Victor Isakov
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Victor Isakov has a total of 29 co-authors. Here are the top co-authors by number of shared publications:
- Jenn-Nan Wang
Institute of Applied Mathematical Sciences, National Taiwan University, Taipei 106, Taiwan
6 shared publications - Jianliang Qian
Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, USA
3 shared publications - Shingyu Leung
Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong
3 shared publications - Nanhee Kim2 shared publications
- Shuai Lu
School of Mathematical Sciences, Fudan University, Shanghai 200433, China
2 shared publications - Aleksander A. Shananin
Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, Moscow, Russia
1 shared publication
Latest Publications
- 22 October 2022
- Published by Walter de Gruyter GmbH in Journal of Inverse and Ill-posed Problems
- Vol. 30 (6) , 807-822
On the inverse gravimetry problem with minimal data
The inverse problem in gravimetry is to find a domain 𝐷 inside the reference domain Ω from measurements of gravitational force outside Ω. We consider the problem in three dimensions where we found that a few parameters of the unknown 𝐷 can be stably determined given data noise in practical situations. An ellipsoid is the best approximation of 𝐷. We prove uniqueness of recovering an ellipsoid in a particular case for the inverse problem from minimal amount of data which are the approximated gravitational force at nine boundary points. In the proofs, we derive and use simple systems of linear and nonlinear algebraic equations for the parameters of an ellipsoid. Similarly, a rectangular parallelepiped 𝐷 is considered. To support our theory, we use numerical examples with different location of measurements points on .- 22 October 2022
- Published by Walter de Gruyter GmbH in Journal of Inverse and Ill-posed Problems
- Vol. 30 (6) , 807-822
On the inverse gravimetry problem with minimal data
The inverse problem in gravimetry is to find a domain 𝐷 inside the reference domain Ω from measurements of gravitational force outside Ω. We consider the problem in three dimensions where we found that a few parameters of the unknown 𝐷 can be stably determined given data noise in practical situations. An ellipsoid is the best approximation of 𝐷. We prove uniqueness of recovering an ellipsoid in a particular case for the inverse problem from minimal amount of data which are the approximated gravitational force at nine boundary points. In the proofs, we derive and use simple systems of linear and nonlinear algebraic equations for the parameters of an ellipsoid. Similarly, a rectangular parallelepiped 𝐷 is considered. To support our theory, we use numerical examples with different location of measurements points on .- 24 January 2022
- Published by Elsevier in Journal of Computational Physics
- Vol. 455
A linearised inverse conductivity problem for the Maxwell system at a high frequency
- 24 January 2022
- Published by Elsevier in Journal of Computational Physics
- Vol. 455
A linearised inverse conductivity problem for the Maxwell system at a high frequency
- Open Access
Uniqueness and increasing stability in electromagnetic inverse source problems
In this paper we study the uniqueness and the increasing stability in the inverse source problem for electromagnetic waves in homogeneous and inhomogeneous media from boundary data at multiple wave numbers. For the unique determination of sources, we consider inhomogeneous media and use tangential components of the electric field and magnetic field at the boundary of the reference domain. The proof relies on the Fourier transform with respect to the wave numbers and the unique continuation theorems. To study the increasing stability in the source identification, we consider homogeneous media and measure the absorbing data or the tangential component of the electric field at the boundary of the reference domain as additional data. By using the Fourier transform with respect to the wave numbers, explicit bounds for analytic continuation, Huygens' principle and bounds for initial boundary value problems, increasing (with larger wave numbers intervals) stability estimate is obtained.
