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Сивкин Владимир Николаевич

Факультет математики

Профиль на hse.ru ↗ тел.: +7(495)772-95-90 | 12755
Публикаций
9
Языков
1
Наград
1
Конференций
0
Профиль Публикации (9) Курсы (5)

Профессиональные интересы

Теория рассеянияобратные задачи

Должности

  • Доцент — Факультет математики
  • Научный сотрудник — Факультет математики

Био

  • · Начал работать в НИУ ВШЭ в 2024 году.
  • · Научно-педагогический стаж: 3 года.

Образование

  • 2023 · PhD: Политехнический институт в Париже
  • 2020 · Специалитет: Московский государственный университет, специальность «Фундаментальные математика и механика», квалификация «Математик. Механик. Преподаватель»

Опыт работы

  • · 2024: Работает в НИУ ВШЭ с года

Награды и поощрения

  • · Благодарность проректора НИУ ВШЭ (октябрь 2025)

Идентификаторы исследователя

Публикации (9)

Multivariate Newton interpolation in downward closed spaces reaches the optimal Bernstein–Walsh approximation rate

2026 · ARTICLE · en

Recent advances in Bernstein—Walsh theory have extended Bernstein’s Theorem to multiple dimensions, stating that a multivariate function can be approximated with a geometric rate in a downward-closed polynomial space if and only if it is analytic in a generalized Bernstein polyellipse. To compute approximations of this class of functions—which we term Bos–Levenberg–Trefethen–(BLT) functions—we extend the classic univariate Newton interpolation algorithm to arbitrary multivariate downward-closed polynomial spaces, while maintaining its quadratic runtime and linear storage complexity. The present generalization supports any choice of (nontensorial) unisolvent interpolation nodes, whose number coincides with the dimension of the chosen downward-closed space. We prove that by selecting Leja nodes, the optimal geometric approximation rates for BLT-functions are achieved and that these rates extend to the derivatives of the interpolants. Choosing a Euclidean degree results in downward-closed spaces whose dimension only grows sub-exponentially with spatial dimension, while delivering approximation rates close to or matching those of the tensorial maximum-degree case, hence mitigating the curse of dimensionality. Importantly, our constructive proof directly inspires an algorithm for multivariate polynomial interpolation. We implemented this algorithm as a Python package and use it here to validate our theoretical findings in numerical experiments. These experiments corroborate the superiority of multivariate Newton interpolation over state-of-the-art alternatives, and they suggest that Leja-ordered Chebyshev–Lobatto nodes offer the same approximation power as Leja nodes.

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

2026 · ARTICLE · en

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 Rd, d ⩾ 2. In this region, we consider a hyper- plane X with sufficiently large distance s from the origin in Rd. 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 → +∞. A numerical imple- mentation is also presented.

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Асимптотика спектров задач Дирихле и Дирихле–Неймана для уравнения Штурма–Лиувилля с интегральным возмущением

2025 · ARTICLE · ru

В статье исследуются задачи Дирихле и Дирихле–Неймана для уравнения Штурма–Лиувилля, возмущенного интегральным оператором с ядром свертки. Найдены точные асимптотические формулы для собственных значений этих задач. Формулы содержат информацию о коэффициентах Фурье потенциала и ядра, а для остаточных членов асимптотики получены оценки, в которых учитывается и скорость убывания с ростом номера собственного значения, и скорость убывания при стремлении норм потенциала и ядра к нулю. Формулы являются новыми и в случае оператора Штурма–Лиувилля, когда ядро свертки равно нулю.

