Quasilinear equations, inverse problems and their applications
Moscow Institute of Physics and Technology, Dolgoprudny
12 Sept. 2016 - 15 Sept. 2016

Planned talks

  • Agaltsov, A. D.
    An explicit algorithm for solving the acoustic tomography problem for a moving fluid
    Abstract, Slides

    We consider a moving fluid in a bounded two-dimensional domain $D$ with variable sound speed $c$, fluid velocity $v$, density $\rho$ and absorption $\alpha$. There are acoustic transducers on the boundary of the domain $D$ which can emit and record time-harmonic acoustic waves. In the acoustic tomography experiment one fixes an emitting transducer, which produces a time-harmonic wave at some fixed frequency $\omega$. This wave propagates through the fluid, and the scattered wave is recorded by the receiving transducers. Next, we change the emitting transducer and, possibly, the frequency, and repeat the experiment. The acoustic tomography problem consists in recovering the fluid parameters from the described measurements of the scattered acoustic waves. This problem is motivated by the applications in medical tomography, where one is interested in recovering the parameters of a body including the blood flows, and by the applications in the ocean tomography, where one is interested in recovering the temperature distribution as well as the currents performing the heat transfer. The proposed algorithm is based, in particular, on the papers [1], [2], [3]. The main ingredient is the solution of a non-local Riemann-Hilbert problem. The algorithm was numerically implemented and studied in some particular cases by Andrey Shurup and Olga Rumyantseva from the acoustics department of Moscow State University. The performed simulations give evidences of a relatively good noise stability and, as a corollary, of a good applicative potential of this approach. Literature: 1. Agaltsov, A. D. and Novikov, R. G., Riemann-Hilbert problem approach for two-dimensional flow inverse scattering, Journal of Mathematical Physics 55 (10), 2014, id103502 2. Agaltsov, A. D., Finding scatteromg data for a time-harmonic wave equation with first-order perturbation from the Dirichlet-to-Neumann map, Journal of Inverse and Ill-Posed Problems 23 (6), 2015, 627-645 3. 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 4 (1), 2016, 4-11

  • Baratchart, L.
    Inverse Potential problems in divergence form: some uniqueness, separation, and recovery issues
    Abstract, Slides

    We derive a topological additive decomposition for $\mathbb R^n$-valued vectors fields of class $L^p$ on a compact Lipschitz hypersurface embedded in $\mathbb R^n$, $1 < p < \infty$, into the trace of a harmonic gradient in the exterior component of the complement of the surface, the trace of a harmonic gradient in the interior component of the complement of the surface, an a tangential vector field which is divergence free on the surface. We apply this decomposition to non-uniqueness issues in inverse potential problems in divergence form, i.e. inverse magnetization problems. We discuss related approximation issues by gradients of discrete potentials.

  • Beklaryan, L. A., Beklaryan, A. L.
    The Questions of Existence of the Periodic and Limited Solutions of Travelling Wave Type
    Abstract, Slides

    Many applied problems lead to studing solutions of travelling waves type for infinite-dimensional dynamic systems. In the report the approach is presented by which the solutions of travelling waves type are studied for a wide class of such systems. The approach will be shown on an example of finite difference analogue of the wave equation with nonlinear potential $$ m_i \ddot{y}_i = y_{i+1} - 2y_i + y_{i-1} +\phi(y_i), \qquad i \in \Bbb Z, \quad t\in \Bbb R.\tag{1} $$ In case of a homogeneous environment (i.e. $m_i=m$, $i \in \Bbb Z$) it is possible to describe all space of the solutions of travelling waves type at the general assumptions on potential in the form of Lipshchitz's condition. The solutions of travelling waves type for system (1) are realised as the solutions of induced one-parametrical family functional-differential equations of pointwise type. In case of the nonhomogeneous environment for system (1) there are not the solutions of travelling waves type, witch is different from stationary, or rectilinear uniform movements. In this connection, the new class of solutions in the form of quasisolutions of travelling waves type is defined.Such quasisolution of travelling waves type for system (1) is "correct" expansion of concept of travelling waves type solution and that coincides with it in case of a homogeneous environment. The quasisolutions of travelling waves type for system (1) are realised as the impulse solutions of induced one-parametrical family functional-differential equations of pointwise type. At the same time, in many cases important presence of solutions of travelling waves type with special properties-it is the periodic and limited solutions of travelling waves type. In frameworks of the offered approach also it is possible to receive the existence conditions both periodic, and the limited solutions of travelling waves type. Bibliography 1. Fraenkel J.I., Kontorova T.A. About the theory of plastic deformation and duality //JETF. (1938) V.8. pp. 89-97. 2. Pustyl'nikov L.D. Infinite-dimensional nonlinear ordinary The differential equations and theory KAM// Uspehi Mathem. Nauk (1997) V.52, N.3 (315). pp. 106-158. 3. Beklaryan L.A. Group singularities of the differential equations with deviating argument and connected to their metric invariants// VINITI. RESULTS OF THE SCIENCE AND ENGINEERING. MODERN MATHEMATICS AND ITS APPLICATIONS (1999.) V.67, pp. 161-182. 4. Beklaryan L.A. Introduction in the theory of the functional - differential equations and their application. The group approach//Modern Mathematics. Fundamental Directions. V.8 (2004.) pp. 3-147. 5. Beklaryan L.A. About quasisolutions of travelling waves// Mathematicheskii Sbornik, (2010) 201:12, 21-68.