Open AccessUniqueness and increasing stability in electromagnetic inverse source problems
In this paper we study the uniqueness and the increasing stability in the inverse source problem for electromagnetic waves in homogeneous and inhomogeneous media from boundary data at multiple wave numbers. For the unique determination of sources, we consider inhomogeneous media and use tangential components of the electric field and magnetic field at the boundary of the reference domain. The proof relies on the Fourier transform with respect to the wave numbers and the unique continuation theorems. To study the increasing stability in the source identification, we consider homogeneous media and measure the absorbing data or the tangential component of the electric field at the boundary of the reference domain as additional data. By using the Fourier transform with respect to the wave numbers, explicit bounds for analytic continuation, Huygens' principle and bounds for initial boundary value problems, increasing (with larger wave numbers intervals) stability estimate is obtained.
- 23 May 2020
- Published by Taylor & Francis in Applicable Analysis
- Vol. 101 (10) , 3550-3562
On increasing stability in an inverse source problem with local boundary data at many wave numbers
To derive the increasing stability of the source term in the Helmholtz equation from local boundary data, we utilize sharp bounds of the analytic continuation for higher wave numbers, the Huygens' principle, and bounds in the lateral Cauchy problem for the wave equation.
- 23 May 2020
- Published by Taylor & Francis in Applicable Analysis
- Vol. 101 (10) , 3550-3562
On increasing stability in an inverse source problem with local boundary data at many wave numbers
To derive the increasing stability of the source term in the Helmholtz equation from local boundary data, we utilize sharp bounds of the analytic continuation for higher wave numbers, the Huygens' principle, and bounds in the lateral Cauchy problem for the wave equation.
- 30 October 2019
- Published by Pleiades Publishing Ltd in Computational Mathematics and Mathematical Physics
- Vol. 59 (10) , 1753-1758
Algorithm for Determining the Volatility Function in the Black–Scholes Model
An algorithm for reconstructing the volatility function in the modified Black–Scholes model is developed. Results of numerical computations are presented. It is shown that adding information...
- 30 October 2019
- Published by Pleiades Publishing Ltd in Computational Mathematics and Mathematical Physics
- Vol. 59 (10) , 1753-1758
Algorithm for Determining the Volatility Function in the Black–Scholes Model
An algorithm for reconstructing the volatility function in the modified Black–Scholes model is developed. Results of numerical computations are presented. It is shown that adding information...
- 1 January 2019
- Published by American Institute of Mathematical Sciences (AIMS) in Inverse Problems and Imaging
- Vol. 13 (5) , 983-1006
On increasing stability of the continuation for elliptic equations of second order without (pseudo)convexity assumptions
We derive bounds of solutions of the Cauchy problem for general elliptic partial differential equations of second order containing parameter (wave number)$ k $ which are getting nearly Lipschitz for large$ k $ . Proofs use energy estimates combined with splitting solutions into low and high frequencies parts, an associated hyperbolic equation and the Fourier-Bros-Iagolnitzer transform to replace the hyperbolic equation with an elliptic equation without parameter$ k $ . The results suggest a better resolution in prospecting by various (acoustic, electromagnetic, etc) stationary waves with higher wave numbers without any geometric assumptions on domains and observation sites.- 1 January 2019
- Published by American Institute of Mathematical Sciences (AIMS) in Inverse Problems and Imaging
- Vol. 13 (5) , 983-1006
On increasing stability of the continuation for elliptic equations of second order without (pseudo)convexity assumptions
We derive bounds of solutions of the Cauchy problem for general elliptic partial differential equations of second order containing parameter (wave number)$ k $ which are getting nearly Lipschitz for large$ k $ . Proofs use energy estimates combined with splitting solutions into low and high frequencies parts, an associated hyperbolic equation and the Fourier-Bros-Iagolnitzer transform to replace the hyperbolic equation with an elliptic equation without parameter$ k $ . The results suggest a better resolution in prospecting by various (acoustic, electromagnetic, etc) stationary waves with higher wave numbers without any geometric assumptions on domains and observation sites.- 29 March 2018
- Published by IOP Publishing in Inverse Problems
- Vol. 34 (5) , 055005
On increasing stability in the two dimensional inverse source scattering problem with many frequencies
In this paper, we will study the increasing stability in the inverse source problem for the Helmholtz equation in the plane when the source term is assumed to be compactly supported in a bounded domain Ω with a sufficiently smooth boundary. Using the Fourier transform in the frequency domain, bounds for the Hankel functions and for scattering solutions in the complex plane, improving bounds for the analytic continuation, and the exact observability for the wave equation led us to our goals which are a sharp uniqueness and increasing stability estimate when the wave number interval is growing.
