61 research outputs found
On the accuracy of solving confluent Prony systems
In this paper we consider several nonlinear systems of algebraic equations
which can be called "Prony-type". These systems arise in various reconstruction
problems in several branches of theoretical and applied mathematics, such as
frequency estimation and nonlinear Fourier inversion. Consequently, the
question of stability of solution with respect to errors in the right-hand side
becomes critical for the success of any particular application. We investigate
the question of "maximal possible accuracy" of solving Prony-type systems,
putting stress on the "local" behavior which approximates situations with low
absolute measurement error. The accuracy estimates are formulated in very
simple geometric terms, shedding some light on the structure of the problem.
Numerical tests suggest that "global" solution techniques such as Prony's
algorithm and ESPRIT method are suboptimal when compared to this theoretical
"best local" behavior
Complete Algebraic Reconstruction of Piecewise-Smooth Functions from Fourier Data
In this paper we provide a reconstruction algorithm for piecewise-smooth
functions with a-priori known smoothness and number of discontinuities, from
their Fourier coefficients, posessing the maximal possible asymptotic rate of
convergence -- including the positions of the discontinuities and the pointwise
values of the function. This algorithm is a modification of our earlier method,
which is in turn based on the algebraic method of K.Eckhoff proposed in the
1990s. The key ingredient of the new algorithm is to use a different set of
Eckhoff's equations for reconstructing the location of each discontinuity.
Instead of consecutive Fourier samples, we propose to use a "decimated" set
which is evenly spread throughout the spectrum
Decimated generalized Prony systems
We continue studying robustness of solving algebraic systems of Prony type
(also known as the exponential fitting systems), which appear prominently in
many areas of mathematics, in particular modern "sub-Nyquist" sampling
theories. We show that by considering these systems at arithmetic progressions
(or "decimating" them), one can achieve better performance in the presence of
noise. We also show that the corresponding lower bounds are closely related to
well-known estimates, obtained for similar problems but in different contexts
Sparse Modelling and Multi-exponential Analysis
The research fields of harmonic analysis, approximation theory and computer algebra are seemingly different domains and are studied by seemingly separated research communities. However, all of these are connected to each other in many ways. The connection between harmonic analysis and approximation theory is not accidental: several constructions among which wavelets and Fourier series, provide major insights into central problems in approximation theory. And the intimate connection between approximation theory and computer algebra exists even longer: polynomial interpolation is a long-studied and important problem in both symbolic and numeric computing, in the former to counter expression swell and in the latter to construct a simple data model. A common underlying problem statement in many applications is that of determining the number of components, and for each component the value of the frequency, damping factor, amplitude and phase in a multi-exponential model. It occurs, for instance, in magnetic resonance and infrared spectroscopy, vibration analysis, seismic data analysis, electronic odour recognition, keystroke recognition, nuclear science, music signal processing, transient detection, motor fault diagnosis, electrophysiology, drug clearance monitoring and glucose tolerance testing, to name just a few. The general technique of multi-exponential modeling is closely related to what is commonly known as the Padé-Laplace method in approximation theory, and the technique of sparse interpolation in the field of computer algebra. The problem statement is also solved using a stochastic perturbation method in harmonic analysis. The problem of multi-exponential modeling is an inverse problem and therefore may be severely ill-posed, depending on the relative location of the frequencies and phases. Besides the reliability of the estimated parameters, the sparsity of the multi-exponential representation has become important. A representation is called sparse if it is a combination of only a few elements instead of all available generating elements. In sparse interpolation, the aim is to determine all the parameters from only a small amount of data samples, and with a complexity proportional to the number of terms in the representation. Despite the close connections between these fields, there is a clear lack of communication in the scientific literature. The aim of this seminar is to bring researchers together from the three mentioned fields, with scientists from the varied application domains.Output Type: Meeting Repor
Algebraic Fourier reconstruction of piecewise smooth functions
Accurate reconstruction of piecewise-smooth functions from a finite number of
Fourier coefficients is an important problem in various applications. The
inherent inaccuracy, in particular the Gibbs phenomenon, is being intensively
investigated during the last decades. Several nonlinear reconstruction methods
have been proposed, and it is by now well-established that the "classical"
convergence order can be completely restored up to the discontinuities. Still,
the maximal accuracy of determining the positions of these discontinuities
remains an open question. In this paper we prove that the locations of the
jumps (and subsequently the pointwise values of the function) can be
reconstructed with at least "half the classical accuracy". In particular, we
develop a constructive approximation procedure which, given the first
Fourier coefficients of a piecewise- function, recovers the locations
of the jumps with accuracy , and the values of the function
between the jumps with accuracy (similar estimates are
obtained for the associated jump magnitudes). A key ingredient of the algorithm
is to start with the case of a single discontinuity, where a modified version
of one of the existing algebraic methods (due to K.Eckhoff) may be applied. It
turns out that the additional orders of smoothness produce a highly correlated
error terms in the Fourier coefficients, which eventually cancel out in the
corresponding algebraic equations. To handle more than one jump, we propose to
apply a localization procedure via a convolution in the Fourier domain
Geometry and Singularities of the Prony mapping
Prony mapping provides the global solution of the Prony system of equations
This system
appears in numerous theoretical and applied problems arising in Signal
Reconstruction. The simplest example is the problem of reconstruction of linear
combination of -functions of the form
, with the unknown parameters $a_{i},\
x_{i},\ i=1,...,n,m_{k}=\int x^{k}g(x)dx.x_{i}.$ The investigation of this type of
singularities has been started in \cite{yom2009Singularities} where the role of
finite differences was demonstrated.
In the present paper we study this and other types of singularities of the
Prony mapping, and describe its global geometry. We show, in particular, close
connections of the Prony mapping with the "Vieta mapping" expressing the
coefficients of a polynomial through its roots, and with hyperbolic polynomials
and "Vandermonde mapping" studied by V. Arnold.Comment: arXiv admin note: text overlap with arXiv:1301.118
The L1-Potts functional for robust jump-sparse reconstruction
We investigate the non-smooth and non-convex -Potts functional in
discrete and continuous time. We show -convergence of discrete
-Potts functionals towards their continuous counterpart and obtain a
convergence statement for the corresponding minimizers as the discretization
gets finer. For the discrete -Potts problem, we introduce an time
and space algorithm to compute an exact minimizer. We apply -Potts
minimization to the problem of recovering piecewise constant signals from noisy
measurements It turns out that the -Potts functional has a quite
interesting blind deconvolution property. In fact, we show that mildly blurred
jump-sparse signals are reconstructed by minimizing the -Potts functional.
Furthermore, for strongly blurred signals and known blurring operator, we
derive an iterative reconstruction algorithm
Setting the cornerstone for the IMRPhenomX family of models for gravitational waves from compact binaries: The dominant harmonic for non-precessing quasi-circular black holes
In this paper we present IMRPhenomXAS, a thorough overhaul of the IMRPhenomD
[1,2] waveform model, which describes the dominant spherical
harmonic mode of non-precessing coalescing black holes in terms of piecewise
closed form expressions in the frequency domain. Improvements include in
particular the accurate treatment of unequal spin effects, and the inclusion of
extreme mass ratio waveforms. IMRPhenomD has previously been extended to
approximately include spin precession [3] and subdominant spherical harmonics
[4], and with its extensions it has become a standard tool in gravitational
wave parameter estimation. Improved extensions of IMRPhenomXAS are discussed in
companion papers [5,6].Comment: 29 pages. 20 figures. Comments and feedback welcome! This paper
corresponds to LIGO DCC P200001
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