arXiv subjects

Vladimir Sivkin

Publications and source records attributed to Vladimir Sivkin.

3 recordsLinked to original sources

arXiv2025

A two-point phase recovering from holographic data on a single plane

We consider a plane wave, a radiation solution, and the sum of these solutions (total solution) for the Helmholtz equation in an exterior region in $\mathbb{R}^d,$ $d\geq 2$. In this region, we consider a hyperplane $X$ with sufficiently large distance $s$ from the origin in ${\mathbb R}^d.$ We give two-point local formulas for approximate recovering the radiation solution restricted to the plane $X$ from the intensity of the total solution at $X$, that is, from holographic data. The recovering is given in terms of the far-field pattern of the radiation solution with a decaying error term as $s \to +\infty.$ A numerical implementation is also presented.

math.AP↗

arXiv2025

Multivariate Newton Interpolation in Downward Closed Spaces Reaches the Optimal Geometric Approximation Rates for Bos--Levenberg--Trefethen Functions

We extend the univariate Newton interpolation algorithm to arbitrary spatial dimensions and for any choice of downward-closed polynomial space, while preserving its quadratic runtime and linear storage cost. The generalisation supports any choice of the provided notion of non-tensorial unisolvent interpolation nodes, whose number coincides with the dimension of the chosen-downward closed space. Specifically, we prove that by selecting Leja-ordered Chebyshev-Lobatto or Leja nodes, the optimal geometric approximation rates for a class of analytic functions -- termed Bos--Levenberg--Trefethen functions -- are achieved and extend to the derivatives of the interpolants. In particular, choosing Euclidean degree results in downward-closed spaces whose dimension only grows sub-exponentially with spatial dimension, while delivering approximation rates close to, or even matching those of the tensorial maximum-degree case, mitigating the curse of dimensionality. Several numerical experiments demonstrate the performance of the resulting multivariate Newton interpolation compared to state-of-the-art alternatives and validate our theoretical results.

math.NA↗

arXiv2024

Multivariate Interpolation in Unisolvent Nodes -- Lifting the Curse of Dimensionality

We extend Newton and Lagrange interpolation to arbitrary dimensions. The core contribution that enables this is a generalized notion of non-tensorial unisolvent nodes, i.e., nodes on which the multivariate polynomial interpolant of a function is unique. By validation, we reach the optimal exponential Trefethen rates for a class of analytic functions, we term Trefethen functions. The number of interpolation nodes required for computing the optimal interpolant depends sub-exponentially on the dimension, hence resisting the curse of dimensionality. Based on these results, we propose an algorithm to efficiently and numerically stably solve arbitrary-dimensional interpolation problems, with at most quadratic runtime and linear memory requirement.

math.NA↗