  • Beklemesheva, K. A., Vasyukov, A. V., Ermakov, A. S., Kazakov, A. O.
    Numerical modeling of non-destructive testing of composites
    Abstract, Slides

    The talk is devoted to numerical modeling of non-destructive testing of a aviation polymer composite covering fragment. Grid-characteristic method is used for solving the system of equations with high space and time resolution. Spatiotemporal dynamic load distribution is calculated for several damage patterns. Elastic waves in composite medium are investigated, including interactions with both external borders of the sample and internal contact boundaries between composite subpackages. A-scans are obtained and compared with experimental data.

  • Belishev, M. I, Simonov, S. A.
    Wave model of the Sturm-Liouville operator with defect indices (1,1) on the semi-axis
    Abstract, Slides

    The notion of a wave spectrum of a symmetric semi-bounded operator was introduced by one of the authors in 2013 [JOT]. A wave spectrum is a topological space, which is canonically determined by the operator. The definition makes the use of a dynamical system associated with the operator: the wave spectrum is constructed from its reachable sets. We describe the wave spectrum of the operator $$L_0=-\frac{d^2}{dx^2}+q,$$ which acts in the space $L_2(0,\infty)$ and has the defect indices $(1,1)$. A functional (wave) model of the operator $L_0^*$ is constructed, the elements of the original $L_2(0,\infty)$ being realized as functions on the wave spectrum. In fact, the model turns out to be identical to the original $L_0^*$. The latter is important for solving inverse problems since the wave model is determined by the inverse data that enables one to recover the original and, thus, find the potential $q$. [JOT] M.I.Belishev, A unitary invariant of a semi-bounded operator in reconstruction of manifolds, Journal of Operator Theory, Volume 69 (2013), Issue 2, 299-326.

  • Boman, J.
    Holmgren theorems for the Radon transform
    Abstract, Slides

    Hörmander's proof of Holmgren's uniqueness theorem was based on a microlocal regularity theorem for solutions of linear PDE's with real analytic coefficients and the following local unique continuation theorem for distributions satisfying an analytic wave front condition. If a distribution vanishes on one side of a $C^1$ hypersurface $S \subset \mathbf R^n$ in some neighborhood of a point $x \in S$, and at least one of the conormals $(x,\xi)$ to $S$ at $x$ is not in $W\!F_A(f)$, then $f=0$ in some neighborhood of $x$. A related local unique continuation theorem for distributions (A local vanishing theorem for distributions, C. R. Acad. Sci. Paris 315 Série I (1992), 1231-1234) reads as follows. Let $S$ be a real analytic submanifold of $\mathbf R^n$ of arbitrary codimension near $x \in S$, and let $f$ be a distribution, defined in some neighborhood of $x$, such that $(x,\xi) \notin W\!F_A(f)$ for every conormal $\xi$ to $S$ at $x$. Assume moreover that the distribution $f$ is flat along $S$ in the sense that the restrictions to $S$ of derivatives of $f$ of all orders are equal to zero. Those restrictions are well defined as distributions on $S$ because of the wave front condition. The conclusion is that $f$ must vanish in some neighborhood of $x$. This result implies unique continuation theorems for Radon transforms, wave equations, and for CR functions.

  • Buchstaber, V. M.
    On a model of Josephson effect, dynamical systems on two-torus and Heun equations
    Abstract, Slides

    We study a family of two-parametric nonlinear equations that arises in the problem of modeling the overdamped Josephson junction in superconductivity. This family is parametrized by the third parameter: the frequency. It originates from quantum mechanics but also arises in several problems of classical mechanics and geometry. It is equivalent to a special three-parametric dynamical systems on two-torus that also arises in the theory of slow-fast systems. We fix a frequency and consider the rotation number of dynamical system as a function of two remaining parameters. The phase-lock areas are the level sets of the rotation number that have non-empty interiors. An important problem is the description of the dynamics of the structure of phase-lock area portrait, as the frequency tends to either zero, or infinity. We present a series of results and conjectures on the geometry of phase-lock areas obtained in collaboration with O.Karpov, S.Tertychnyi and A.Glutsyuk. These results were obtained via complexification, which allowed to use the methods of investigation of equations in the complex domain. This was realized via reduction of the nonlinear equations under question to appropriate second order linear ordinary differential equations on functions of complex variable: a certain subfamily of double confluent Heun equations. The first result is the quantization effect of the rotation number: the phase-lock areas exist only for integer rotation numbers (joint result with O.Karpov and S.Tertychnyi). A series of joint results with S.Tertychnyi relates the geometry of phase-lock areas to the existence of entire (or polynomial) solutions of the double confluent Heun equations. A joint result with A.Glutsyuk implies a conjecture due to S.Tertychnyi and the speaker about the description of the parameter values corresponding to the adjacencies of the phase-lock areas (equivalently, to the double confluent Heun equations having entire solutions). We also present a recent joint result with A.Glutsyuk describing the double confluent Heun equations having monodromy eigensolution with a given eigenvalue. The latter result implies a description of boundaries of phase-lock areas as solutions of explicit analytic functional equations.

