multigrid. The multigrid method is applied to the Helmholtz problem formulated on a complex contour and uses GMRES as a smoother substitute at each level. A one-dimensional model is analyzed both in a continuous and discrete way. It is shown that the Krylov convergence rate of the continuous problem is independent of the wave number. The discrete problem, however, can deviate significantly from this bound due to a pitchfork in the spectrum. It is further shown in numerical experiments that the convergence rate of the Krylov method approaches the continuous bound as the grid distance

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

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Long-time solution of the time-dependent Schrodinger equation for an atom in an electromagnetic field using complex coordinate contours

gets small.

Complex Additive Geometric Multilevel Solvers for Helmholtz Equations on Spacetrees

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

A High Arithmetic Intensity Krylov Subspace Method Based on Stencil Compiler Programs.

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A polynomial multigrid smoother for the iterative solution of the heterogeneous Helmholtz problem

-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...

A Preconditioned Iterative Solver for the Scattering Solutions of the Schrödinger Equation

We demonstrate that exterior complex scaling (ECS) can be used to impose outgoing wave boundary conditions exactly on solutions of the time-dependent Schrodinger equation for atoms in intense electromagnetic pulses using finite grid methods. The procedure is formally exact when applied in the appropriate gauge and is demonstrated in a calculation of high harmonic generation in which multiphoton resonances are seen for long pulse durations. However, we also demonstrate that while the application of ECS in this way is formally exact, numerical error can appear for long time propagations that can only be controlled by extending the finite grid. A mathematical analysis of the origins of that numerical error, illustrated with an analytically solvable model, is also given.

Iterative and multigrid methods for wave problems with complex-valued boundaries

We introduce a family of implementations of low-order, additive, geometric multilevel solvers for systems of Helmholtz equations arising from Schrödinger equations. Both grid spacing and arithmetics may comprise complex numbers, and we thus can apply complex scaling to the indefinite Helmholtz operator. Our implementations are based on the notion of a spacetree and work exclusively with a finite number of precomputed local element matrices. They are globally matrix-free. Combining various relaxation factors with two grid transfer operators allows us to switch from additive multigrid over a hierarchical basis method into a Bramble-Pasciak-Xu (BPX)-type solver, with several multiscale smoothing variants within one code base. Pipelining allows us to realize full approximation storage (FAS) within the additive environment where, amortized, each grid vertex carrying degrees of freedom is read/written only once per iteration. The codes realize a single-touch policy. Among the features facilitated by matrix-free FAS is arbitrary dynamic mesh refinement (AMR) for all solver variants. AMR as an enabler for full multigrid (FMG) cycling—the grid unfolds throughout the computation—allows us to reduce the cost per unknown. The present work primary contributes toward software realization and design questions. Our experiments show that the consolidation of single-touch FAS, dynamic AMR, and vectorization-friendly, complex scaled, matrix-free FMG cycles delivers a mature implementation blueprint for solvers of Helmholtz equations in general. For this blueprint, we put particular emphasis on a strict implementation formalism as well as some implementation correctness proofs.

Latency hiding of global reductions in pipelined Krylov methods

Stencil calculations and matrix-free Krylov subspace solvers represent important components of many scientific computing applications. In these solvers, stencil applications are often the dominant part of the computation; an efficient parallel implementation of the kernel is therefore crucial to reduce the time to solution. Inspired by polynomial preconditioning, we remove upper bounds on the arithmetic intensity of the Krylov subspace building block by replacing the matrix with a higher-degree matrix polynomial. Using the latest state-of-the-art stencil compiler programs with temporal blocking, reduced memory bandwidth usage and, consequently, better utilization of SIMD vectorization and thus speedup on modern hardware, we are able to obtain performance improvements for higher polynomial degrees than simpler cache-blocking approaches have yielded in the past, demonstrating the new appeal of polynomial techniques on emerging architectures. We present results in a shared-memory environment and an extension to a distributed-memory environment with local shared memory.