Computation of Electromagnetic Fields Scattered From Objects With Uncertain Shapes Using Multilevel Monte Carlo Method

Computational tools for characterizing electromagnetic scattering from objects with uncertain shapes are needed in various applications ranging from remote sensing at microwave frequencies to Raman spectroscopy at optical frequencies. Often, such computational tools use the Monte Carlo (MC) method to sample a parametric space describing geometric uncertainties. For each sample, which corresponds to a realization of the geometry, a deterministic electromagnetic solver computes the scattered fields. However, for an accurate statistical characterization the number of MC samples has to be large. In this work, to address this challenge, the continuation multilevel Monte Carlo (CMLMC) method is used together with a surface integral equation solver. The CMLMC method optimally balances statistical errors due to sampling of the parametric space, and numerical errors due to the discretization of the geometry using a hierarchy of discretizations, from coarse to fine. The number of realizations of finer discretizations can be kept low, with most samples computed on coarser discretizations to minimize computational cost. Consequently, the total execution time is significantly reduced, in comparison to the standard MC scheme.Comment: 25 pages, 10 Figure

Distributed-memory parallelization of an explicit time-domain volume integral equation solver on Blue Gene/P

Two distributed-memory schemes for efficiently parallelizing the explicit marching-on in-time based solution of the time domain volume integral equation on the IBM Blue Gene/P platform are presented. In the first scheme, each processor stores the time history of all source fields and only the computationally dominant step of the tested field computations is distributed among processors. This scheme requires all-to-all global communications to update the time history of the source fields from the tested fields. In the second scheme, the source fields as well as all steps of the tested field computations are distributed among processors. This scheme requires sequential global communications to update the time history of the distributed source fields from the tested fields. Numerical results demonstrate that both schemes scale well on the IBM Blue Gene/P platform and the memory efficient second scheme allows for the characterization of transient wave interactions on composite structures discretized using three million spatial elements without an acceleration algorithm

Explicit Solution of the Time Domain Volume Integral Equation Using a Stable Predictor-Corrector Scheme

An explicit marching-on-in-time (MOT) scheme for solving the time domain volume integral equation is presented. The proposed method achieves its stability by employing, at each time step, a corrector scheme, which updates/corrects fields computed by the explicit predictor scheme. The proposedmethod is computationally more efficient when compared to the existing filtering techniques used for the stabilization of explicit MOT schemes. Numerical results presented in this paper demonstrate that the proposed method maintains its stability even when applied to the analysis of electromagnetic wave interactions with electrically large structures meshed using approximately half a million discretization elements

High-order, Dispersionless "Fast-Hybrid" Wave Equation Solver. Part I: O(1)\mathcal{O}(1) Sampling Cost via Incident-Field Windowing and Recentering

This paper proposes a frequency/time hybrid integral-equation method for the time dependent wave equation in two and three-dimensional spatial domains. Relying on Fourier Transformation in time, the method utilizes a fixed (time-independent) number of frequency-domain integral-equation solutions to evaluate, with superalgebraically-small errors, time domain solutions for arbitrarily long times. The approach relies on two main elements, namely, 1) A smooth time-windowing methodology that enables accurate band-limited representations for arbitrarily-long time signals, and 2) A novel Fourier transform approach which, in a time-parallel manner and without causing spurious periodicity effects, delivers numerically dispersionless spectrally-accurate solutions. A similar hybrid technique can be obtained on the basis of Laplace transforms instead of Fourier transforms, but we do not consider the Laplace-based method in the present contribution. The algorithm can handle dispersive media, it can tackle complex physical structures, it enables parallelization in time in a straightforward manner, and it allows for time leaping---that is, solution sampling at any given time $T$ at $\mathcal{O}(1)$-bounded sampling cost, for arbitrarily large values of $T$, and without requirement of evaluation of the solution at intermediate times. The proposed frequency-time hybridization strategy, which generalizes to any linear partial differential equation in the time domain for which frequency-domain solutions can be obtained (including e.g. the time-domain Maxwell equations), and which is applicable in a wide range of scientific and engineering contexts, provides significant advantages over other available alternatives such as volumetric discretization, time-domain integral equations, and convolution-quadrature approaches.Comment: 33 pages, 8 figures, revised and extended manuscript (and now including direct comparisons to existing CQ and TDIE solver implementations) (Part I of II

A transient boundary element method model of Schroeder diffuser scattering using well mouth impedance

Room acoustic diffusers can be used to treat critical listening environments to improve sound quality. One popular class is Schroeder diffusers, which comprise wells of varying depth separated by thin fins. This paper concerns a new approach to enable the modelling of these complex surfaces in the time domain. Mostly, diffuser scattering is predicted using steady-state, single frequency methods. A popular approach is to use a frequency domain Boundary Element Method (BEM) model of a box containing the diffuser, where the mouth of each well is replaced by a compliant surface with appropriate surface impedance. The best way of representing compliant surfaces in time domain prediction models, such as the transient BEM is, however, currently unresolved. A representation based on surface impedance yields convolution kernels which involve future sound, so is not compatible with the current generation of time-marching transient BEM solvers. Consequently, this paper proposes the use of a surface reflection kernel for modelling well behaviour and this is tested in a time domain BEM implementation. The new algorithm is verified on two surfaces including a Schroeder diffuser model and accurate results are obtained. It is hoped that this representation may be extended to arbitrary compliant locally reacting materials

Efficiency improvement of the frequency-domain BEM for rapid transient elastodynamic analysis

The frequency-domain fast boundary element method (BEM) combined with the exponential window technique leads to an efficient yet simple method for elastodynamic analysis. In this paper, the efficiency of this method is further enhanced by three strategies. Firstly, we propose to use exponential window with large damping parameter to improve the conditioning of the BEM matrices. Secondly, the frequency domain windowing technique is introduced to alleviate the severe Gibbs oscillations in time-domain responses caused by large damping parameters. Thirdly, a solution extrapolation scheme is applied to obtain better initial guesses for solving the sequential linear systems in the frequency domain. Numerical results of three typical examples with the problem size up to 0.7 million unknowns clearly show that the first and third strategies can significantly reduce the computational time. The second strategy can effectively eliminate the Gibbs oscillations and result in accurate time-domain responses