Quasilinear Equations, Inverse Problems and Their Applications

Moscow Institute of Physics and Technology, Dolgoprudny

5 Dec. 2017 - 7 Dec. 2017

**Agaltsov, A. D.**

Inverse scattering without phase information

Abstract(Joint work with A. Gillman, T. Hohage and R. G. Novikov) We consider the phaseless inverse scattering problem for the Schrodinger equation at fixed energy. The problem consists in determination of an unknown potential under the assumption that only the modulus of the scattering amplitude can be measured directly in an experiment. This limitation is related to the probabilistic interpretation of wavefunction which goes back to Max Born (1926). We show that a single experiment is not sufficient for the unique determination of the potential, but that it is possible to get rid of this non-uniqueness and recover the potential by making two additional experiments with known background potentials. We introduce an iterative reconstruction algorithm, estimate reconstruction errors and give numerical examples. The talk is based on the results of the work [2] and of the previous works [1,3]. [1] A. D. Agaltsov, R. G. Novikov, Error estimates for phaseless inverse scattering in the Born approximation at high energies, The Journal of Geometric Analysis, doi:10.1007/s12220-017-9872-6; arXiv:1604.06555v2 [2] A. D. Agaltsov, A. Gillman, T. Hohage, R. G. Novikov, An interative approach to monochromatic phaseless inverse scattering, in preparation [3] R. G. Novikov, Explicit formulas and global uniqueness for phaseless inverse scattering in multidimensions, The Journal of Geometric Analysis 26, 346-359, 2016

**Aristova, E. N.**, Karavaeva, N. I.

Bicompact schemes for numerical solving of transport problems by HOLO algorithms

AbstractThe integral-differential transport equation is common description of neutron or uncharged particles transport processes. One of the ways to accelerate iterations due to presence of an integral term in the equation is implementation of HOLO algorithms. At least two (or more) systems of equations are solved: high order (HO) transport equation with respect to distribution function in phase space and time and low order (LO) equations which are direct consequences of the transport equation on some conditions. The first HOLO algorithm was quasi-diffusion method (QDM) suggested by V.Ya.Goldin in 1964. At the present time HOLO algorithms find application for transport problems, plasma etc. In QDM low order equations derived from the transport equation by integration over angle variables defining a direction of particle propagation. These equations are written with respect to macroscopic parameters such as flux and vector flux. At present work a bicompact schemes of fours approximation order on space and third approximation order on time are used for both HO and LO systems of equations. Bicompact schemes of different approximation orders in space and time (not less than 4 and 3 respectively) for transport equation are investigated in B.V.Rogov works. A bicompact scheme for the quasi-diffusion system of equations with approximation orders matched with these ones for transport equation are presented in this work. The stencil of bicompact scheme includes two points in each space direction and two points in time. The bicompact scheme is constructed in semi-discrete way (method of lines). Time derivatives are remained in differential form, finite differences are used only on space variables. The high order of approximation in space is possible due to widening list of unknown variables and finding not only nodal values of unknown functions but spatial cell averaged values too. SDIRK methods are used for numerical integration of the semi-discrete system over time. Unlike to schemes for transport equation, a boundary problems appear in schemes for the system of quasi-diffusion equations. Economic algorithm for numerical solving of this boundary problem has been suggested.

**Balashov, M. V.**

The Lipschitz property of the metric projection in the Hilbert space

AbstractWe shall consider when the metric projection on a closed (nonconvex in general) subset from a real Hilbert space is Lipschitz continuous with respect to the point. We shall specify the class of subsets with the Lipschitz property of the metric projection.

**Beklemysheva, K. A.**

Numerical modeling of a low-velocity impact on hybrid composites

AbstractPolymer composites that are being used in aviation are often exposed to low-velocity strikes - hail, gravel, hits during maintenance, etc.). These strikes don’t leave visible marks or dents, but lead to the emergence of barely visible impact damage, which severely reduces the residual strength of the whole part. In this work, we propose the application of an existing set of programs based on the grid-characteristic method for the investigation of various parameters that can influence the final strength of hybrid composites with respect to low-speed impacts. The grid-characteristic method proposed in this work was previously used for numerical simulation of polymer composites and helped to obtain a good correspondence to the experiment. Its application for the analysis of failure processes in hybrid composites will allow to modify the structure of the hybrid composite and increase its strength at low-velocity impacts.

**Belishev, M. I.**

On algebras of 3d harmonic quaternion fields**Boman, J.**

Injectivity of the Radon transform with restricted data

AbstractFor a compactly supported function $f$ in the plane we denote by $Rf(L)$ the value of the Radon transform $R f$ at the line $L$, i.e., $R f(L) = \int_L f\, ds$, where $ds$ denotes arc length measure on the line $L$. Given a compact set $K$ in the plane one can ask for which open and connected sets $E$ of lines $L$ it is true that a function $f$, supported in $K$, is uniquely determined by $R f(L)$ for all $L \in E$. This question, as well as a couple of closely related questions, will be discussed in the talk.

