James Bremer

A Nonlinear Optimization Procedure for Generalized Gaussian Quadratures

We present a new nonlinear optimization procedure for the computation of generalized Gaussian quadratures for a broad class of square integrable functions on intervals. While some of the components of this algorithm have been previously published, we present a simple and robust scheme for the determination of a sparse solution to an underdetermined nonlinear optimization problem which replaces the continuation scheme of the previously published works. The new algorithm successfully computes generalized Gaussian quadratures in a number of instances in which the previous algorithms fail. Four applications of our scheme to computational physics are presented: the construction of discrete plane wave expansions for the Helmholtz Greens function, the design of linear array antennae, the computation of a quadrature for the discretization of Laplace boundary integral equations on certain domains with corners, and the construction of quadratures for the discretization of Laplace and Helmholtz boundary integral equations on smooth surfaces.

Universal quadratures for boundary integral equations on two-dimensional domains with corners

We describe the construction of a collection of quadrature formulae suitable for the efficient discretization of certain boundary integral equations on a very general class of two-dimensional domains with corner points. The resulting quadrature rules allow for the rapid high-accuracy solution of Dirichlet boundary value problems for Laplaces equation and the Helmholtz equation on such domains under a mild assumption on the boundary data. Our approach can be adapted to other boundary value problems and certain aspects of our scheme generalize to the case of surfaces with singularities in three dimensions. The performance of the quadrature rules is illustrated with several numerical examples.

A fast direct solver for the integral equations of scattering theory on planar curves with corners

We describe an approach to the numerical solution of the integral equations of scattering theory on planar curves with corners. It is rather comprehensive in that it applies to a wide variety of boundary value problems; here, we treat the Neumann and Dirichlet problems as well as the boundary value problem arising from acoustic scattering at the interface of two fluids. It achieves high accuracy, is applicable to large-scale problems and, perhaps most importantly, does not require asymptotic estimates for solutions. Instead, the singularities of solutions are resolved numerically. The approach is efficient, however, only in the low- and mid-frequency regimes. Once the scatterer becomes more than several hundred wavelengths in size, the performance of the algorithm of this paper deteriorates significantly. We illustrate our method with several numerical experiments, including the solution of a Neumann problem for the Helmholtz equation given on a domain with nearly 10000 corner points.

A Nyström method for weakly singular integral operators on surfaces

We describe a modified Nystrom method for the discretization of the weakly singular boundary integral operators which arise from the formulation of linear elliptic boundary value problems as integral equations. Standard Nystrom and collocation schemes proceed by representing functions via their values at a collection of quadrature nodes. Our method uses appropriately scaled function values in lieu of such representations. This results in a scheme which is mathematically equivalent to Galerkin discretization in that the resulting matrices are related to those obtained by Galerkin methods via conjugation with well-conditioned matrices, but which avoids the evaluation of double integrals. Moreover, we incorporate a new mechanism for approximating the singular integrals which arise from the discretization of weakly singular integral operators which is considerably more efficient than standard methods. We illustrate the performance of our method with numerical experiments.

Efficient discretization of Laplace boundary integral equations on polygonal domains

We describe a numerical procedure for the construction of quadrature formulae suitable for the efficient discretization of boundary integral equations over very general curve segments. While the procedure has applications to the solution of boundary value problems on a wide class of complicated domains, we concentrate in this paper on a particularly simple case: the rapid solution of boundary value problems for Laplaces equation on two-dimensional polygonal domains. We view this work as the first step toward the efficient solution of boundary value problems on very general singular domains in both two and three dimensions. The performance of the method is illustrated with several numerical examples.

Biorthogonal diffusion wavelets for multiscale representations on manifolds and graphs

Recent work by some of the authors presented a novel construction of a multiresolution analysis on manifolds and graphs, acted upon by a given symmetric Markov semigroup {Tt}t≥0, for which Tt has low rank for large t. This includes important classes of diffusion-like operators, in any dimension, on manifolds, graphs, and in nonhomogeneous media. The dyadic powers of an operator are used to induce a multiresolution analysis, analogous to classical Littlewood-Paley and wavelet theory, while associated wavelet packets can also be constructed. This extends multiscale function and operator analysis and signal processing to a large class of spaces, such as manifolds and graphs, with efficient algorithms. Powers and functions of T (notably its Greens function) are efficiently computed, represented and compressed. This construction is related and generalizes certain Fast Multipole Methods, the wavelet representation of Calderon-Zygmund and pseudo-differential operators, and also relates to algebraic multigrid techniques. The original diffusion wavelet construction yields orthonormal bases for multiresolution spaces {Vj}. The orthogonality requirement has some advantages from the numerical perspective, but several drawbacks in terms of the space and frequency localization of the basis functions. Here we show how to relax this requirement in order to construct biorthogonal bases of diffusion scaling functions and wavelets. This yields more compact representations of the powers of the operator, better localized basis functions. This new construction also applies to non self-adjoint semigroups, arising in many applications.

