We apply a fully connected neural network to determine the shape of the lacunae in the solutions ... more We apply a fully connected neural network to determine the shape of the lacunae in the solutions of the wave equation. Lacunae are the regions of quietness behind the trailing fronts of the propagating waves. The network is trained using a computer simulated data set containing a sufficiently large number of samples. The network is then shown to correctly reconstruct the shape of lacunae including the configurations when it is fully enclosed.
This monograph presents a mathematical perspective on synthetic aperture imaging of the Earth’s s... more This monograph presents a mathematical perspective on synthetic aperture imaging of the Earth’s surface from satellites. Its main focus is on the accurate quantitative description of the distortions of SAR images due to the ionosphere and on the development and analysis of the means for mitigating those distortions (Chapter 3). The discussion of transionospheric SAR imaging also includes the case of a turbulent ionosphere (Chapter 4) and the case of a gyrotropic ionosphere (Chapter 5).
Faraday rotation (FR) affects the low-frequency transionospheric radar by creating cross-talk bet... more Faraday rotation (FR) affects the low-frequency transionospheric radar by creating cross-talk between polarizations. The baseline part of FR can be compensated for by applying an appropriate linear transformation - rotation with a known FR angle. Yet the differential Faraday rotation (dFR), which is a frequency-dependent part of FR, persists and introduces distortions into the observations. We build a simplified model with two polarimetric scattering channels that allows us to evaluate the effect of dFR on the accuracy of PolInSAR reconstruction. We also assess the severity of distortions due to dFR for the future BIOMASS mission and several other spaceborne radar systems.
Radar interferometry is an advanced remote sensing technology that utilizes complex phases of two... more Radar interferometry is an advanced remote sensing technology that utilizes complex phases of two or more radar images of the same target taken at slightly different imaging conditions and/or different times. Its goal is to derive additional information about the target, such as elevation. While this kind of task requires centimeter-level accuracy, the interaction of radar signals with the target, as well as the lack of precision in antenna position and other disturbances, generate ambiguities in the image phase that are orders of magnitude larger than the effect of interest.Yet the common exposition of radar interferometry in the literature often skips such topics. This may lead to unrealistic requirements for the accuracy of determining the parameters of imaging geometry, unachievable precision of image co-registration, etc. To address these deficiencies, in the current work we analyze the problem of interferometric height reconstruction and provide a careful and detailed account ...
The nonlinear Schr#odinger equation #NLS# is the standard model for propagation of intense laser ... more The nonlinear Schr#odinger equation #NLS# is the standard model for propagation of intense laser beams in Kerr media. The NLS is derived from the nonlinear Helmholtz equation #NLH# by employing the paraxial approximation and neglecting the backscattered waves. In this study we use a fourth-order #nite-di#erence method supplemented by special two-way arti#cial boundary conditions #ABCs# to solve the NLH as a boundary value problem. Our numerical methodology allows for a direct comparison of the NLH and NLS models and for an accurate quantitative assessment of the backscattered signal.
A Theoretical Introduction to Numerical Analysis, 2006
In this chapter, we provide a very brief account of the classical potential theory and show how i... more In this chapter, we provide a very brief account of the classical potential theory and show how it can help reduce a given boundary value problem to an equivalent integral equation at the boundary of the original domain. We also address the issue of discretization for the corresponding integral equations, and identify the difficulties that limit the class of problems solvable by the method of boundary elements.
For the analysis of the SAR data inversion algorithm in Chapters 2 through 5, we have employed th... more For the analysis of the SAR data inversion algorithm in Chapters 2 through 5, we have employed the start-stop approximation, which is considered standard in the literature, see, e.g., [25, 40, 76, 79] and also [86]. It assumes that the radar antenna is at standstill while it sends the interrogating pulse toward the target and receives the scattered response, after which the antenna moves down the flight track to the position where the next pulse is emitted and received.
: Since the 1950s, the work of Professor Victor S. Ryaben'kii has been of central importance ... more : Since the 1950s, the work of Professor Victor S. Ryaben'kii has been of central importance for the formation and development of numerical methods as a mathematical discipline. International conference Difference Schemes and Applications in honor of his ninetieth birthday took place in May of 2013 at the Keldysh Institute of Applied Mathematics, Russian Academy of Sciences (RAS), in Moscow, Russia. The conference brought together a number of leading experts in computational mathematics and related areas. It provided a forum for discussing the recent progress in numerical solution of partial differential equations (PDEs), and for reviewing the promising new trends in this and other fields. During three working days, about sixty oral and poster presentations were given, discussing the following subjects: * Numerical analysis of PDEs and scientific computation; * Differential and difference equations; * Difference potentials, artificial boundary conditions; * Inverse problems, mat...
We propose an efficient high order accurate boundary algorithm for the numerical solution of unst... more We propose an efficient high order accurate boundary algorithm for the numerical solution of unsteady exterior initial boundary problems for the three-dimensional wave equation. The algorithm relies on the method of difference potentials combined with the Huygens’ principle.
We consider problems that involve the propagation of waves over large regions of space with smoot... more We consider problems that involve the propagation of waves over large regions of space with smooth, but not necessarily constant, material characteristics, separated by interfaces of arbitrary shape. The external boundaries can also be arbitrarily shaped. We present a numerical methodology for solving such problems that provides high order accuracy. It is based on Calderon’s operators and the method of difference potentials and overcomes the difficulties inherent in more traditional approaches.