**Chistyakov, V.**, Kolubin, S. A.

Quasilinear integral equation in dynamics of rigid body projectile motion in medium conditions close to real ones

AbstractDual-projective variables were used to trasform radically dynamical equations of projectile motion in medium with density gradient and sophisticated dependence of aerodynamical forces and torques on speed. Account of all the factors above leads to integral equation relatively second derivative of tangent line intercept by inclination. In frames of perturbative approach these equations may be led to quasilinear and solved by iterative method. And inverse Lagrange transformation allows to receive parametrical equation of the projectile trajectory.

**Demchenko, M. N.**

Local reconstruction of solution of the wave equation from boundary data

AbstractWe consider the problem of determining solution of the wave equation in a domain from Cauchy data given on the boundary. The case of local data is studied: the solution and its normal derivative are given on some part of the boundary on a finite time interval. We provide a reconstruction algorithm based on analytic expressions. Possible reformulations of the problem in consideration are the source identification problem and the problem of recovering initial data in the IBVP for the wave equation. Such problems arise in geophysics (ground-penetrating radar), photoacoustic tomography with limited data, and source identification in the linearized theory of shallow water waves.

**Demidov, A. S.**

On the search algorithm for all essentially different solutions of the inverse problem for the Grad-Shafranov equation

AbstractFor suppression of plasma instabilities in a tokamak is necessary to know the distribution of electric current $j:\overline{\omega}\ni (x,y)\mapsto j(x,y)\ge 0$ flowing through the cross section $\omega$ plasma discharge. In the simplest case, $j(x,y)=f_{u}(x,y),$ where $f_{u}(x,y)=f\bigl(u(x,y)\bigl).$ What as regards the functions $f$ and $u$, it is known only that \begin{equation}\label{0.1new} \tag{1} \left|\int_{\omega}f\bigl(u(x,y)\bigl)\,dxdy- 1\right| \le \mu\ll 1\,, \end{equation} \begin{equation}\label{LapluF} \tag{2} \Delta u=f\bigl(u(x,y)\bigl)\ge 0\,,\quad u\bigl|_{\gamma=\partial\omega}=0,\quad \sup\limits_{P\in\gamma}\Bigl|\frac{\partial u}{\partial\nu}(P)-~\Phi(P)\Bigl|\le\lambda\sup\limits_{P\in\gamma}\Bigl|\Phi(P)\Bigl|\,. \end{equation} Here $1/\lambda\gg 1,$ and $\Phi$ is a given function. It has been shown [1] that for some $\Phi$ there are essentially different solutions $f_u^1$ and $f_u^2$ of the problem \eqref{0.1new}-\eqref{LapluF} in the sense that \begin{equation*}\label{0.6} \Bigl|\frac{\|f^1_u\|-\|f^2_u\|} {\max\limits\{\,\|f^1_u\|\,,\,\|f^2_u\|\,\}}\Bigl|\ge \alpha\sim~0.1\div 0.2\,,\quad\text{where}\quad \|f^j_u\|\overset{def}{=} \max\limits_{(x,y)\in\omega}|f^j_u(x,y)|\,, \end{equation*} and \begin{equation*}\label{eq6new} (\hat x,\hat y)\in \mathop{\rm absmax} f^1_u \qquad \Rightarrow %\Longrightarrow \qquad (\hat x,\hat y)\in \mathop{\rm absmin} f^2_u\,. \end{equation*} The report will consider the finding of all essentially different solutions to the problem \eqref{0.1new}-\eqref{LapluF}. [1] A. S. Demidov and V. V. Savelyev (2010), Essentially different current distributions in the inverse problem for the Grad-Shafranov equation, Russian Journal of Mathematical Physics, Vol. 17, No 1, pp. 56-65.

**Dymarskii, Y.**

Fibration of the space of potentials and periodic solutions of Korteweg–de Vries equation

AbstractWe consider the space of stationary Schrodinger Operators on a circle and describe the fibration of this space to hypersurfaces of constant spectral n-th lacuna. Solutions of KdV-equations lie on the fibers, which is homotopic to circle. Therefore, the degree of projection of solution to circle is determined. We prove that the degree is equal to the number n of the eigenvalue.

**Dryuma, V. S.**

On solutions of the flow equations of incompressible liquids

AbstractAn examples of exact solutions of the Euler system of equations are obtained. Their properties are discussed.

**Favorskaya, A. V.**

About the development of strategies for minimizing of risks of a different nature by full-wave numerical simulation

AbstractFull-wave numerical simulation by grid-characteristic method allows to analyze dynamic wave patterns and dynamics of destructions of various structures. This analysis enables geophysicists to study the causes of the appearance of certain effects on seismograms, which is an alternative to solving inverse problems. Also, analysis of dynamic wave patterns can be used to improve algorithms for solving inverse problems. Analysis of the dynamics of destruction of various buildings enables engineers to optimize their structures to minimize the risks of their destruction as a result of negative impacts of natural and man-made character.