  • Chistyakov, V.
    On rotational dynamics of a rigid body around non-principal central axis under combinated friction acting

    The dynamics is studying for rigid body rotating around fixed axis $Oz$ being central but not principal. Therefore the inertial torques $M_x$ and $M_y$ arose depending both on mass geometry $J_{xz}$, $J_{yz}$ and on angular velocity $\omega$ and acceleration $\varepsilon$. Dry friction acting on axis’s supports with coefficient $\delta$ leads to that the value of $\varepsilon$ serves as the reason and result of the motion simultaneously. There were integrated numerically and/or analytically the dynamical equations of free and forced motion including rotational harmonic and inharmonic oscillations too. The results obtained are comparing with those following from the standard linear equations. The work is fulfilled in frames of the RBRF grant 16-08-00997 «Investigation of nonlinear multiple controlled mechanical systems by means of mathematical and computer modelling»

  • Galchenkova, M., Demidov, A., Kochurov, A.
    A flat approximation of inverse MEG-problems
    Abstract, Slides

    We get the explicit solution of a flat approximation of inverse MEG-problems.

  • Dobrokhotov, S., Nazaikinskii, V. and Tirozzi, B.
    Asymptotics of wave propagation and run-up in the framework of Shallow Water Equations
    Abstract, Slides

    We consider 2D shallow water equations with spatially localized sources. First we construct the effective formulas for the wave propagation far from the coast taking into account focal points on the fronts. Then we show that the coastline in linear theory is a caustic of a special type as well, and describe the behavior of the asymptotic solution near the coastline. The passage to the nonlinear run-up problem is based on the Carrier--Greenspan transform, which allows one to compute the uprush distance on the beach. We discuss the use of obtained formulas for the reconstruction of the source. All these results are described by several closed-form analytical expressions and illustrated by pictures based on these expressions. No special prior knowledge is needed for understanding the talk.

  • Dryuma, V.
    On integration of the equations of flows of incompressible liquids
    Abstract, Slides

    The Navier-Stokes and the Euler systems of equations are considered. An examples of non singular solutions are constructed and their properties are discussed.

  • Dymarskii, Ya.
    Fibration of the periodical eigenfunctions manifold into hypersurfaces
    Abstract, Slides

    We consider the space of stationary Schrodinger Operators on a circle. For each natural $n$, this space of operators generates the manifold of eigenfunctions with exactly $2n$ zeros. We provide the description of analytical and smooth structures on this manifold. We also describe this manifold as a fibration of hypersurfaces of costant eigenvalue as well as hypersurfaces of constant oriented length of the spectral gap.

  • Elaeva, M., Zhukov, M., Shiryaeva, E.
    Interaction of weak and strong discontinuities for two-component mixture separation
    Abstract, Slides

    The problem of zone electrophoresis is studied. Mathematical model is first-order PDE with discontinuous initial data. The discontinuities are defined in different space points. In this case the weak and strong discontinuities interact with each other. In particular the interaction of weak discontinuities (rarefaction waves) appears. In this case we have the Goursat problem with initial date on characteristics. Using the hodograph method the solution is constructed analytically in the form of implicit relations. A numerical algorithm is described that reduces the PDE’s to some ODE’s. This algorithm recovers the explicit form of solution. The numerical results for zonal electrophoresis equations with discontinuous initial data are presented.

  • Epstein, Ch. L.
    Degenerate Diffusions in Population Biology
    Abstract, Slides

    Models that describe the behavior of evolving populations and financial markets are usually phrased in the language of stochastic differential equations, which in turn define Markov processes. The underlying variables in these processes are confined to subsets that typically have non-smooth boundaries, like simplices and orthants. The PDEs that generate these processes are degenerate elliptic operators, where the degeneracies reflect important empirical properties of the underlying processes. In this talk I will describe a class of such operators, called Kimura diffusions that are naturally defined on manifolds with corners. I will then detail recent progress on the regularity theory of solutions to the elliptic and parabolic problems, and the tools used to establish these results. This is recent joint work of CE, along with Rafe Mazzeo, Camelia Pop and Jon Wilkening.

  • Favorskaya, A.
    Numerical modeling of seismic wave processes using grid-charactreistic method
    Abstract, Slides

    Grid-characteristic method is a numerical method developed for physically and mathematically correct investigation of wave processes in heterogeneous media. There are several modifications of this method aimed to solve different kinds of practical problems. This work is aimed to study seismic waves in the Earth.

  • Filatova, V., Pestov, L., Nosikova, V.
    Application of Reverse Time Migration (RTM) procedure for ultrasound tomography problem
    Abstract, Slides

    We consider the medical ultrasound imaging problem [1]. For simulating data we use a special version of time reversal continuation to obtain ultraweak inclusions in a complicated acoustical model. The results of RTM procedure [2] for ultrasound imaging data are presented. This paper was supported by the RFBR grant 16-31-00265. 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 2. D. Rocha, N. Tanushev, P. Sava Acoustic wavefield imaging using the energy norm // 2015 SEG Annual Meeting, 18-23 October, New Orleans, Louisiana, P. 49-68

  • Gasnikov, A. V.
    Inverse problem in mathematical modelling of computer networks
    Abstract, Slides

    Based on the talk "Proceedings MIPT. 2016. V. 8. no. 3 arXiv:1603.07701" we plane to describe efficent primal-dual algorithm for Minimal mutual information model. This model allow to calculate demand matrix in computer network when one can observe only traffic on links. We consider large computer network with more than 10 000 vertices (nodes) and 100 000 edges (links).

  • Gindikin, S. G.
    Complex integral geometry and around
    Abstract, Slides

    This talk is a reminisce about several years of a joint work with Gena Henkin. I want recall not so much our specific results as many conceptual discussions about multidimensional complex analysis, including a complex approach to Radons transform, possibilities to develop a holomorphic language for higher cohomology Cauchy-Riemann, explicit formulas for residues at multiple polar singularities etc.

