Fast Ewald summation for electrostatic potentials with arbitrary periodicity

A unified treatment for fast and spectrally accurate evaluation of electrostatic potentials subject to periodic boundary conditions in any or none of the three space dimensions is presented. Ewald decomposition is used to split the problem into a real space and a Fourier space part, and the FFT based Spectral Ewald (SE) method is used to accelerate the computation of the latter. A key component in the unified treatment is an FFT based solution technique for the free-space Poisson problem in three, two or one dimensions, depending on the number of non-periodic directions. The cost of calculations is furthermore reduced by employing an adaptive FFT for the doubly and singly periodic cases, allowing for different local upsampling rates. The SE method will always be most efficient for the triply periodic case as the cost for computing FFTs will be the smallest, whereas the computational cost for the rest of the algorithm is essentially independent of the periodicity. We show that the cost of removing periodic boundary conditions from one or two directions out of three will only marginally increase the total run time. Our comparisons also show that the computational cost of the SE method for the free-space case is typically about four times more expensive as compared to the triply periodic case. The Gaussian window function previously used in the SE method, is here compared to an approximation of the Kaiser-Bessel window function, recently introduced. With a carefully tuned shape parameter that is selected based on an error estimate for this new window function, runtimes for the SE method can be further reduced. Keywords: Fast Ewald summation, Fast Fourier transform, Arbitrary periodicity, Coulomb potentials, Adaptive FFT, Fourier integral, Spectral accuracy.Comment: 21 pages, 11 figure

Numerical Analysis of the Non-uniform Sampling Problem

We give an overview of recent developments in the problem of reconstructing a band-limited signal from non-uniform sampling from a numerical analysis view point. It is shown that the appropriate design of the finite-dimensional model plays a key role in the numerical solution of the non-uniform sampling problem. In the one approach (often proposed in the literature) the finite-dimensional model leads to an ill-posed problem even in very simple situations. The other approach that we consider leads to a well-posed problem that preserves important structural properties of the original infinite-dimensional problem and gives rise to efficient numerical algorithms. Furthermore a fast multilevel algorithm is presented that can reconstruct signals of unknown bandwidth from noisy non-uniformly spaced samples. We also discuss the design of efficient regularization methods for ill-conditioned reconstruction problems. Numerical examples from spectroscopy and exploration geophysics demonstrate the performance of the proposed methods

Time-parallel iterative solvers for parabolic evolution equations

We present original time-parallel algorithms for the solution of the implicit Euler discretization of general linear parabolic evolution equations with time-dependent self-adjoint spatial operators. Motivated by the inf-sup theory of parabolic problems, we show that the standard nonsymmetric time-global system can be equivalently reformulated as an original symmetric saddle-point system that remains inf-sup stable with respect to the same natural parabolic norms. We then propose and analyse an efficient and readily implementable parallel-in-time preconditioner to be used with an inexact Uzawa method. The proposed preconditioner is non-intrusive and easy to implement in practice, and also features the key theoretical advantages of robust spectral bounds, leading to convergence rates that are independent of the number of time-steps, final time, or spatial mesh sizes, and also a theoretical parallel complexity that grows only logarithmically with respect to the number of time-steps. Numerical experiments with large-scale parallel computations show the effectiveness of the method, along with its good weak and strong scaling properties