**Filatova, V. M.**, Pestov L., Nosikova V.

Combined method for detecting small inclusions in the ultrasound tomography problem

AbstractThe paper is devoted to the problem of determining small sound speed fluctuations in medical ultrasound tomography problem. We studied two-dimensional sound speed model based on model of breast, considered in the papers of N. Duric [1]. The sound speed in the fatty tissue and the glandular tissue are assumed to be known but the boundary “FG” between them is unknown as well as inclusions. The problem we are numerically solved is to find “FG” boundary and inclusions visualization. Moreover we recover sound speed in inclusions. The method is based on 1) - determination of the internal boundary, 2) - visualization of inclusions using method of Reverse Time Migration and 3) - determination of sound speed in inclusions. The results of numerical simulation are presented. This work was supported by the Russian Science Foundation under grant 16-11-10027. 1. Sandhu GY, Li C, Roy O, Schmidt S, Duric N. Frequency domain ultrasound waveform tomography: breast imaging using a ring transducer // Physics in Medicine & Biology. 2015. 60, P. 5381-5398

**Fomochkina, A.**, Bukchin, B.

Application of parallel calculations to speed up the determination of earthquake parameters by a systematic exploration of their values

AbstractWe consider the source of an earthquake in approximation of instant point shift dislocation. Such a source is given by its depth, the focal mechanism determined by three angles (strike, dip and slip) and the seismic moment characterizing the earthquake intensity. We determine the source depth and focal mechanism by a systematic exploration of 4D parametric space, and seismic moment - by solving the problem of minimization of the misfit between observed and calculated surface wave spectra for every combination of all other parameters. But this approach in the case of sufficiently high detail of the exploration requires considerable computer time. To speed up the determination of earthquake parameters we apply the methods of parallel calculations. The results of use of such approach to study the deep Okhotsk sea earthquake (2013) are presented.

**Galchenkova, M.**, Demidov, A. S.

The inverse Magnetoencephalography problem and its flat approximation

AbstractContrary to the prevailing opinion about the incorrectness of the inverse MEEG-problem, we prove that its uniqueness solution in the framework of the electrodynamic system of Maxwell equations~[1]. The solution of this problem is the distribution of ${\bf y} \mapsto {\bf q}({\bf y})$ current dipoles of brain neurons that occupies the region $Y \subset \mathbb{R}^3 $. It is uniquely determined by the non-invasive measurements of the electric and magnetic fields induced by the current dipoles of neurons on the patient's head. The solution can be represented in the form ${\bf q}={\bf q}^*+\rho\delta\Big|_{\partial Y}$, where ${\bf q}^*$ is the usual function defined in $Y,$ and $\rho\delta\Big|_{\partial Y} $ is a $\delta$-function on the boundary of the domain $Y$ with a certain density $\rho.$ However, in cases where the conductivity is assumed to be everywhere the same (in the brain, skull, ambient air) and, in addition, it is not possible (or impossible) to record {\it in time} the electric and magnetic inductions, it is impossible to completely find ${\bf q}$. Nevertheless, it is still possible to obtain partial information about the distribution of ${\bf q}: Y\ni {\bf y} \mapsto {\bf q}({\bf y})$. This question is considered in detail in a flat model situation. A.S. Demidov (2017) Unique solvability of the inverse MEEG-problem (to appear) T.A. Stroganova et al. (2011) EEG alpha activity in the human brain during perception of an illusory kanizsa square, Neuroscience and Behavioral Physiology, V. 41 (2), 130-139. M.H. Hamalainen et al. (1993) Magnetoencephalography -- theory, instrumentation, and applications to noninvasive studies of the working human brain, Reviews of Modern Physics, V. 65, No 2,413-497. A.S. Demidov (1973) Elliptic pseudodifferential boundary value problems with a small parameter in the coefficient of the leading operator, Math. USSR-Sb., 20:3, 439-463. A.S. Demidov (1975) Asymptotics of the solution of the boundary value problem for elliptic pseudo-differential equations with a small parameter with the highest operator, Trudy Moskov. Math. obshchestva, 32, Moscow University Press, M. 119-146 (In Russian). A.S. Demidov, M.A. Galchenkova, A.S. Kochurov (2015) On inverse problem magneto-ence-phalography, Quasilinear equations, inverse problems and their applications. Moscow, 30.11.2015--02.12.2015, Conference handbook and proceedings, p. 22.

**Golubev, V.**, Favorskaya, A.

Seismic imaging in elastic media: novel approaches**Goncharov, F. O.**

Injectivity and non-injectivity for weighted Radon transforms

AbstractWe consider weighted Radon transforms $R_W^d$ (and, in particular, weighted ray transforms along straight lines) with strictly positive weights $W$. In this talk we recall the results on injectivity and non-injectivity for $R^d_W$ obtained by Quinto (1983), Markoe, Quinto (1985), Finch (1986), Boman (1993) and we present the recent progress in this direction. In particular, we present several recent examples of non-injectivity for $R^d_W$ which significantly extend the aforementioned old results. The talk is based, in particular, on the works [1], [2], [3]. 1. Goncharov F.O., Novikov R.G., An example of non-uniqueness for the weighted Radon transforms along hyperplanes in multidimensions, hal-01583755v2, 2017. 2. Goncharov F.O., Novikov R.G., An example of non-uniqueness for Radon transforms with continuous positive rotation invariant weights, hal-01593781v1, 2017. 3. Goncharov F.O., Novikov R.G., An example of non-uniqueness for weighted ray transforms in multidimensions (in preparation), 2017.