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Phase retrieval and phaseless inverse scattering with background information

2024 · ARTICLE · en

We consider the problem of finding a compactly supported potential in the multidimensional Schr & ouml;dinger equation from its differential scattering cross section (squared modulus of the scattering amplitude) at fixed energy. In the Born approximation this problem simplifies to the phase retrieval problem of reconstructing the potential from the absolute value of its Fourier transform on a ball. To compensate for the missing phase information we use the method of a priori known background scatterers. In particular, we propose an iterative scheme for finding the potential from measurements of a single differential scattering cross section corresponding to the sum of the unknown potential and a known background potential, which is sufficiently disjoint. If this condition is relaxed, then we give similar results for finding the potential from additional monochromatic measurements of the differential scattering cross section of the unknown potential without the background potential. The performance of the proposed algorithms is demonstrated in numerical examples. In the present work we significantly advance theoretically and numerically studies of Agaltsov et al (2019 Inverse Problems 35 24001) and Novikov and Sivkin (2021 Inverse Problems 37 055011).

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Multipoint formulas in inverse problems and their numerical implementation

2023 · ARTICLE · en

We present the first numerical study of multipoint formulas for finding leading coefficients in asymptotic expansions arising in potential and scattering theories. In particular, we implement different formulas for finding the Fourier transform of potential from the scattering amplitude at several high energies. We show that the aforementioned approach can be used for essential numerical improvements of classical results including the slowly convergent Born-Faddeev formula for inverse scattering at high energies. The approach of multipoint formulas can be also used for recovering the x-ray transform of potential from boundary values of the scattering wave functions at several high energies. Determination of total charge (electric or gravitational) from several exterior measurements is also considered. In addition, we show that the aforementioned multipoint formulas admit an efficient regularization for the case of random noise. In particular, we proceed from theoretical works (Novikov 2020 Inverse Problems 36 095001; 2021 Russ. Math. Surv. 76 723-5).

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Approximate Lipschitz stability for phaseless inverse scattering with background information

2023 · ARTICLE · en

We prove approximate Lipschitz stability for monochromatic phaseless inverse scattering with background information in dimension d = 2. Moreover, these stability estimates are given in terms of non-overdetermined and incomplete data. Related results for reconstruction from phaseless Fourier transforms are also given. Prototypes of these estimates for the phased case were given in [R. G. Novikov, Approximate Lipschitz stability for non-overdetermined inverse scattering at fixed energy, J. Inverse Ill-Posed Probl. 21(2013), no. 6, 813-823].

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Fixed-distance multipoint formulas for the scattering amplitude from phaseless measurements

2022 · ARTICLE · en

We give new formulas for finding the complex (phased) scattering amplitude at fixed frequency and angles from absolute values of the scattering wave function at several points x (1), horizontal ellipsis , x ( m ). In dimension d > 2, for m > 2, we significantly improve previous results in the following two respects. First, geometrical constraints on the points needed in previous results are significantly simplified. Essentially, the measurement points x ( j ) are assumed to be on a ray from the origin with fixed distance tau = |x ( j+1) - x ( j )|, and high order convergence (linearly related to m) is achieved as the points move to infinity with fixed tau. Second, our new asymptotic reconstruction formulas are significantly simpler than previous ones. In particular, we continue studies going back to Novikov (2015 Bull. Sci. Math. 139 923-936).

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Phaseless inverse scattering with background information

2021 · ARTICLE · en

We consider phaseless inverse scattering for the multidimensional Schrodinger equation with unknown potential v using the method of known background scatterers. In particular, in dimension d > 2, we show that |f (1)|(2) at high energies uniquely determines v via explicit formulas, where f (1) is the scattering amplitude for v + w (1), w (1) is an a priori known nonzero background scatterer, under the condition that supp v and supp w (1) are sufficiently disjoint. If this condition is relaxed, then we give similar formulas for finding v from |f|(2), |f (1)|(2), where f is the scattering amplitude for v. In particular, we continue studies of Novikov (2016 J. Geom. Anal. 26 346-59) and Leshem et al (2016 Nat. Commun. 7 1-6).

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Error estimates for phase recovering from phaseless scattering data.

2020 · ARTICLE · en

We study the simplest explicit formulas for approximate finding the complex scattering amplitude from modulus of the scattering wave function. We obtain detailed error estimates for these formulas in dimensions d = 3 and d = 2.

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Курсы (5)