Moments of spectral functions: Monte Carlo evaluation and verification

The subject of the present study is the Monte Carlo path-integral evaluation of the moments of spectral functions. Such moments can be computed by formal differentiation of certain estimating functionals that are infinitely-differentiable against time whenever the potential function is arbitrarily smooth. Here, I demonstrate that the numerical differentiation of the estimating functionals can be more successfully implemented by means of pseudospectral methods (e.g., exact differentiation of a Chebyshev polynomial interpolant), which utilize information from the entire interval $(-\beta \hbar / 2, \beta \hbar/2)$. The algorithmic detail that leads to robust numerical approximations is the fact that the path integral action and not the actual estimating functional are interpolated. Although the resulting approximation to the estimating functional is non-linear, the derivatives can be computed from it in a fast and stable way by contour integration in the complex plane, with the help of the Cauchy integral formula (e.g., by Lyness' method). An interesting aspect of the present development is that Hamburger's conditions for a finite sequence of numbers to be a moment sequence provide the necessary and sufficient criteria for the computed data to be compatible with the existence of an inversion algorithm. Finally, the issue of appearance of the sign problem in the computation of moments, albeit in a milder form than for other quantities, is addressed.Comment: 13 pages, 2 figure

Slip distribution of the 2010 Mentawai earthquake from inversion of tsunami waveforms and tsunami field survey data

We study the 2010 Mentawai earthquake, a tsunami earthquake that occurred seaward of the southern Mentawai islands of Sumatra, and produced a locally devastating tsunami, with runup commonly in excess of 6 m. As a unique tsunami earthquake case, there is a significant discrepancy between the observed small GPS displacement and\ud the very large tsunami runup (maximum value > 16 m), which cannot be explained by the conventional GPS or seismic inversion model. The goal of this work is to infer the slip distribution of this earthquake from the available waveforms recorded by nearby tide gauges or from the tsunami height and runup data collected in a field survey, using two\ud inversion models based on Green???s fuctions technique. We subdivide the fault plane into 18 subfaults and compute the corresponding numerical Green???s functions by integrating shallow-water equations via a finite-difference method. The slip distributions inversed by these two models were compared. The limitations of these two methods are discussed

Thermoacoustic tomography with detectors on an open curve: an efficient reconstruction algorithm

Practical applications of thermoacoustic tomography require numerical inversion of the spherical mean Radon transform with the centers of integration spheres occupying an open surface. Solution of this problem is needed (both in 2-D and 3-D) because frequently the region of interest cannot be completely surrounded by the detectors, as it happens, for example, in breast imaging. We present an efficient numerical algorithm for solving this problem in 2-D (similar methods are applicable in the 3-D case). Our method is based on the numerical approximation of plane waves by certain single layer potentials related to the acquisition geometry. After the densities of these potentials have been precomputed, each subsequent image reconstruction has the complexity of the regular filtration backprojection algorithm for the classical Radon transform. The peformance of the method is demonstrated in several numerical examples: one can see that the algorithm produces very accurate reconstructions if the data are accurate and sufficiently well sampled, on the other hand, it is sufficiently stable with respect to noise in the data

High-contrast Imaging from Space: Speckle Nulling in a Low Aberration Regime

High-contrast imaging from space must overcome two major noise sources to successfully detect a terrestrial planet angularly close to its parent star: photon noise from diffracted star light, and speckle noise from star light scattered by instrumentally-generated wavefront perturbation. Coronagraphs tackle only the photon noise contribution by reducing diffracted star light at the location of a planet. Speckle noise should be addressed with adaptative-optics systems. Following the tracks of Malbet, Yu and Shao (1995), we develop in this paper two analytical methods for wavefront sensing and control that aims at creating dark holes, i.e. areas of the image plane cleared out of speckles, assuming an ideal coronagraph and small aberrations. The first method, speckle field nulling, is a fast FFT-based algorithm that requires the deformable-mirror influence functions to have identical shapes. The second method, speckle energy minimization, is more general and provides the optimal deformable mirror shape via matrix inversion. With a NxN deformable mirror, the size of matrix to be inverted is either N^2xN^2 in the general case, or only NxN if influence functions can be written as the tensor product of two one-dimensional functions. Moreover, speckle energy minimization makes it possible to trade off some of the dark hole area against an improved contrast. For both methods, complex wavefront aberrations (amplitude and phase) are measured using just three images taken with the science camera (no dedicated wavefront sensing channel is used), therefore there are no non-common path errors. We assess the theoretical performance of both methods with numerical simulations, and find that these speckle nulling techniques should be able to improve the contrast by several orders of magnitude.Comment: 31 pages, 8 figures, 1 table. Accepted for publication in ApJ (should appear in February 2006

Numerical evaluation of two and three parameter Mittag-Leffler functions

The Mittag-Leffler (ML) function plays a fundamental role in fractional calculus but very few methods are available for its numerical evaluation. In this work we present a method for the efficient computation of the ML function based on the numerical inversion of its Laplace transform (LT): an optimal parabolic contour is selected on the basis of the distance and the strength of the singularities of the LT, with the aim of minimizing the computational effort and reduce the propagation of errors. Numerical experiments are presented to show accuracy and efficiency of the proposed approach. The application to the three parameter ML (also known as Prabhakar) function is also presented.Comment: Accepted for publication in SIAM Journal on Numerical Analysi