**Hohage, T.**

Source conditions and stability estimates**Holodov, A. S.**

On the approximation of strongly oscillating data of large volume**Ivanov, A.**, Khokhlov, N.

Applying perfectly matched layer boundary condition to wave propagation modeling in layered elastic media

AbstractApplication of FDTD method on staggered grid to elastic and acoustic waves propagation modeling is considered. Due to axial symmetry of task, equations are rewritten in cylindrical coordinates, because it will reduce dimensionality and computation will be performed faster. PML boundary condition is also should be rewritten in those coordinates. Correct formulation of CPML (convolution PML) in case of axial symmetry is considered. We propose method of sequential correction of computed derivatives according to boundary condition. This procedure independent on type of differential equation and can be applied to any operator.

**Ivanov, G. E.**

Weak convexity of functions and the infimal convolution

AbstractIn terms of sup-inf convolutions we characterize the class CWC(f) of convolutionally weakly convex functions, generated by the function f. Motivated by Moreau-Yosida and Lasry-Lions regularizations, we consider a subclass of CWC(f), namely, the class of regularly convolutionally weakly convex functions and show that it is exactly the same as previously known class WC(f).

**Jollivet, A.**

Generalized stability estimates in inverse transport theory**Khubiev, K. U.**

Boundary value problem for characteristically loaded hyperbolic-parabolic equation

AbstractIn the paper we establish a theorem of solution uniqueness and existence for a non-local problem for characteristicly loaded mixed hyperbolic-parabolic equation.

**Kirzhinov, R. A.**

On solution uniqueness of a non-local problem for a second-order equation of mixed type

AbstractIn the paper we obtained conditions for unique solution existence to the study problem in the rectanglular domain. To prove problem solution uniqueness we used Green function. The solution has been found explicitly.

**Klemashev, N. I.**, Shananin, A. A.

Testing economic data for consistency with the model of two rational representative consumers

AbstractWe provide a review of the applications of the generalized non-parametric method to studying the economic data for the case when the observed economic behavior is not consistent with the classical Pareto's model of a single rational representative consumer, but may be consistent with the more general model of two representative consumers. We start with the recent result about NP-completeness of a problem of testing economic data for consistency with the model of two or more temporary dictators. This model is a particular case of the general model with many representative consumers. We then overview two cases where we could revealed two rational representative consumers while not having the general algorithm for this. One case is about studying the budget statistics of UK in 1975 to 1999 and revealing the emergence of the class of the richest households. The second case is about studying the Chinese stock market crash in the Summer 2015 and revealing the class of minor investors or speculators. That analysis allowed us to reveal a single stock that explains the increased irrationality on the Chinese stock market in the Summer 2015.

**Klibanov, M. V.**

Phaseless Inverse Scattering and Global Convergence for Inverse Problems

AbstractInverse Scattering Problems without the phase information arise in applications to imaging of nanostructures. In this case the wavelength of the incident wave is of the order of one micron size. On this range of wavelengths only the intensity of the complex valued wave field can be measured. The phase cannot be measured. We will present uniqueness theorems and reconstruction methods (joint with V.G. Romanov) for these problems. In fact, these results, along with some results of R.G. Novikov, address, for the first time, a well known unsolved problem posed in 1977 by French mathematicians K. Shadan and P. Sabatier. The second topic, which is closely connected with the first one, is the topic of globally convergent numerical methods for Coefficient Inverse Problems with single measurement data. Global convergence addresses the following crucial question: “How to obtain at least one point in a sufficiently small neighborhood of the exact coefficient without any advanced knowledge of this neighborhood?” In the last several years, the research group of the Speaker has achieved a significant progress in this topic. These results will be presented. In particular, we will present results for experimental data. These data were collected by our group.

**Korotyaev, E.**

Estimates for Laplacians and Schroedinger operators on the lattice

AbstractShortly we consder the Schroedinger operators on the lattice with dim > 2. We obtain the dispersion estimates for the Laplacian, obtain the trace formulas for the case of complex potentials and estimate the complex eigenvalues in terms of potentials.

**Kulaev, R. Ch.**, Pogrebkov, A. K., Shabat, A. B.

Darboux system and its generalizations

AbstractMethod of the Inverse scattering transform gave numerous results that developed spectral theory of third order differential equations. In our talk we present two versions of application of this theory to generalized Darboux equations, that comes from the classical problem on orthogonal coordinate systems in R^3.

**Lapin, A.**

Domain Decomposition and Uzawa-Type Iterative Methods for an Elliptic Variational Inequality**Leonov, A. S.**, Bakushinsky A. B.

Fast numerical method of solving 3D coefficient inverse problem for wave equation from integral data

AbstractAn inverse coefficient problem for time-dependent wave equation in three dimensions is under consideration. We are looking for a spatially varying coefficient of this equation knowing special time integrals of the scattered wave field in an observation domain. The inverse problem has applications to the reconstruction of the refractive index of an inhomogeneous medium, as well as to acoustic sounding, medical imaging etc. In the article, a new linear 3D Fredholm integral equation of the first kind is introduced from which it is possible to find the unknown coefficient. We present and substantiate a numerical algorithm for solving this integral equation. The algorithm does not require large computational resources and big-time implementation. It is based on the use of fast Fourier transform under some a priori assumptions about unknown coefficient and observation region of the scattered wave field. Typical results of solving this 3D inverse problem on a personal computer for simulated data demonstrate the capabilities of the proposed algorithm.

