We consider the problem of sparse interpolation of an approximate multivariate black-box polynomial in floating-point arithmetic. That is, both the inputs and outputs of the black-box polynomial have some error, and all numbers are represented in standard, fixed-precision, floating point arithmetic. By interpolating the black box evaluated at random primitive roots of unity, we give efficient and numerically robust solutions. We note the similarity between the exact Ben-Or/Tiwari sparse interpolation algorithm and the classical Prony’s method for interpolating a sum of exponential functions, and exploit the generalized eigenvalue reformulation of Prony’s method. We analyze the numerical stability of our algorithms and the sensitivity of the solutions, as well as the expected conditioning achieved through randomization. Finally, we demonstrate the effectiveness of our techniques in practice through numerical experiments and applications. 1.

. In many areas of science and engineering it is of interest to find the shape of an object or region from indirect measurements which can actually be distilled into moments of the underlying shapes we seek to reconstruct. In this paper, we describe a theoretical framework for the reconstruction of a class of planar semi-analytic domains from their moments. A part of this class, known as quadrature domains, can approximate, arbitrarily closely, any bounded domain in the complex plane, and is therefore of great practical importance. We provide an exact reconstruction algorithm of quadrature domains. Some numerical demonstrations of the proposed algorithms will be presented. In addition, relations of the present theory to computer-assisted tomography and a geophysical inverse problem will be briefly discussed. 1. Introduction The theoretical subject of this paper is the truncated L problem of moments in two variables and some of its ramifications. The practical aspects of the paper are ...

This paper discusses the problem of recovering a planar polygon from its measured complex moments. These moments correspond to an indicator function defined over the polygon's support. Previous work on this problem gave necessary and sufficient conditions for such successful recovery process and focused mainly on the case of exact measurements being given. In this paper

We consider the problem of reconstruction of a non-linear finite-parametric model M = Mp(x), with p = (p1,...,pr) a set of parameters, from a set of measurements µj(M). In this paper µj(M) are always the moments mj(M) = ∫ x j Mp(x)dx. This problem is a central one in Signal Processing, Statistics, and in many other applications. We concentrate on a direct (and somewhat “naive”) approach to the above problem: we simply substitute the model function Mp(x) into the measurements µj and compute explicitly the resulting “symbolic ” expressions of µj(Mp) in terms of the parameters p. Equating these “symbolic ” expressions to the actual measurement results, we produce a system of nonlinear equations on the parameters p, which we consequently try to solve. The aim of this paper is to review some recent results (mostly of [11, 13, 18,

We consider the problem of efficiently interpolating an “approximate” black-box polynomial p(x) that is sparse when represented in the Chebyshev basis. Our computations will be in a traditional floating-point environment, and their numerical sensitivity will be investigated. As well, we consider the related problem of interpolating a sparse linear combination of (approximate) trigonometric functions. The costs of all our algorithms will be sensitive to the sparsity of the output. 1

The recovery of signal parameters from noisy sampled data is a fundamental problem in digital signal processing. In this paper, we consider the following spectral analysis problem: Let f be a real–valued sum of complex exponentials. Determine all parameters of f, i.e., all different frequencies, all coefficients, and the number of exponentials from finitely many equispaced sampled data of f. This is a nonlinear inverse problem. In this paper, we present new results on an approximate Prony method (APM) which is based on [1]. In contrast to [1], we apply matrix perturbation theory such that we can describe the properties and the numerical behavior of the APM in detail. The first part of APM estimates the frequencies and the second part solves an overdetermined linear Vandermonde–type system in a stable way. We compare the first part of APM also with the known ESPRIT method. The second part is related to the nonequispaced fast Fourier transform (NFFT). Numerical experiments show the performance of our method.

We address in this paper the following two closely related problems: 1. How to represent functions with singularities (up to a prescribed accuracy) in a compact way? 2. How to reconstruct such functions from a small number of measurements? The stress is on a comparison of linear and nonlinear approaches. As a model case we use piecewise-constant functions on [0, 1], in particular, the Heaviside jump function Ht = χ [0,t]. Considered as a curve in the Hilbert space L 2 ([0, 1]) it is completely characterized by the fact that any two its disjoint chords are orthogonal. We reinterpret this fact in a context of step-functions in one or two variables. Next we study the limitations on representability and reconstruction of piecewise-constant functions by linear and semi-linear methods. Our main tools in this problem are Kolmogorov’s n-width and ɛ-entropy, as well as Temlyakov’s (N, m)-width. On the positive side, we show that a very accurate non-linear reconstruction is possible. It goes through a solution of certain specific non-linear systems of algebraic equations. We discuss the form of these systems and methods of their solution, stressing their relation to Moment Theory and Complex Analysis. Finally, we informally discuss two problems in Computer Imaging which are parallel to the problems 1 and 2 above: compression of still images and video-sequences on one side, and image reconstruction from indirect measurement (for example, in Computer Tomography), on the other. This research was supported by the ISF, Grant No. 304/05, and by

In this paper, we discuss the numerical solution of two nonlinear approximation problems. Many applications in electrical engineering, signal processing, and mathematical physics lead to the following problem: Let h be a linear combination of exponentials with real frequencies. Determine all frequencies, all coefficients, and the number of summands, if finitely many perturbed, uniformly sampled data of h are given. We solve this problem by an approximate Prony method (APM) and prove the stability of the solution in the square and uniform norm. Further, an APM for nonuniformly sampled data is proposed too. The second approximation problem is related to the first one and reads as follows: Let ϕ be a given 1–periodic window function as defined in Section 4. Further let f be a linear combination of translates of ϕ. Determine all shift parameters, all coefficients, and the number of translates, if finitely many perturbed, uniformly sampled data of f are given. Using Fourier technique, this problem is transferred into the above parameter estimation problem for an exponential sum which is solved by APM. The stability of the solution is discussed in the square and uniform norm too. Numerical experiments show the performance of our approximation methods.

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Abstract. In many areas of science and engineering it is of interest to find the shape of an object or region from indirect measurements which can actually be distilled into moments of the underlying shapes we seek to reconstruct. In this paper, we describe a theoretical framework for the reconstruction of a class of planar semi-analytic domains from their moments. A part of this class, known as quadrature domains, can approximate, arbitrarily closely, any bounded domain in the complex plane, and is therefore of great practical importance. We provide an exact reconstruction algorithm of quadrature domains. Some numerical demonstrations of the proposed algorithms will be presented. In addition, relations of the present theory to computer-assisted tomography and a geophysical inverse problem will be briefly discussed. 1.