Dmitry Batenkov

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

On the Accuracy of Solving Confluent Prony Systems

k

Moment inversion problem for piecewise D-finite functions

Fourier coefficients of a piecewise-

Boosting productivity with the Boost Graph Library

C^{2d+1}

Open BEAGLE: a generic framework for evolutionary computations

function, recovers the locations of the jumps with accuracy

Automatic animation of high resolution images

sim k^{-(d+2)}

Analyzing EEG data

, and the values of the function between the jumps with accuracy

Web maps of renewable energy

sim k^{-(d+1)}

Hands-on introduction to genetic programming

(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.

Real-time detection with webcam

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 Pronys algorithm and the ESPRIT method are suboptimal when compared to this theoretical “best local” behavior.