**Lopushanski, M. S.**

Properties of weakly convex sets in asymmetric normed spaces

AbstractIn this work we present different results for weakly convex sets is spaces with asymmetric seminorm. We present the theorem for the well-posedness of the closest points problem and the Separation Theorem for weakly and strongly convex sets w.r.t. a quasiball. We also proved the proximal normal and Frechet normal regularities for weakly convex sets in spaces with asymmetric seminorm.

**Mikhailov, V. S.**, Mikhailov A. S.

The Boundary Control method and De Branges spaces

AbstractWe establish the relationship between the Boundary Control method for dynamic inverse problems and the method of De Branges on the examples of dynamical systems for Schrodinger and Dirac operators on a half-line and semi-infinite discrete Schrodinger operator. For each of the system we construct the De Branges space and describe in natural dynamic terms its attributes: the set of function the space consists of, the scalar product, the reproducing kernel.

**Molchan, G.**

The Inviscid Burgers Equation with Fractional Brownian Initial Data: The Dimension of Regular Lagrangian Points

AbstractYa. Sinai and U. Frisch initiated in 1992 the study of fractal properties of solutions of the inviscid Burgers equation with a stochastic initial velocity u(x). One of the problems that arose was the Hausdorff dimension of regular Lagrangian points S. These points describe the initial locations of those fluid particles which have not collided until a fixed time. The original model of u(x) was Fractional Brownian motion, H-FBM, with Hurst parameter $0 \le H \le 1$. Sinai (1992) showed that $dim S=1/2$ for the Brownian motion case (H=1/2). Handa (1993) found simple arguments to derive a lower bound of dim S, namely dim S >=H. The exact equality dim S=H is known as a conjecture since 1992. We represent a complete proof of the conjecture as a result of analysis of more general problem related to so called persistence/survival exponents of H-FBM

**Molchanov, E. G.**

Finding of substitution elasticity in the inverse problem arising in analysis of economic statistics**Novikov, R. G.**

Weighted ray transforms and applications

AbstractWe consider the problem of inversion of weighted ray transforms. This problem arises in different tomographies and, in particular, in single photon emission computed tomography and positron emission tomography. We present old and very recent results on this problem.

**Omelchenko, S. S.**

Solving entropy-regularized optimal transport problem with small parameter of regularization

AbstractThe most popular method of solving entropy-regularized optimal transport problem is the Sinkhorn method. But it can fail to work when the entropy-regularization parameter is small. In the talk will be introduced a new accelerated gradient method with adaptive choice of gradient's Lipschitz constant. This method has similar to Sinkhorn's method complexity of each iteration, but is faster and more stable numerically, when the regularization parameter is small. There will be given an illustration of the advantage of method by numerical experiments for the some applications.

**Pankratov, L. S.**

Nonlinear flow through doudle porosity media in variadle exponent sobolev spaces

AbstractWe studied the asymptotic behavior of the solution of a nonlinear parabolic equation with nonstandard growth in a ε-periodic fractured medium, where ε is a parameter that characterizes the scale of the microstructure tending to zero. We consider a double porosity type model describing the flow of a compressible fluid in a heterogeneous anisotropic porous medium obeying the nonlinear Darcy law. We assume that the permeability ratio of matrix blocks to fractures is of order $ε^{p_ε(x)}$, where $p_ε(x)$ is a continuous positive function. We obtained the convergence of the solution and a macroscopic model of the problem was constructed using the notion of two-scale convergence combined with the variational homogenization method in the framework of Sobolev spaces with variable exponents.

**Petrosyan, N. S.**

Asymptotic time-behavior of the solution of the mixed problem for quasilinear scalar conservation law

AbstractWe study the behavior as the time increases indefinitely of the generalized entropy solution of the mixed problem in the quarter of plane for the quasilinear scalar first-order conservation law with a strictly convex flux function. We suppose that the initial function is measurable, bounded and has a limiting mean value that is uniform with respect to translations, the boundary function is constant.

**Petrov, D. I.**

Application of numerical methods in seismic problems in the Arctic

AbstractThe aim of this work is numerical simulation of wave propagation in the Arctic. The main goal of the research is studying the influence of the Arctic geological medium on longitudinal and transverse velocity components by carrying out numerical experiments. Keywords: grid-characteristic method, numerical modeling, Arctic seismic exploration

**Podoroga, A. V.**, Tikhonov, I. V.

The general principal of the stabilization for periodic solutions of the quasi-linear equation of traffic flow**Polovinkin, E. S.**

Calculation of Subdifferentials for the Difference of Two Convex Functions

AbstractIt is shown that for some classes of functions all epiderivatives and subdifferentials of F. H. Clarke type and P. Michel – J.-P. Penot type and others coincide. Several rules of calculation of subdifferentials for the difference of two convex functions are obtained. Some examples are considered.

