arXiv subjects

Nadezda Sukhorukova

Publications and source records attributed to Nadezda Sukhorukova.

21 records · Page 2Linked to original sources

arXiv2015

Chebyshev approximation for multivariate functions

In this paper, we derive optimality conditions (Chebyshev approximation) for multivariate functions. The theory of Chebyshev (uniform) approximation for univariate functions is very elegant. The optimality conditions are based on the notion of alternance (maximal deviation points with alternating deviation signs). It is not very straightforward, however, how to extend the notion of alternance to the case of multivariate functions. There have been several attempts to extend the theory of Chebyshev approximation to the case of multivariate functions. We propose an alternative approach, which is based on the notion of convexity and nonsmooth analysis.

math.OC↗

arXiv2015

Linear least squares problems involving fixed knots polynomial splines and their singularity study

In this paper, we study a class of approximation problems, appearing in data approximation and signal processing. The approximations are constructed as combinations of polynomial splines (piecewise polynomials), whose parameters are subject to optimisation, and so called prototype functions, whose choice is based on the application, rather than optimisation. The corresponding optimisation problems can be formulated as Linear Least Squares Problems (LLSPs). If the system matrix is non-singular, then the corresponding problem can be solved using the normal equations method, while for singular cases slower (but more robust) methods have to be used. In this paper we develop sufficient conditions for non-singularity. These conditions can be verified much faster than the direct singularity verification of the system matrices. Therefore, the algorithm efficiency can be improved by choosing a better suited method for solving the corresponding LLSPs.

math.OC↗

arXiv2014

Characterization theorem for best polynomial spline approximation with free knots

In this paper, we derive a necessary condition for a best approximation by piecewise polynomial functions. We apply nonsmooth nonconvex analysis to obtain this result, which is also a necessary and sufficient condition for inf-stationarity in the sense of Demyanov-Rubinov. We start from identifying a special property of the knots. Then, using this property, we construct a characterization theorem for best free knots polynomial spline approximation, which is stronger than the existing characterisation results when only continuity is required.

math.OC↗