  • Golse, F.
    On the Mean-Field and Semiclassical Limits of the N-Body Schrödinger Equation
    Abstract, Slides

    The mean-field limit of the dynamics of large particle systems replaces the N-body motion equation with a single-body equation, driven by the potential created by the particle density itself. In classical mechanics, the mean-field equation is the Vlasov equation, while in quantum mechanics, it is either the Hartree equation or the Hartree-Fock equation depending on whether the particles considered are bosons or fermions. In this talk, we discuss whether the mean-field limit is uniform in the Planck constant, i.e. whether the mean-field and the semiclassical limit commute. (Work in collaboration with C. Mouhot and T. Paul).

  • Golubev, V.
    Simulation of 3-D seismic responses from curvilinear geological boundaries
    Abstract, Slides

    Seismic survey process is the common technique for understanding the structure of subsurface area. It is an important step in the estimation of potential locations of oil and gas deposits and evaluation of their reserves. Nowadays there are a lot of different approaches for numerical simulation of this process. During the past decades a rapid grows in the direction of full-wave modeling was observed. In this article the method for simulation of seismic processes in geological media with curvilinear boundaries was successfully applied to the real-scale models of Arkhangelsk region.

  • Goncharov, F. O.
    An analog of Chang inversion formula for weighted Radon transforms in multidimensions
    Abstract, Slides

    We consider weighted Radon transforms in multidimensions. We introduce an analog of Chang approximate inversion formula for such transforms and describe all weights for which this formula is exact. In addition, we indicate possible tomographical applications of inversion methods for weighted Radon transforms in 3D. This talk is based on the recent work [F.O. Goncharov, R.G. Novikov, Eurasian Journal of Mathematical and Computer Applications 4(2), 2016, 23 – 32].

  • Grigorievykh, D.
    Simulation of large elastic-plastic deformation using markers method on rectangular grids

    The processes of deformation and fracture of metals, glass and ice have been researched by experimental data. Implemented numerical algorithm using Euler structural grid, the border of the Lagrangian markers and grid-characteristic method. A numerical simulation of the following tasks are presented: a blow to the glass, ice explosion, puncture of metal plate.

  • Grinevich, P. G., Novikov, R. G.
    Moutard transform for the generalized analytic functions with contour singularities and 2-D inverse scattering problem for large data

    In 1988 P.G. Grinevich and S.P. Novikov showed that the fixed-energy inverse scattering problem for the two-dimensional Schrodinger operator with large potentials requires the study of generalized analytic functions with very special contour singularities. We show that exactly these singularities can be naturally generated or removed by applying the Moutard-type transformations, therefore we obtain an efficient tool for studying large date inverse problems.

  • Hasanov, A.
    Solvability results for direct and inverse problems related to Euler-Bernoulli equation

    In this study, two classes of inverse problems of identifying of unknown temporal and spatial loads are considered for Euler-Bernoulli beam equation $$\rho(x)w_{tt}+\mu(x)w_{t}+(EI(x)w_{xx})_{xx}-T_ru_{xx}=F(x)G(t),\\ (x,t)\in \Omega_T:=(0,l)\times (0,T).$$ In the first class of identification problems either the measured deflection $w(x,t)$ or slope $w_x(x,t)$ is assumed to be known at an appropriate boundary (at $x=0$ or $x=l$). In the second class of identification problems the same measured measured data are assumed to be known at some interior point(s). Note that in the case of long pipes and bridges, it is relevant, desirable and feasible to measure deflection, slope or bending moment near internal supports. The general purpose of this study is to develop mathematical concepts and tools that are capable to provide also effective numerical algorithms for numerical solution of the considered class of direct and inverse problems. We develop weak and regular weak solution theory for direct problems, deriving some necessary a priori estimates. Using these estimates we prove compactness of input-output operators. Tikhonov regularization method combined with the adjoint and transmission adjoint problems, allows to prove the Fréchet differentiability of cost functionals explicitly deriving them via the corresponding weak solutions of adjoint problems and the known spatial or temporal loads. Moreover, we prove that these gradients are Lipschitz continuous, which allows use of gradient types iteration convergent algorithms. Some applications of the proposed theory are presented. References: [1] Hasanov, A. (2009). Identification of an unknown source term in a vibrating cantilevered beam from final overdetermination, Inverse Problems, 25, 115015. [2] Hasanov, A., Kawano, A. (2016). Identification of unknown spatial load distributions in a vibrating Euler-Bernoulli beam from limited measured data, Inverse Problems (Online First). [3] Hasanov, A., Baysal, O. (2016). Identification of unknown temporal and spatial load distributions in a vibrating Euler-Bernoulli beam from Dirichlet boundary measured data, Automatica, 71(2016) 106-117. DOI: 10.1016/j.automatica.2016.04.034.

  • Ivanov, A., Khokhlov, N.
    Applying OpenCL technology for seismic modelling using grid-characteristic methods
    Abstract, Slides

    In recent years, there are more and more intensive usage of technologies for high-performance computing on graphics devices. These technologies are well suited to the seismic problems in elastic media, as they require large amount of computing resources. Solving these problems requires the numerical solution of hyperbolic systems of equations. We used grid-characteristic method, which works well for this type of problems. This method responds well to parallelization, because explicit method and large computational grids are used. The algorithm was implemented using OpenCL technology. Percentage of peak performance for specific devices, performance of the algorithm in GFLOPS units and speed-up of OpenCL implementation were compared with a single-core version. In addition, NVIDIA GPU tests of OpenCL implementation was compared with the same implementation on CUDA. NVIDIA and AMD devices were used for testing. Differences in the effectiveness of the single-precision and double precision calculations were considered. Intel Xeon E5-2697 was used as a device for testing single-core CPU implementation.