**Karov, D. D.**, Puro, A. E.

Optical Polarimetric Tomography Residual Stresses with Beam Deflection (Grin Articles)

AbstractGRIN structures, like rod lenses, laser rods, ball lenses, fiber preforms, optical waveguides, diffractive gratings, etc. are widely used in many fields of optics and optoelectronics. Most of them have an axially symmetric refractive index distribution. Only some works devoted to reconstruction of residual stresses by using polarimetric tensor tomography with beam deflection has been published during this time. All these investigations are based on the measurement of the change in the polarization of light passed through a birefringent medium. Birefringence in GRIN lenses is very weak thus the linearized quasi - isotropic approximation can used as the bases of tensor tomography. In this case, we can separate optical problem on two parts: determination of the ray paths and determination of the change in the polarization of light passed through a birefringent medium. A surprisingly large diversity of optical profiling techniques has been developed during last decades, and first problem has numerous technical solution [1]. Solution of the second problem can be written as two linearized integrals. One of them is connected with the transversal interaction of the two-dimensional vector field and the other with the transversal interaction of the two-dimensional tensor field. From a mathematical point of view we have specal case of tesor tomography with beam deflection [2]. The reconstruction algorithm is based on cicular harmonic decomposition (Cormac – type inversion)[3]. This method of reconstruction is applicable to axially symmetric refractive index and axisymmetric attenuation. In uor case complete reconstruction of the tensor stress field by using sample transilluminatuion in a system of parallel planes is possible only based on additional information. In particula case , the consept of fictitious temperature is used when determinig the residual stress in GRIN lenses. In contrast to the straight light rays polarimetric tomography the optical problem and the inverse thermoelastic problem must be solved together [4]. References 1. Yablon A. D. Recent Progress in Optical Fiber Refractive Index Profiling Published in: OSA/OFC/NFOEC 2011. Los Angeles. USA. DOI: 101364/ OFC.2011 OMF1 2. Monard F. Efficient tensor tomography in fan-beam coordinates, Inverse Probl. Imaging, 10 (2016), pp. 433–459. 3. Puro A., Garin A. Cormack-type inversion of attenuated Radon transform. Inverse problems. 2013. V.29. N.6 14p. doi:10.1088/0266-5611/29/6/065004 4. Puro A., Karov D. Inverse problem of thermoelasticity of fiber gratings. Journal of thermal stresses. 2016. V.39. № 5. P. 500 – 512.

**Sakbaev, V. Zh.**

On the variational description of the averaging quantum dynamical maps

AbstractThe dynamics of quantum system with degenerated Hamiltonian will be studied. To this end the approximating sequence of regularized Hamiltonians and corresponding sequence of dynamical semigroups acting in the space of quantum states will be considered. The regularization semigroups will be investigated as the random variable with values in the space of strongly continuous operator-valued function. The mean values of random semigroups and its properties are investigated. The trajectories of the family of averaging dynamical maps are described by the solutions of some variational problems.

**Shananin, A. A.**

Inverse problems in economic measurement

AbstractThe report discusses mathematical problems of economic measurement in the conditions of globalization of world markets.

**Sharafutdinov, V. A.**

Orthogonality relations for a stationary flow of ideal fluid

AbstractThe work relates to the long standing open question: Do the stationary Euler equations have a non-trivial smooth compactly supported solution in $R^3$? If such a solution exists, it satisfies an infinite series of integral equations that are called orthogonality relations. For example, if $v=(v_1,v_2,v_3)$ is the velocity vector field, then the functions $v_1,v_2,v_3$ are pairwise $L^2$-orthogonal to each other and have coincident $L^2$-norms.

**Shifrin, E. G.**

Solving the inverse problem of aerpdynamic designing by transformation to a well-posed problem in the hodograph variables

AbstractThe aerodynamic designing providing the non-separated flow of an ideal perfect or imperfect gas is presented. The Chaplygin transformation of the plane, or axially symmetric potential transonic flow makes it well-posed. The model of ideal gas is adequate by virtue of the existence continuous boundary layer condition. The examples discussed: supersonic Laval nozzles, nozzles with the central body, supersonic nozzle lattices of turbines, subcritical wing profiles, inlet valves of piston engines. Designing turbojets and rocket nozzles is discussed as well.

**Shurup, A. S.**, Rumyantseva, O. D.

Numerical modeling of functional algorithms for purposes of medical and ocean acoustic tomography problems

