Approximation of discontinuity lines of a noisy function of two variables

In the paper, the methods are constructed and investigated of localization (determination of the location) of the line in a neighborhood of which a measured function of two variables is smooth, whereas it has a discontinuity of the first kind on this line. Instead of the exact function, there is available some of its approximations in L ₂ and the perturbation level. The problem under consideration belongs to the class of nonlinear ill-posed problems, and to solve it some regularization algorithms are needed. A simplified theoretical approach is suggested to the problem of localization of the discontinuity line of a noisy function, when the conditions on the exact function are imposed on an arbitrary small strip intersecting the discontinuity line. Some averaging methods are constructed, and estimates for the precision of the line localization are obtained.     

Learning a function from noisy samples at a finite sparse set of points

In learning theory the goal is to reconstruct a function defined on some (typically high dimensional) domain Ω, when only noisy values of this function at a sparse, discrete subset ω⊂Ω are available.In this work we use Koksma–Hlawka type estimates to obtain deterministic bounds on the so-called generalization error. The resulting estimates show that the generalization error tends to zero when the noise in the measurements tends to zero and the number of sampling points tends to infinity sufficiently fast.     

Hilbert subsets ands-integral points

A classical tool for studying Hilbert's irreducibility theorem is Siegel's finiteness theorem forS-integral points on algebraic curves. We present a different approach based ons-integral points rather thanS-integral points. Given an integers>0, an elementt of a fieldK is said to bes-integral if the set of placesv ∈M _K for which |t|_v > l is of cardinality ≤s (instead of contained inS for “S-integral”). We prove a general diophantine result fors-integral points (Th.1.4). This result, unlike Siegel's theorem, is effective and is valid more generally for fields with the product formula. The main application to Hilbert's irreducibility theorem is a general criterion for a given Hilbert subset to contain values of given rational functions (Th.2.1). This criterion gives rise to very concrete applications: several examples are given (§2.5). Taking advantage of the effectiveness of our method, we can also produce elements of a given Hilbert subset of a number field with explicitely bounded height (Cor.3.7). Other applications, including the case thatK is of characteristicp>0, will be given in forthcoming papers ([8], [9]).     

Localization method for lines of discontinuity of an approximately defined function of two variables

A function of two variables with lines of discontinuity of the first kind is considered. It is assumed that outside the discontinuity lines the function is smooth and has a bounded partial derivative. An approximation to the function in L ₂ and a perturbation level are known. The problem in question belongs to a class of nonlinear ill-posed problems, which are solved by constructing some regularizing algorithms. We propose a simple theoretical approach to solving the problem of localizing the discontinuity lines of a function that is noisy in the space L ₂. Some conditions on the exact function are imposed ??in the small.?? Methods of averaging are constructed, and error estimates of localizing the lines (in the small) are obtained.     

Localization of minimax points

In their proof of Gilbert–Pollak conjecture on Steiner ratio, Du and Hwang (Proceedings 31th FOCS, pp. 76–85 (1990); Algorithmica 7:121–135, 1992) used a result about localization of the minimum points of functions of the type max _y∈Y f(·, y). In this paper, we present a generalization of such a localization in terms of generalized vertices, when we minimize over a compact polyhedron, and Y is a compact set. This is also a strengthening of a result of Du and Pardalos (J. Global Optim. 5:127–129, 1994). We give also a random version of our generalization.     

Localization of the discontinuity line of the right-hand side of a differential equation

We propose a new approach to studying the inverse problems for differential equations with constant coefficients. Its application is illustrated by an example of some partial differential equation with three independent variables. The right-hand side of the equation is assumed to be a function discontinuous in spatial variables. In the inverse problem, it is required to find some hull containing the discontinuity line of the right-hand side. An algorithm for constructing such a hull is obtained: It is a square whose sides are tangent to the discontinuity line.     

New methods for localizing discontinuities of a noisy function

For a problem of localizing singularities (discontinuities of the first kind) of a noisy function in L _p (1 ≤ p < ∞), new classes of regularizing methods are constructed. The methods determine the number of discontinuities and approximate their positions. The upper and lower bounds of the localizing singularities and the separability threshold are also obtained. It is proved that the methods are order-optimal by accuracy and separability on some classes of functions with discontinuities.     