Diffusion-driven multiscale analysis on manifolds and graphs: top-down and bottom-up constructions

Classically, analysis on manifolds and graphs has been based on the study of the eigenfunctions of the Laplacian and its generalizations. These objects from differential geometry and analysis on manifolds have proven useful in applications to partial differential equations, and their discrete counterparts have been applied to optimization problems, learning, clustering, routing and many other algorithms.1−7 The eigenfunctions of the Laplacian are in general global: their support often coincides with the whole manifold, and they are affected by global properties of the manifold (for example certain global topological invariants). Recently a framework for building natural multiresolution structures on manifolds and graphs was introduced, that greatly generalizes, among other things, the construction of wavelets and wavelet packets in Euclidean spaces.8,9 This allows the study of the manifold and of functions on it at different scales, which are naturally induced by the geometry of the manifold. This construction proceeds bottom-up, from the finest scale to the coarsest scale, using powers of a diffusion operator as dilations and a numerical rank constraint to critically sample the multiresolution subspaces. In this paper we introduce a novel multiscale construction, based on a top-down recursive partitioning induced by the eigenfunctions of the Laplacian. This yields associated local cosine packets on manifolds, generalizing local cosines in Euclidean spaces.10 We discuss some of the connections with the construction of diffusion wavelets. These constructions have direct applications to the approximation, denoising, compression and learning of functions on a manifold and are promising in view of applications to problems in manifold approximation, learning, dimensionality reduction.

On the numerical evaluation of the singular integrals of scattering theory

In a previous work, the authors introduced a scheme for the numerical evaluation of the singular integrals which arise in the discretization of certain weakly singular integral operators of acoustic and electromagnetic scattering. That scheme is designed to achieve high-order algebraic convergence and high-accuracy when applied to operators given on smoothly parameterized surfaces. This paper generalizes the approach to a wider class of integral operators including many defined via the Cauchy principal value. Operators of this type frequently occur in the course of solving scattering problems involving boundary conditions on tangential derivatives. The resulting scheme achieves high-order algebraic convergence and approximately 12 digits of accuracy.

Measurement of zone plate efficiencies in the extreme ultraviolet and applications to radiation monitors for absolute spectral emission

The diffraction efficiencies of a Fresnel zone plate (ZP), fabricated by Xradia Inc. using the electron-beam writing technique, were measured using polarized, monochromatic synchrotron radiation in the extreme ultraviolet wavelength range 3.4-22 nm. The ZP had 2 mm diameter, 3330 zones, 150 nm outer zone width, and a 1 mm central occulter. The ZP was supported by a 100 nm thick Si3N4 membrane. The diffraction patterns were recorded by CMOS imagers with phosphor coatings and with 5.2 μm or 48 μm pixels. The focused +n orders (n=1-4), the diverging -1 order, and the undiffracted 0 order were observed as functions of wavelength and off-axis tilt angle. Sub-pixel focusing of the +n orders was achieved. The measured efficiency in the +1 order was in the 5% to 30% range with the phase-shift enhanced efficiency occurring at 8.3 nm where the gold bars are partially transmitting. The +2 and higher order efficiencies were much lower than the +1 order efficiency. The efficiencies were constant when the zone plate was tilted by angles up to ±1° from the incident radiation beam. This work indicates the feasibility and benefits of using zone plates to measure the absolute EUV spectral emissions from solar and laboratory sources: relatively high EUV efficiency in the focused +1 order, good out-of-band rejection resulting from the low higher-order efficiencies and the ZP focusing properties, insensitivity to (unfocused) visible light scattered by the ZP, flat response with off-axis angle, and insensitivity to the polarization of the radiation based on the ZP circular symmetry. EUV sensors with Fresnel zone plates potentially have many advantages over existing sensors intended to accurately measure absolute EUV emission levels, such as those implemented on the GOES N-P satellites that use transmission gratings which have off-axis sensitivity variations and poor out-of-band EUV and visible light rejection, and other solar and laboratory sensors using reflection gratings which are subject to response variations caused by surface contamination and oxidation.

Optimization of the GOES-I Imager's radiometric accuracy: drift and 1/ f noise suppression

The raw output of many scanning radiometers is a small, rapidly varying signal superimposed on a large background that varies more slowly, due to thermal drifts and 1/f noise. To isolate the signal, it is necessary to perform a differential measurement: measure a known reference and subtract it from each of the raw outputs, cancelling the common-mode background. Calibration is also a differential measurement: the difference between two outputs is divided by the difference between the two known references that produced them to determine the gain. The GOES-I Imager views space as its background subtraction reference and a full-aperture blackbody as its second reference for calibration. The background suppression efficiency of a differential measurement algorithm depends on its timing. The Imager measures space references before and after each scan line and performs interpolated background subtraction: a unique, linearly weighted average of the two references is subtracted from each scene sample in that line, cancelling both constant bias and linear drift. Our model quantifies the Gaussian noise and 1/f noise terms in the noise equivalent bandwidth, which is minimized to optimize the algorithm. We have obtained excellent agreement between our analytical predictions and Monte Carlo computer simulations.