AbstractAcoustic tomography is the unique approach for studying natural media that are transparent to acoustic waves when direct measurement of medium properties is difficult or impossible. The main characteristics, which influence on propagation of acoustic waves and, as result, which can be reconstructed by acoustic tomography methods, are spatial distributions of sound speed, medium density, absorption coefficient, and vector field of flows. To use algorithms, which allow reconstruction of all mentioned medium characteristics for a wide class of scatterers in a joint tomography scheme, is perspective for solution of the inverse scattering problems in acoustic applications. Most of the known methods for the solution of acoustic tomography problems are approximate. The linear approximation is generally applied with iteration procedures and regularizations. The general perturbation theory is also considered. On the other hand, there are quite mathematically rigorous (at least, for a rather wide class of scatterers) functional-analytical methods for solving the inverse problems; these methods were initially developed in quantum mechanics. Nowadays, detail investigations based on numerical modeling are required to understand applicability of these methods for acoustics inverse problems. Possibilities of the functional algorithms [1–5] for the purposes of 2D and 3D acoustic tomography are discussed in the present report. These algorithms take into account the multiple scattering processes and do not require either linearization of the model with iterations or an additional regularization. The joint reconstruction of sound speed, absorption coefficient, medium density and vector field of flows is considered in 2D problem. In comparison with previously reported results, reconstruction of frequency dependence of the absorption coefficient is regarded. The frequency dependence of the absorption coefficient is an additional informative parameter, which can be used in medical applications for early diagnostics of breast cancer. Results of 3D tomography reconstruction which takes into account multichannel scattering effects are also presented. These results are perspective for the ocean acoustic tomography which takes into account nonadiabatic propagation of acoustic modes in ocean waveguides. Results of the numerical simulation which show high resolution and good interference resistance of the considered functional algorithms are presented. Thus, the algorithms can be acceptable for practical purposes. The work was supported by RFBR grants № 16-29-02097 ofi_m and № 16-02-00680. Grinevich P.G., Novikov R.G. Analogues of multisoliton potentials for the two-dimensional Schrödinger equations and a nonlocal Riemann problem // Soviet Math. Dokl. 1986. V. 33. N 1. P. 9–12. Novikov R.G. The inverse scattering problem on a fixed energy level for the two-dimensional Schrödinger operator // Journal of Functional Analysis. 1992. V. 103. N 2. P. 409–463. Novikov R.G., Santacesaria M. Monochromatic reconstruction algorithms for two-dimensional multi-channel inverse problems // International Mathematics Research Notices. 2013. V. 2013. N 6. P. 1205–1229. Agaltsov A.D., Novikov R.G. Riemann–Hilbert problem approach for two-dimensional flow inverse scattering // J. Math. Phys. 2014. V. 55. N 10. P. 103502-1–103502-25. Agaltsov A.D. On the reconstruction of parameters of a moving fluid from the Dirichlet-to-Neumann map // Eurasian Journal of Mathematical and Computer Applications. 2016. V. 4. N 1. P. 4–11.

**Stognii, P. V.**

Numerical modeling of wave processes in the Arctic with ice formations being present

AbstractThis work discusses seismic replies in the Arctic shelf in models with ice cover and toroses. Much attention is payed to divisible waves in different media.

**Taimanov, I.**

The Moutard transformation of two-dimensional Dirac operators and conformal geometry of surfaces in the four-space**Utkin, P. S.**, Sidorenko, D. A.

Multiscale modeling of shock wave – dense particles cloud interaction

AbstractThe work is devoted to the mathematical modeling of shock wave – dense particles cloud interaction. The problem is considered on different scales. The detailed gas dynamics simulations using Cartesian grid method are used to get information about the collective effects during shock wave – particles interaction. The simulations using heterogeneous media mechanics approach (Baer-Nunziato equations) are used to describe the large scale effects such as particles cloud motion. The results of modeling using both approaches are compared with each other and with the available experimental data.

**Vedenyapin, V.**, Adjiev S. Z., Kazantseva V. V., Fimin N. N., Melihov I. V., Negmatov M. A.

The Hamilton–Jacobi Method in the Non Hamiltonian Situation and Boltzmann extremals

AbstractThe hydrodynamic substitution, which is well known in the theory of the Vlasov equation [1–3], has recently been applied to the Liouville equation and Hamiltonian mechanics [4–8]. In [4–6], Kozlov outlined the simplest derivation of the Hamilton–Jacobi(HJ) equation, and the hydrodynamic substitution simply related this derivation to the Liouville equation[7, 8,10]. The hydrodynamic substitution also solves the interesting geometric problem of how a surface of any dimension subject to an arbitrary system of nonlinear ordinary differential equations moves in Euler coordinates (in Lagrangian coordinates, the answer is obvious). This has created prerequisites for generalizing the HJ method to the non Hamiltonian situation. The H-theorem is proved for generalized equations of chemical kinetics, and important physical examples of such generalizations are considered: a discrete model of the quantum kinetic equations (the Uehling–Uhlenbeck equations) and a quantum Markov process (a quantum random walk). The time means are shown to coincide with the Boltzmann extremes for these equations and for the Liouville equation[9,11]. This give possibility to prove existence of analogues of action-angles variables in non hamiltonian situation. 1. A. A. Vlasov, Statistical Distribution Functions (Nauka, Moscow, 1966) [in Russian]. 2. D. Bom, General Theory of Collective Coordinates (Wiley, New York, 1959; Mir, Moscow, 1964). 3. V. V. Vedenyapin, A. V. Synitsyn, and E. I. Dulov, Kinetic Boltzmann, Vlasov, and Related Equations (Elsevier, Amsterdam, 2011). 4. V. V. Kozlov, Vestn. Mosk. Univ. Ser. Mat. Mekh., No. 6, 10–22 (1983). 5. V. V. Kozlov, Symmetry, Topology, and Resonances in Hamiltonian Mechanics (Udmurt. Gos. Univ., Izhevsk, 1995) [in Russian]. 6. V. V. Kozlov, General Vortex Theory (Udmurt. Gos. Univ., Izhevsk, 1998) [in Russian]. 7. V. V. Vedenyapin and N. N. Fimin, Lioville equation, hydrodynamic substitution and Hamilton-Jacoby method. Dokl. Math. 86, 697–699 (2012). 8. V. V. Vedenyapin and M. A. Negmatov, On the Topology of Steady State Solutions of Hydrodynamic and Vortex Consequences of the Vlasov Equation and the Hamilton–Jacobi Method Dokl. Math. 87, 240–244 (2013). 9. V. V. Vedenyapin and S. Z. Adzhiev. Entropy in the sense of Boltzmann and Poincare Russian Math. Surveys 69:6 995–1029. Uspekhi Mat. Nauk 69:6 45–80 (2014). 10. Веденяпин В.В., Негматов М.А., Фимин Н.Н., Уравнения типа Власова и Лиувилля и их микроскопические, энергетические и гидродинамические следствия. Изв. РАН. Сер. матем., 81:3 (2017), 45–82. 11. SZ Adzhiev, IV Melikhov, VV Vedenyapin. The H-theorem for the physico-chemical kinetic equations with explicit time discretization. Physica A: Statistical Mechanics and its Applications. Volume 481, 1 September 2017, Pages 60–69.