Nonparametric reconstruction of a multifractal function from noisy data

We estimate a real-valued function f of d variables, subject to additive Gaussian perturbation at noise level ${\varepsilon > 0}$ , under L ^π -loss, for π ≥ 1. The main novelty is that f can have an extremely varying local smoothness, exhibiting a so-called multifractal behaviour. The results of Jaffard on the Frisch–Parisi conjecture suggest to link the singularity spectrum of f to Besov properties of the signal that can be handled by wavelet thresholding for denoising purposes. We prove that the optimal (minimax) rate of estimation of multifractal functions with singularity spectrum d(H) has explicit representation ${\varepsilon^{2v(d({\bullet}),\pi)}}$ , with $$ v(d({\bullet}),\pi)=\min_{H}\frac{H+\left(d-d(H)\right)/\pi}{2H+d}.$$ The minimum is taken over a specific domain and the rate is corrected by logarithmic factors in some cases. In particular, the usual rate ${\varepsilon^{2s/(2s+d)}}$ is retrieved for monofractal functions (with spectrum reduced to a single value s) irrespectively of π. More interestingly, the sparse case of estimation over single Besov balls has a new interpretation in terms of multifractal analysis.     

Sets of points of discontinuity and differentiability of certain operators

An investigation of the structure of the set of points off-continuity of an operator A (f-continuity is an analog of the Lipschitz condition).Translated from Matematicheskie Zametki, Vol. 9, No. 3, pp. 323–331, March, 1971.     

Approximation by Bernstein polynomials at the discontinuity points of derivatives

We obtain asymptotic formulas for the deviation of Bernstein polynomials from functions at the first-kind discontinuity points of the highest derivatives of odd order.     

Convergence of eigenfunction expansions at points of discontinuity of the coefficients of a differential operator

The question of the convergence of expansions in the eigenfunctions of a differential operator with discontinuous coefficients at a point x₀ of discontinuity of the coefficients is studied. Given an arbitrary function f(x) in the class L₂, a corresponding function \(\tilde f_{x_o } (x)\) is constructed which is such that at the point x₀ the eigenfunction expansion of f(x) diverges with the expansion of \(\tilde f_{x_o } (x)\) into a Fourier trigonometric series.     

Localization of favorite points for diffusion in a random environment

For a diffusion _Xt$X_t$ in a one-dimensional Wiener medium W$W$, it is known that there is a certain process _(br(W))r≥0${\left(b_r\left(W\right)\right)}_{r\ge 0}$ that depends only on the environment, so that _Xt−_blogt(W)$X_t-b_{logt}\left(W\right)$ converges in distribution as t→∞$t\to \infty$. The paths of b$b$ are step functions. Denote by _FX(t)$F_X\left(t\right)$ the point with the most local time for the diffusion at time t$t$. We prove that, modulo a relatively small time change, the paths of the processes _(br(W))r≥0${\left(b_r\left(W\right)\right)}_{r\ge 0}$, _{(FX(er))r≥0}${\left(F_X\left(e^r\right)\right)}_{r\ge 0}$ are close after some large r$r$.     

More on planar point subsets with a specified number of interior points

An interior point of a finite planar point set is a point of the set that is not on the boundary of the convex hull of the set. For any integer k ≥ 1, let g(k) be the smallest integer such that every set P of points in the plane with no three collinear points and with at least g(k) interior points has a subset containing precisely k interior point of P. We prove that g(k) ≥ 3k for k ≥ 3, which improves the known result that g(k) ≥ 3k ? 1 for k ≥ 3.     

Approximation by Bernstein polynomials at the points of discontinuity of the derivatives

Mathematical Notes - It is proved that, in the asymptotic formulas for the deviations of Bernstein polynomials from functions at the points of discontinuity of the first kind of the highest...     

Localization of blow-up points for a parabolic system with a nonlinear boundary condition

The authors localize the blow-up points of positive solutions of the systemu _t =Δu,v _t =Δv with conditions at the boundary of a bounded smooth domain Θ under some restrictions off andg and the initial data (Δu ₀, Δν₀>c>0). If Θ is a ball, the hypothesis on the initial data can be removed. Supported by Universidad de Buenos Aires under grant EX071 and CONICET.