1,397 research outputs found
Data-driven basis for reconstructing the contrast in inverse scattering: Picard criterion, regularity, regularization, and stability
We consider the inverse medium scattering of reconstructing the medium
contrast using Born data, including the full aperture, limited-aperture, and
multi-frequency data. We propose data-driven basis functions for these inverse
problems based on the generalized prolate spheroidal wave functions and related
eigenfunctions. Such data-driven eigenfunctions are eigenfunctions of a Fourier
integral operator; they remarkably extend analytically to the whole space, are
doubly orthogonal, and are complete in the class of band-limited functions. We
first establish a Picard criterion for reconstructing the contrast using the
data-driven basis, where the reconstruction formula can also be understood from
the viewpoint of data processing and analytic extrapolation. Another salient
feature associated with the generalized prolate spheroidal wave functions is
that the data-driven basis for a disk is also a basis for a Sturm-Liouville
differential operator. With the help of Sturm-Liouville theory, we estimate the
approximation error for a spectral cutoff approximation of
functions. This yields a spectral cutoff regularization strategy for noisy data
and an explicit stability estimate for contrast in () in the
full aperture case. In the limited-aperture and multi-frequency cases, we also
obtain spectral cutoff regularization strategies for noisy data and stability
estimates for a class of contrast
Accelerated gradient methods for the X-ray imaging of solar flares
In this paper we present new optimization strategies for the reconstruction
of X-ray images of solar flares by means of the data collected by the Reuven
Ramaty High Energy Solar Spectroscopic Imager (RHESSI). The imaging concept of
the satellite is based of rotating modulation collimator instruments, which
allow the use of both Fourier imaging approaches and reconstruction techniques
based on the straightforward inversion of the modulated count profiles.
Although in the last decade a greater attention has been devoted to the former
strategies due to their very limited computational cost, here we consider the
latter model and investigate the effectiveness of different accelerated
gradient methods for the solution of the corresponding constrained minimization
problem. Moreover, regularization is introduced through either an early
stopping of the iterative procedure, or a Tikhonov term added to the
discrepancy function, by means of a discrepancy principle accounting for the
Poisson nature of the noise affecting the data
Acoustic source localization : exploring theory and practice
Over the past few decades, noise pollution became an important issue in modern society. This has led to an increased effort in the industry to reduce noise. Acoustic source localization methods determine the location and strength of the vibrations which are the cause of sound based onmeasurements of the sound field. This thesis describes a theoretical study of many facets of the acoustic source localization problem as well as the development, implementation and validation of new source localization methods. The main objective is to increase the range of applications of inverse acoustics and to develop accurate and computationally efficient methods for each of these applications. Four applications are considered. Firstly, the inverse acoustic problem is considered where the source and the measurement points are located on two parallel planes. A new fast method to solve this problem is developed and it is compared to the existing method planar nearfield acoustic holography (PNAH) from a theoretical point of view, as well as by means of simulations and experiments. Both methods are fast but the newmethod yields more robust and accurate results. Secondly, measurements in inverse acoustics are often point-by-point or full array measurements. However a straightforward and cost-effective alternative to these approaches is a sensor or array which moves through the sound field during the measurement to gather sound field information. The same numerical techniques make it possible to apply inverse acoustics to the case where the source moves and the sensors are fixed in space. It is shown that the inverse methods such as the inverse boundary element method (IBEM) can be applied to this problem. To arrive at an accurate representation of the sound field, an optimized signal processing method is applied and it is shown experimentally that this method leads to accurate results. Thirdly, a theoretical framework is established for the inverse acoustical problem where the sound field and the source are represented by a cross-spectral matrix. This problem is important in inverse acoustics because it occurs in the inverse calculation of sound intensity. The existing methods for this problem are analyzed from a theoretical point of view using this framework and a new method is derived from it. A simulation study indicates that the new method improves the results by 30% in some cases and the results are similar otherwise. Finally, the localization of point sources in the acoustic near field is considered. MUltiple SIgnal Classification (MUSIC) is newly applied to the Boundary element method (BEM) for this purpose. It is shown that this approach makes it possible to localize point sources accurately even if the noise level is extremely high or if the number of sensors is low
A non-perturbative estimate of the heavy quark momentum diffusion coefficient
We estimate the momentum diffusion coefficient of a heavy quark within a pure
SU(3) plasma at a temperature of about 1.5Tc. Large-scale Monte Carlo
simulations on a series of lattices extending up to 192^3*48 permit us to carry
out a continuum extrapolation of the so-called colour-electric imaginary-time
correlator. The extrapolated correlator is analyzed with the help of
theoretically motivated models for the corresponding spectral function.
Evidence for a non-zero transport coefficient is found and, incorporating
systematic uncertainties reflecting model assumptions, we obtain kappa = (1.8 -
3.4)T^3. This implies that the "drag coefficient", characterizing the time
scale at which heavy quarks adjust to hydrodynamic flow, is (1.8 - 3.4)
(Tc/T)^2 (M/1.5GeV) fm/c, where M is the heavy quark kinetic mass. The results
apply to bottom and, with somewhat larger systematic uncertainties, to charm
quarks.Comment: 18 pages. v2: clarifications adde
A Maximum Entropy Method of Obtaining Thermodynamic Properties from Quantum Monte Carlo Simulations
We describe a novel method to obtain thermodynamic properties of quantum
systems using Baysian Inference -- Maximum Entropy techniques. The method is
applicable to energy values sampled at a discrete set of temperatures from
Quantum Monte Carlo Simulations. The internal energy and the specific heat of
the system are easily obtained as are errorbars on these quantities. The
entropy and the free energy are also obtainable. No assumptions as to the
specific functional form of the energy are made. The use of a priori
information, such as a sum rule on the entropy, is built into the method. As a
non-trivial example of the method, we obtain the specific heat of the
three-dimensional Periodic Anderson Model.Comment: 8 pages, 3 figure
Estimates for the SVD of the truncated fourier transform on L2(cosh(b.)) and stable analytic continuation
The Fourier transform truncated on [−c, c] is usually analyzed when acting on L2(−1/b, 1/b) and its right-singular vectors are the prolate spheroidal wave functions. This paper considers the operator acting on the larger space L2(cosh(b·)) on which it remains injective. We give nonasymptotic upper and lower bounds on the singular values with similar qualitative behavior in m (the index), b, and c. The lower bounds are used to obtain rates of convergence for stable analytic continuation of possibly nonbandlimited functions which Fourier transform belongs to L2(cosh(b·)). We also derive bounds on the sup-norm of the singular functions. Finally, we provide a numerical method to compute the SVD and apply it to stable analytic continuation when the function is observed with error on an interval
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