- 29 March 2018
- Published by IOP Publishing in Inverse Problems
- Vol. 34 (5) , 055005
On increasing stability in the two dimensional inverse source scattering problem with many frequencies
In this paper, we will study the increasing stability in the inverse source problem for the Helmholtz equation in the plane when the source term is assumed to be compactly supported in a bounded domain Ω with a sufficiently smooth boundary. Using the Fourier transform in the frequency domain, bounds for the Hankel functions and for scattering solutions in the complex plane, improving bounds for the analytic continuation, and the exact observability for the wave equation led us to our goals which are a sharp uniqueness and increasing stability estimate when the wave number interval is growing.
- Open Access
- 1 January 2018
- Published by American Institute of Mathematical Sciences (AIMS) in Inverse Problems and Imaging
- Vol. 12 (4) , 955-970
Inverse source problems without (pseudo) convexity assumptions
We study the inverse source problem for the Helmholtz equation from boundary Cauchy data with multiple wave numbers. The main goal of this paper is to study the uniqueness and increasing stability when the (pseudo)convexity or non-trapping conditions for the related hyperbolic problem are not satisfied. We consider general elliptic equations of the second order and arbitrary observation sites. To show the uniqueness we use the analytic continuation, the Fourier transform with respect to the wave numbers and uniqueness in the lateral Cauchy problem for hyperbolic equations. Numerical examples in 2 spatial dimension support the analysis and indicate the increasing stability for large intervals of the wave numbers, while analytic proofs of the increasing stability are not available.
- 1 January 2018
- Published by American Institute of Mathematical Sciences (AIMS) in Inverse Problems and Imaging
- Vol. 12 (4) , 955-970
Open AccessInverse source problems without (pseudo) convexity assumptions
We study the inverse source problem for the Helmholtz equation from boundary Cauchy data with multiple wave numbers. The main goal of this paper is to study the uniqueness and increasing stability when the (pseudo)convexity or non-trapping conditions for the related hyperbolic problem are not satisfied. We consider general elliptic equations of the second order and arbitrary observation sites. To show the uniqueness we use the analytic continuation, the Fourier transform with respect to the wave numbers and uniqueness in the lateral Cauchy problem for hyperbolic equations. Numerical examples in 2 spatial dimension support the analysis and indicate the increasing stability for large intervals of the wave numbers, while analytic proofs of the increasing stability are not available.
- Open Access
On increasing stability in the two dimensional inverse source scattering problem with many frequencies
In this paper, we will study increasing stability in the inverse source problem for the Helmholtz equation in the plane when the source term is assumed to be compactly supported in a bounded domain $\Omega$ with sufficiently smooth boundary. Using the Fourier transform in the frequency domain, bounds for the Hankel functions and for scattering solutions in the complex plane, improving bounds for the analytic continuation, and exact observability for wave equation led us to our goals which are a sharp uniqueness and increasing stability estimate with larger wave numbers interval.Open AccessOn increasing stability in the two dimensional inverse source scattering problem with many frequencies
In this paper, we will study increasing stability in the inverse source problem for the Helmholtz equation in the plane when the source term is assumed to be compactly supported in a bounded domain $\Omega$ with sufficiently smooth boundary. Using the Fourier transform in the frequency domain, bounds for the Hankel functions and for scattering solutions in the complex plane, improving bounds for the analytic continuation, and exact observability for wave equation led us to our goals which are a sharp uniqueness and increasing stability estimate with larger wave numbers interval. - 25 February 2017
- Published by Springer Nature
- p. 309-354
Inverse Parabolic Problems