  • Khokhlov, N., Kassirova, O.
    Using MPI technology for parallelization grid-characteristic method on multi-block structured grids

    In this talk we proposed the algorithm of decomposition multi-block structured to calculate on multiprocessor computer systems with distributed memory using MPI technology. Special feature of the algorithm is the preservation of the structural partitioning within each block. An analytical evaluation of the effectiveness of the algorithm are obtained. As an example, consider the problem of the passing of dynamic wave disturbances through the building, which is represented by a set of structured grid.

  • Klemashev, N. A.
    Applications of the generalized nonparametric method to the analysis of a stock market crash
    Abstract, Slides

    We present the results of analysis of Chinese stock market crash in 2015 obtained with the generalized nonparametric method. The method was originally developed in order to find a generalized solution to the inverse problem of the demand analysis in case it does not have a classical solution. Our analysis shows in particular how one can reveal a few stocks trading in which have signs of fraud or inside trading.

  • Klibanov, M. V.
    Phaseless inverse scattering problems and global convergence
    Abstract, Slides

    Phaseless Inverse Scattering Problems (PISPs) arise in the case when the wavelength of electromagnetic wave is too short. For example, imaging of nanostructures of hundreds of nanometers size as well as imaging of biological cells require the wavelength of the micron size or less. While it is possible to measure the intensity of an electromagnetic wave for this kind of wavelengths, it is impossible to measure the phase, at least with a reasonable accuracy. Mathematically this means that the modulus of the scattering complex valued wave can be measured, while the phase cannot be measured. Recently a significant progress has been achieved for PISPs in joint works of M.V. Klibanov and V.G. Romanov. In addition, significant results were obtained by R.G. Novikov. In this talk we will present two sorts of results: 1. Uniqueness theorems. 2. Reconstruction procedure of Klibanov and Romanov. 3. Numerical studies. As to the numerical aspect, we have discovered that the best way is to have two-stage reconstruction procedure. On the first stage phase at the measurement surface should be recovered, as well as locations and shapes of target abnormalities. On the second stage abnormality/background contrast should be calculated via a globally convergent numerical method for a Coefficient Inverse Problem. Thus, if time will allow, we will also talk about that method and will present reconstruction results for experimental backscattering data.

  • Lukashov, A. L., Akar, H.
    Loewner Evolution as Itô Diffusion
    Abstract, Slides

    F. Bracci, M.D. Contreras, S. Díaz Madrigal [1] proved that any evolution family of order $d$ is described by a generalized Loewner chain. G. Ivanov and A. Vasil'ev [2] considered randomized version of the chain and found a substitution which transforms it to an Itô diffusion.We generalize their result to vector randomized Loewner chain and prove there are no other possibilities to transform such Loewner chains to Itô diffusions. 1. F.Bracci, M.D.Contreras, S.Díaz-Madrigal, Evolution families and the Loewner equation I: the unit disc. J. Reine Angew. Math. 672 (2012), 1-37. 2. G. Ivanov, A. Vasil'ev, Löwner evolution driven by a stochastic boundary point. Anal. Math. Phys. 1 (2011), 387-412.

  • Michel, V.
    The two dimensional conductivity inverse problem
    Abstract, Slides

    Let $\gamma$ be a (smooth) real curve and $N$ an operator acting on (smooth) functions defined on $γ$. $γ$ is assumed to be the boundary of an unknown open two dimensional real manifold $M$ equipped with an unknown conductivity tensor $\sigma$ and $N$ is assumed to be the DirichletNeuman operator associated to $(M,\sigma)$, that is the operator which to a (smooth) function $u$ on $\gamma$ associates restriction to $\gamma$ of the normal derivative of the solution of the Dirichlet problem $U|_\gamma = u$, $d\sigma(dU)=0$. Our concern is about the reconstruction of $(M,\sigma)$ from $(\gamma,N)$. Using results of Henkin, Michel, Novikov and Santecesaria in different join works and some new material, we propose an effective process to design $(M,\sigma)$ as a $\mathbb CP^3$-embedded Riemann surface equipped with a known isotropic conductivity.

  • Miryaha, V.
    Discontinuous Galerkin method for ice specimen strength investigation
    Abstract, Slides

    This paper discusses numerical modeling of various ice strength measurment experiments including uniaxial compression and bending as well as comparison of data obtained from numerical experiments with field ones. Numerical simulation is based on dynamic continuum mechanics system of equations with ice modelled as elastoplastic medium with brittle and crushing fracture dynamic criteria. Simulation software developed by the authors is based on discontinuous Galerkin method and runs on high-performance systems with distributed memory. Estimation of explicit values used by mathematical models poses a major problem because it's impossible to measure some of them in field experiments directly due to multiple physical processes interference. It's only possible to directly measure their total influence in practice. However, this problem may be solved by comparison of numerical experiment with field experiment data. As a result of this work, elastoplastic ice model adequancy is discussed and some missing physical properties are obtained from numerical experiments.