We apply a fully connected neural network to determine the shape of the lacunae in the solutions ... more We apply a fully connected neural network to determine the shape of the lacunae in the solutions of the wave equation. Lacunae are the regions of quietness behind the trailing fronts of the propagating waves. The network is trained using a computer simulated data set containing a sufficiently large number of samples. The network is then shown to correctly reconstruct the shape of lacunae including the configurations when it is fully enclosed.
This monograph presents a mathematical perspective on synthetic aperture imaging of the Earth’s s... more This monograph presents a mathematical perspective on synthetic aperture imaging of the Earth’s surface from satellites. Its main focus is on the accurate quantitative description of the distortions of SAR images due to the ionosphere and on the development and analysis of the means for mitigating those distortions (Chapter 3). The discussion of transionospheric SAR imaging also includes the case of a turbulent ionosphere (Chapter 4) and the case of a gyrotropic ionosphere (Chapter 5).
Faraday rotation (FR) affects the low-frequency transionospheric radar by creating cross-talk bet... more Faraday rotation (FR) affects the low-frequency transionospheric radar by creating cross-talk between polarizations. The baseline part of FR can be compensated for by applying an appropriate linear transformation - rotation with a known FR angle. Yet the differential Faraday rotation (dFR), which is a frequency-dependent part of FR, persists and introduces distortions into the observations. We build a simplified model with two polarimetric scattering channels that allows us to evaluate the effect of dFR on the accuracy of PolInSAR reconstruction. We also assess the severity of distortions due to dFR for the future BIOMASS mission and several other spaceborne radar systems.
Radar interferometry is an advanced remote sensing technology that utilizes complex phases of two... more Radar interferometry is an advanced remote sensing technology that utilizes complex phases of two or more radar images of the same target taken at slightly different imaging conditions and/or different times. Its goal is to derive additional information about the target, such as elevation. While this kind of task requires centimeter-level accuracy, the interaction of radar signals with the target, as well as the lack of precision in antenna position and other disturbances, generate ambiguities in the image phase that are orders of magnitude larger than the effect of interest.Yet the common exposition of radar interferometry in the literature often skips such topics. This may lead to unrealistic requirements for the accuracy of determining the parameters of imaging geometry, unachievable precision of image co-registration, etc. To address these deficiencies, in the current work we analyze the problem of interferometric height reconstruction and provide a careful and detailed account ...
The nonlinear Schr#odinger equation #NLS# is the standard model for propagation of intense laser ... more The nonlinear Schr#odinger equation #NLS# is the standard model for propagation of intense laser beams in Kerr media. The NLS is derived from the nonlinear Helmholtz equation #NLH# by employing the paraxial approximation and neglecting the backscattered waves. In this study we use a fourth-order #nite-di#erence method supplemented by special two-way arti#cial boundary conditions #ABCs# to solve the NLH as a boundary value problem. Our numerical methodology allows for a direct comparison of the NLH and NLS models and for an accurate quantitative assessment of the backscattered signal.
A Theoretical Introduction to Numerical Analysis, 2006
In this chapter, we provide a very brief account of the classical potential theory and show how i... more In this chapter, we provide a very brief account of the classical potential theory and show how it can help reduce a given boundary value problem to an equivalent integral equation at the boundary of the original domain. We also address the issue of discretization for the corresponding integral equations, and identify the difficulties that limit the class of problems solvable by the method of boundary elements.
For the analysis of the SAR data inversion algorithm in Chapters 2 through 5, we have employed th... more For the analysis of the SAR data inversion algorithm in Chapters 2 through 5, we have employed the start-stop approximation, which is considered standard in the literature, see, e.g., [25, 40, 76, 79] and also [86]. It assumes that the radar antenna is at standstill while it sends the interrogating pulse toward the target and receives the scattered response, after which the antenna moves down the flight track to the position where the next pulse is emitted and received.
: Since the 1950s, the work of Professor Victor S. Ryaben'kii has been of central importance ... more : Since the 1950s, the work of Professor Victor S. Ryaben'kii has been of central importance for the formation and development of numerical methods as a mathematical discipline. International conference Difference Schemes and Applications in honor of his ninetieth birthday took place in May of 2013 at the Keldysh Institute of Applied Mathematics, Russian Academy of Sciences (RAS), in Moscow, Russia. The conference brought together a number of leading experts in computational mathematics and related areas. It provided a forum for discussing the recent progress in numerical solution of partial differential equations (PDEs), and for reviewing the promising new trends in this and other fields. During three working days, about sixty oral and poster presentations were given, discussing the following subjects: * Numerical analysis of PDEs and scientific computation; * Differential and difference equations; * Difference potentials, artificial boundary conditions; * Inverse problems, mat...
We propose an efficient high order accurate boundary algorithm for the numerical solution of unst... more We propose an efficient high order accurate boundary algorithm for the numerical solution of unsteady exterior initial boundary problems for the three-dimensional wave equation. The algorithm relies on the method of difference potentials combined with the Huygens’ principle.
We consider problems that involve the propagation of waves over large regions of space with smoot... more We consider problems that involve the propagation of waves over large regions of space with smooth, but not necessarily constant, material characteristics, separated by interfaces of arbitrary shape. The external boundaries can also be arbitrarily shaped. We present a numerical methodology for solving such problems that provides high order accuracy. It is based on Calderon’s operators and the method of difference potentials and overcomes the difficulties inherent in more traditional approaches.
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Papers by Semyon Tsynkov