**Volosova, N. K.**, Volosova, A. K., Volosov, K. A.

Inverse problems and the Korteweg-de-Vries equation AbstractWe used relatively new method of ``not fixed constructive changes of variables'' dynamic systems for the nonlinear partial differential Harry Dym and Korteweg-de Vries (PDE HD and KdV) equations. The necessary condition for the stability of the phase trajectories and condition of reduction the volume of the phase flow was formulated. It was proved that two necessary solvability conditions of the system PD of the first order has one common non-trivial factor. Nonlinear PDE HD and KdV equations have a new feature, namely: after the change of variables in the equation for the function of the first derivative in new variables, this equation is accurately separated from all another equations. This is nonlinear PDE, which is important for further researches. The study of dynamic systems in neighborhoods of the fixed points was performedd. We found two new wide classes of exact solutions both for KdV and HD equations. The method gives a possibility to construct exact and approximate solutions.

**Yagola, A.**, Lukyanenko D. V., Wang Y.

Regularized Inversion of Full Tensor Magnetic Gradient Data

AbstractFeatures of numerical solution of the three-dimensional ill-posed problem devoted to the inversion of full tensor magnetic gradient data are considered. This problem is simulated by a system of three-dimensional Fredholm integral equations of the first kind. The Tikhonov regularization is applied to solve this ill-posed problem. The conjugate gradient method is used as a minimization method. The choice of the regularization parameter is realized according to the generalized discrepancy principle with consideration of round-off errors capable of affecting the final result of calculations significantly. Authors thank the Russian Foundation for Basic Research and the National Science Foundation of China for partial financial support (grant 17-51-53002- NSFC-a) REFERENCES A.N.Tikhonov, A.V.Goncharsky, V.V.Stepanov, A.G.Yagola. Numerical methods for the solution of ill-posed problems. - Kluwer Academic Publishers, Dordrecht, 1995, p. 1-253. A.N.Tikhonov, A.S.Leonov, A.G.Yagola. Nonlinear ill-posed problems. V.1 - Chapman and Hall, London, 1998, p. 1-222. A.N.Tikhonov, A.S.Leonov, A.G.Yagola. Nonlinear ill-posed problems. V.2 - Chapman and Hall, London, 1998, p. 223-387. D.V. Lukyanenko, A.G. Yagola, N.A. Evdokimova. Application of inversion methods in solving ill-posed problems for magnetic parameter identification of steel hull vessel. – Journal of Inverse and Ill-Posed Problems, 2011, v. 18, issue 9, pp. 1013–1029. D.V. Lukyanenko, A.G. Yagola. Some methods for solving of 3D inverse problem of magnetometry. - Eurasian Journal of Mathematical and Computer Applications, 2016, v.4, N 3, pp. 4-14.

**Zhang, S.**

Farsighted v.s. myopic optimal pricing and emission reduction strategy in a green supply chain under emission permits trading

AbstractAs an effective tool to reduce the emission of greenhouse gas (GHG), the market-based emission permits trading scheme has been developing prosperously in recent years. In this study, we examine the impacts of emission permits trading and foresight on low carbon supply chain behaviors. In a manufacturer-retailer channel, the unit production cost will decrease with the accumulation of production because of the presence of learning by doing, and the market demand is influenced by the emission level per unit production due to the improvement of consumers' environmental awareness. Our results show that the emission permits price and unit production emission play an important role in the choice of channel members' pricing strategies. It is necessary for the farsighted decision maker to be patient and to be confident of the future from the profit perspective. For emission reduction strategy, the integrated channel's emission reduction level is much higher than the one in the decentralized channel. Additionally, when the emission permits price is high enough, there will be a free-riding in the decentralized channel, i.e. the retailer acquires the extra benefit from the manufacturer's emission reduction behavior.