  • Muratov, M. V.
    Modeling of dynamical processes in multi-dimensional exploration seismology in fracture media problems

    The aim of the work for results in this report is the study of wave responses from fracture formations in geological media by methods of numerical modeling. The computations were produced with use of grid-characteristic method on unstructured triangle (2D-case) and tetrahedral (3D-case) meshes. Fractures were set discretely as boundaries and contact boundaries. Different contact and boundary conditions were studied with aim of their influence to response detection. The computations were produced with use of high-performance computational systems. The responses from clusters (formations) of subvertical macrofractures were obtained. The formations of different geometry (vertical and horizontal dimensions, density, depth) were involved into consideration. Also there was study of geometrical characteristics dispersions and response structure dependence on initial state form. In this work authors made the range of conclusions: 1) in case of fluid-filling fractures the brightest response one can observe on X-components of seismogram, which give us the main information about fracture formation structure; 2) the cluster (formation) of subvertical macrofractures produces intensive multiphase wave response, forming in result of refractions between fractures; 3) the response of diffracted wave from fracture cluster is rather stable to dispersions of geometry.

  • Novikov, R. G.
    Inverse scattering in multidimensions
    Abstract, Slides

    We give a short review of old and recent results on inverse scattering in multidimensions related with works of G.M. Henkin. This talk is based, in particular, on references [1]-[4]. References [1] G.M. Henkin, R.G. Novikov, The ∂¯-equation in the multidimensional inverse scattering problem, Russ. Math. Surv. 42(3), 109–180, 1987. [2] G.M. Henkin, N.N Novikova, The reconstruction of the attracting potential in the Sturm-Liouville equation through characteristics of negative discrete spectrum, Stud. Appl. Math. 97, 17–52, 1996. [3] R.G. Novikov, The ∂¯-approach to monochromatic inverse scattering in three dimensions, J. Geom. Anal. 18(2), 612–631, 2008. [4] R.G. Novikov, Formulas for phase recovering from phaseless scattering data at fixed frequency, Bull. Sci. Math. 139(8), 923–936, 2015.

  • Novikov, R. G.
    Introductory lecture for students «Tomography and integral geometry»
    Abstract, Slides

    Tomography is known first of all as a research domain related with the problem of determining the structure of an object from X-ray photographs. At present, in addition to this X-ray tomography, several other types of tomography are also known, where instead of incident X-rays some other types of radiation are used. These problems arise in medicine, biology, different domains of physics, industry, etc . On the mathematical level, these problems are often reduced to studies of classical Radon transforms and their different generalizations ( or, by other words, to problems of integral geometry). The objective of this lecture is to give an introduction to this research domain.

  • Ovsienko, V. Y.
    Discrete Boussinesq equation and projective differential geometry

    A discrete analog of the Boussinesq equation, called the Pentagram Map, has the following three properties: - this is a dynamical system with discrete time, - it is Hamiltonian and integrable in the Liouville-Arnold sense, - it naturally arises in classical projective geometry. This talk is an overview of (sometimes unexpected) relations of this subject to various topics of geometry, algebra and combinatorics.

  • Pestov, L. N.
    Boundary rigidity of simple manifolds
    Abstract, Slides

    Let $(M,g)$ be a smooth compact $n$-dimensional Riemannian manifold with the boundary $\partial M$ and $d_g$ be the distance between the boundary points. Boundary rigidity is the problem of injectivity of the map $g \to d_g$. A manifold (in some class of manifolds) is called boundary rigid, if $d_g$ determines metric $g$ up to an isometry, which is identically at the boundary. There are some examples boundary rigidity manifolds : two-dimensional manifolds of non-positive curvature (C. Croke, [1]), manifolds whose metrics close to flat (D. Burago, S. Ivanov [2]), two-dimensional simple manifolds (L. Pestov, G. Uhlmann [3]). R. Michel’s conjecture [4] assumes that simple manifolds are boundary rigid. In this talk this conjecture is proved for any dimension. The proof is based on Pestov’s differential identity [5]. This work was supported by the Russian Science Foundation under grant 16-11-10027. 1. C. Croke. Rigidity for surfaces of non-positive curvature // Comment. Math. Helv. 65 (1990), P. 150-169. 2. D. Burago and S. Ivanov. Boundary rigidity and filling volume minimality of metrics close to a flat one, manuscript, 2005. 3. L. Pestov and G. Uhlmann. Two dimensional simple compact manifolds with boundary are boundary rigid // Annals of Math. 161 (2005), P. 1093-1110. 4. R. Michel. Sur la rigite impose par la longueur des geodesiques // Invent. Math. 65 (1981), no. 1, P. 71-84. 5. L.Pestov and V. Sharafutdinov. Integral geometry of tensor fields on manifold of negative curvature. // Siberian Math. J. 29 (1988), no. 3, 427-441.

  • Petrosyan, N. S.
    Asymptotics of solutions of the Cauchy problem for a quasilinear first order equation with several space variables
    Abstract, Slides

    We investigate the convergence as time increases indefinitely of the solutions of the Cauchy problem for a scalar quasilinear first-order conservation law with several space variables to the solutions of the one-dimensional problem of decay of discontinuity (the Riemann problem).

  • Petrov, I. B.
    The problem of numerical modeling of dynamic processes in solid heterogeneous environments
    Abstract, Slides

    Wave processes modeling are widely used in many areas of science. Physically correct problem definitions and examples of numerical modeling using up-to -date numerical methods including grid-characteristic method in geophysics, seismic prospecting, global seismic, medicine, aircraft and railway industry will be discussed in my presentation. Grid-characteristic method is a numerical method for solving hyperbolic systems of equations (for example, elastic and acoustic wave equations). This method allows to calculate the wave processes in heterogeneous media accurately and physically correctly. Grid-characteristic method permits to use the correct boundary and interface conditions in integral regions. Problems of seismic prospecting, earthquake stability, global seismic on Earth and Mars, medicine, railway ultrasonic non-destructive testing, aircraft composites modeling, and other applications had been solved using the developed grid-characteristic method. In this presentation the numerical solution of collision problems and robot-technique modeling are included as well.

