Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

Please note that all times are shown in the time zone of the conference. The current conference time is: 9th Dec 2022, 12:17:17am CET

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Session Overview
MS 33b: fast solvers for wave problems
Tuesday, 13/July/2021:
4:00pm - 6:00pm

Session Chair: Oscar Bruno
Session Chair: Olaf Steinbach
Virtual location: Zoom 5

Session Abstract

Problems in wave propagation have remained some of the most relevant and chal-

lenging in science and engineering. The importance of computational methods in

these areas continues to accelerate, as they span vast areas of application, from

photonics to acoustics, communications, optics, and seismology, among many

others. Significant progress in the field of computational wave scattering has

resulted in recent years from consideration of fast, high-order solvers based on

discretizations of various types, including finite elements, finite differences and

spectral methods, and relying on both differential and integral formulations. The

resulting methods have provided important new mathematical tools applicable to

a wide range of application areas. In many cases, the resulting methods have pro-

vided solution to previously intractable problems. We believe that the proposed

ICOSAHOM minisymposium, which brings together some of the most important

researchers in the area, will help disseminate and advance the field, and it will ul-

timately give rise to significant progress in this important area of computational


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4:00pm - 4:30pm

Hybrid frequency-time analysis and numerical methods for time-dependent wave propagation

Thomas Geoffrey Anderson1, Oscar Bruno2, Mark Lyon3

1University of Michigan; 2California Institute of Technology; 3University of New Hampshire

A brief introduction to recent developments in both the analysis of and numerical methods for time-dependent wave scattering, and the connections therebetween.

On the numerical side, we propose a frequency/time hybrid integral-equation method for transient wave scattering. The method uses Fourier-time transformation, resulting in required solution of a fixed set (with size independent of the desired solution time) of frequency-domain integral equations to evaluate transient solutions for arbitrarily long times. Two main concepts are introduced, namely 1) A smoothly-windowed time-partitioning methodology that enables accurate band-limited representations for arbitrary long time signals, and 2) A novel Fourier transform approach which delivers dispersionless spectrally-accurate solutions. The proposed algorithm is computationally parallelizable and exhibits high-order convergence for scattering from complex geometries while, crucially, enabling time-parallel solution with an O(1)-cost of sampling at large times T.

On the analysis side, some recent results in scattering theory are outlined. It becomes useful to study temporal decay of wave solutions (including in “trapping” scenarios), a classical question treated by the well-known Lax-Phillips scattering theory. We develop (computationally-amenable) “domain-of-dependence” bounds on solutions to wave scattering problems and establish rapid decay estimates using only (existing) Helmholtz resolvent estimates on the real frequency axis, for geometries that have previously posed as barriers to proving rapid decay. This includes the first rapid decay rates for wave scattering for connected “trapping” obstacles and, additionally, for scattering in contexts where periodic trapped orbits span the full volume of a physical cube.

4:30pm - 5:00pm

Integral equation methods and adjoint techniques for nanophotonic device design

Emmanuel Garza, Constantine Sideris

University of Southern California

Nanophotonic devices are a novel class of components capable of wielding light at wavelength scales. These devices promise to provide the next generational leap in data transfers by integrating with traditional electronic circuits. The small feature size of photonic devices require for full-wave considerations, as opposed to purely geometrical optics methods. At the same time, these devices can span dozens to hundreds of wavelengths. Hence, in order to design complex nanophotonic devices that can have unintuitive geometrical features, highly efficient computational methods are required for both simulation and optimization of these complex dielectric structures. In this talk, we present a framework for designing three-dimensional nanophotonic devices using integral equation methods. This framework is based on three main components: (1) a high-order integral equation method based on Chebyshev expansions, (2) the windowed Green function (WGF) method to simulate infinitely long waveguide structures, and (3) an adjoint computation of the gradient with respect to the simulation parameters.

We show the advantages of using the adjoint method for the discretized integral equation system, where the Hermitian transpose can be explicitly computed without forming the matrix, making it suitable for matrix-free methods to solve the adjoint problem. This adjoint approach to compute the gradient results in a computational cost of two system solves---one for the direct system and one for the adjoint---plus M sparse operations, where M is the number of optimization parameters. Hence, this technique is highly advantageous for problems with a large number of design parameters. Our implementation interfaces directly with CAD software, which, using spline boundaries and transfinite patches, provides a direct way to parametrize highly-complex prototypes that can be further optimized. Furthermore, by performing the far interactions of the integral operators on a GPU, we obtain speedups of up to 1,700x compared to a single CPU core. We present results for optimized designs obtained using this framework, which include mode splitters, grating couplers and waveguide tapers.

5:00pm - 5:30pm

Interpolated Factored Green Function" Method for accelerated solution of Scattering Problems

Oscar Bruno

Caltech, United States of America

We present a novel "Interpolated Factored Green Function" method (IFGF) for the accelerated evaluation of the integral operators in scattering theory and other areas. Like existing acceleration methods in these fields, the IFGF algorithm evaluates the action of Green function-based integral operators at a cost of O(N\log N) operations for an N-point surface mesh. Importantly, the proposed method does not utilize previously-employed acceleration elements such as the Fast Fourier transform (FFT), special-function expansions, high-dimensional linear-algebra factorizations, translation operators, equivalent sources, or parabolic scaling. Instead, the IFGF strategy, which leads to an extremely simple algorithm, capitalizes on slow variations inherent in a certain Green-function "analytic factor", which is analytic up to and including infinity, and which therefore allows for accelerated evaluation of fields produced by groups of sources on the basis of a recursive application of classical interpolation methods. In particular, the IFGF method runs on a small memory footprint, and, as it does not utilize the Fast Fourier Transforms (FFT), it is better suited than other methods for efficient parallelization in distributed-memory computer systems. Related integral equation techniques and associated device-optimization problems will be mentioned. (IFGF work in collaboration with graduate student Christoph Bauinger. Device-optimization work in collaboration with former postdoc Constantine Sideris and former students Emmanuel Garza and Agustin Fernandez-Lado.)

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