  • Petrov, D. I., Stognii, P. V., and Khokhlov, N. I.
    High-order schemes in numerical problems of seismic exploration in the Arctic
    Abstract, Slides

    The aim of this work is numerical simulation of wave propagation in media with linear-elastic and acoustic layers. The complete system of equations describing the state of a linearly elastic body and a system of equations describing the acoustic field are solving. The use of the grid-characteristic method provides correctly describing the contact and boundary conditions, including the contact condition of between acoustic and linear-elastic layers. Keywords: numerical modeling, Arctic shelf seismic prospecting, icebergs, grid-characteristic method

  • Podoroga, A., Tikhonov, I.
    On stability of special solutions for quasi-linear equations of traffic flow
    Abstract, Slides

    We discuss a quasi-linear equation of traffic flow with fundamental diagram of Nagel-Schreckenberg. For this equation a Cauchy problem on the ring road is considered. We present a new result on the stability of solutions for this problem.

  • Pogrebkov, A. K.
    Highest Hirota equations and their consequences
    Abstract, Slides

    Relation of integrability with commutators identities on associative algebras is used to construct higher Hirota difference equations. Differential-difference and differential integrable equations that result from continuous limits of the higher Hirota difference equation are presented.

  • Polterovich, V.
    Schumpeterian dynamics: a survey of different approaches
    Abstract, Slides

    The process of productivity growth of production units due to both technology innovations and imitation of technologies from more advanced agents is called Schumpeterian dynamics. Different mechanisms of innovation and imitation generate various patterns of Schumpeterian dynamics described by a wide range of non-linear equations, including Burgers - type equations, Kolmogorov-Petrovskii-Piskunov-type equations, Boltzmann equation, etc. I discuss the economic essence of these mechanisms and recent results of their investigations. Some unsolved related problems are formulated as well.

  • Polyakov, P. L.
    Explicit Hodge decomposition on Riemann surfaces

    Theorem. For any form $\phi\in Z^{(0,1)}\left(V\right)$ there exists a unique Hodge decomposition: $$ \phi=\bar\partial R_1[\phi]+ H_1[\phi], \tag{1} $$ where $H_1$ is the orthogonal projection operator from $Z^{(0,1)}\left(V\right)$ onto the subspace ${\cal H}^{(0,1)}\left(V\right)$ of antiholomorphic $(0,1)$-forms on $V$, $R_1=\bar\partial^*G_1$, $\bar\partial^*:{\cal E}^{(0,1)}(V)\to {\cal E}^{(0,0)}(V)$, where $\bar\partial^*=-*\bar\partial*$ is the Hodge dual operator for $\bar\partial$, $*$ is the Hodge operator, and $G_1$ is the Hodge-Green operator for Laplacian $\triangle=\bar\partial\bar\partial^*+\bar\partial^*\bar\partial$ on $V$. Hodge Theorem was proved by W.V.D. Hodge using the Fredholm's theory of integral equations. The proof was later modified by H. Weyl using his method of orthogonal projection, and by K. Kodaira, who also used the method of orthogonal projection. However, the Hodge Theorem, as it is formulated above, is not explicit enough for some applications. This disadvantage was pointed out by Griffiths and Harris in their textbook on Algebraic Geometry, where the authors remarked that the "Hilbert space method has the disadvantage of not giving us the Green's operator in the form of an integral operator" with "a beautiful kernel on $M\times M$ with certain singularities along the diagonal". Specific application of an explicit Hodge-type decomposition that Gennadi had in mind was an explicit solution of the inverse conductivity problem on a bordered Riemann surface, in which the conductivity function has to be reconstructed from the Dirichlet-to-Neumann map on the boundary of the surface (see papers by Calderon [C], and Henkin-Novikov [HN] in a more general setting going back to Gelfand [Ge]). In [HP] we made the first step toward explicit solution of the inverse conductivity problem by constructing an explicit Hodge-type decomposition for $\bar\partial$-closed residual currents of homogeneity zero on reduced complete intersections in $\mathbb CP^n$. The main results that are discussed in the present talk are some applications of this decomposition for $\bar\partial$-closed forms on an arbitrary Riemann surface. [C] A.P. Calderon, On an inverse boundary problem, In Seminar on Numerical Analysis and Its Applications to Continuum Physics, Soc. Brasiliera de Matematica, (1980), 61-73. [Ge] I.M. Gelfand, Some problems of Functional Analysis and Algebra, in Proc. Int. Congr. Math. (Amsterdam 1954), 253-276. [HN] G.M. Henkin, R. G. Novikov, On the reconstruction of conductivity of a bordered two-dimensional surface in $\mathbb R^3$ from electrical current measurements, on its boundary, J. Geom. Anal. 21 (2011), no. 3, 543-587. [HP] G.M. Henkin, P.L. Polyakov, Explicit Hodge-type decomposition on projective complete intersections, The Journal of Geometric Analysis, 26(1), 2016, 672-713, DOI 10.1007/s12220-015-9643-1.

  • Rigat, S.
    Fokas Methods applied to a Boundary Valued Problem for Conjugate Conductivity Equations
    Abstract, Slides

    In this talk, we give explicit integral formulas for the solution of planar conjugate conductivity equations in a domain of the right half-plane with conductivity $\sigma(x,y)=x^p$ for $p\in \mathbb Z^*$. The representations are obtained via a Riemann-Hilbert problem on the complex plane when $p$ is even and on a two-sheeted Riemann surface when $p$ is odd. They involve the Dirichlet and Neumann data on the boundary of the domain. We also show how to make the conversion from one type of conditions to the other by using the so-called global relation in the case where geometry of the domain is simple (a disc). This is a joint work with S. Chaabi and F. Wielonsky.

  • Romanov, A. V.
    On normally hyperbolic inertial manifolds of evolutionary equations

    For 3D reaction--diffusion equations, we study the problem of existence or nonexistence of an inertial manifold that is normally hyperbolic or absolutely normally hyperbolic. We present a system of two coupled equations with a cubic nonlinearity which does not admit a normally hyperbolic inertial manifold. An example separating the classes of such equations admitting an inertial manifold and a normally hyperbolic inertial manifold is constructed. Similar questions concerning both absolutely normally hyperbolic inertial manifolds and more general evolutionary equations are discussed.

  • Romanov, V. G.
    Some geometric aspects in inverse problems
    Abstract, Slides

    We consider inverse problems related to recovering coefficients in partial differential equations of the second order. It is supposed that some measurements of solutions to direct problems are produced on convenient sets. A study of some inverse problems for hyperbolic equations leads to geometric problems: recovering a function from its integrals along geodesic lines of the Riemannian metric or recovering the Riemannian metric inside a domain from given distances between arbitrary points of the domain boundary. Our main goal here is to demonstrate how such geometrical problems arise for equations of parabolic and elliptic types.

  • Shabat, A. B.
    Inverse scattering problems revisited
    Abstract, Slides

    Connections ISP (Inverse scattering problem) with the matrix Riemann-Hilbert problem will be established and a Uniqueness Theorem for ISP with potentials which are Borel finite measures will be proved.

  • Shananin, A. A.
    Inverse problems in resource distribution models
    Abstract, Slides

    The problem of modeling the substitution of production factors in the models of resource allocation Houthakker-Johansen. The connection conditions correct type Houthakker-Johansen models with inverse problem for the generalized Radon transform with incomplete data and Bernstein's theorems on completely monotone functions and separate analyticity.

  • Shurup, A. S., Rumyantseva, O. D.
    Acoustic tomography of scalar and vector inhomogeneities based on the Novikov-Agaltsov algorithm
    Abstract, Slides

    The results of numerical modeling the reconstruction of scalar and vector inhomogeneities are presented for purposes of acoustic tomography problems. In comparison with previously reported results, the joint reconstruction of sound speed, absorption coefficient, density and vector field of flows has been implemented. The considered acoustic tomography scheme is based on the two-dimensional functional-analytic Novikov-Agaltsov algorithm. The appropriate noise stability of this algorithm and its capability for the reconstruction of quite strong scatterers, which are beyond the first Born approximation, make it perspective to use the Novikov-Agaltsov algorithm for the development of practical schemes of acoustic tomography in different applications.

  • Sobolevski, A. N.
    Optimal transportation and geometry of Wasserstein spaces
    Abstract, Slides

    It is well-known that solution of the Monge-Kanorovich transportation problem endows the collection of measures of fixed common mass, supported on a given metric space, with a metric structure. Moreover, in terms of this metric (often called the Wasserstein metric) the set of measures may be described formally as an infinite-dimensional Riemannian manifold. This discovery, made around 2000, allowed to develop the theory in several important directions, one of which is the introduction of Wasserstein barycenters, a kind on nonlinear average in measure spaces. In the talk, based on joint work with Alexei Kroshnin, we present natural generalisation of the notion of barycenter and results on continuity and consistency of generalised barycenters (convergence of empirical barycenter to the distribution barycenters of a random measure).

  • Stognii, P. V., Petrov, D. I., Petrov, I. B.
    Numerical modeling of various ice formations in north seas and their influence on seismic replies
    Abstract, Slides

    Numerical simulation of wave propagation in the Arctic with the presence of different ice formations has been researched. The grid-characteristic method, which provides correctly describing the contact and boundary conditions, was used in calculations. All the computations have been done for 3D case in order to best of all approximate real conditions.

  • Suetin, S. P.
    On the Nuttall partition of a three-sheeted Riemann surface and limit zero distribution of Hermite-Padé polynomials
    Abstract, Slides

    Let $f \in \mathscr H(\infty)$ be a multivalued analytic function with a finite set $\Sigma\subset\mathbb C$ of branch points. For each $n\in\mathbb N$ let $P_{n,0}$, $P_{n,1}$ be Padé polynomials for $f$ of degree $n$, i.e., $$ (P_{n,0}+P_{n,1}f)(z)=O(z^{-n-1}), \quad z\to\infty. $$ From Stahl's Theorem it follows that Padé approximants $[n/n]_f:=-P_{n,0}/P_{n,1}$ converge in capacity as $n\to\infty$ to a single-valued branch of $f$ in Stahl's domain which is the ``maximal'' domain over all domains where $f$ has analytic and single valued continuation. There is also limit distribution of the zeros of Padé polynomials $P_{n,j}$. In particular, these zeros concentrated near some of the branch points of $f$ which in the sense are ``visible'' for Padé polynomials. But other branch points are ``invisible'' for them. We consider the problem of recovering of these ``invisible'' branch points form the same germ of function $f\in\mathscr H(\infty)$. The approach is based on the construction of Hermite--Padé polynomials of type I, i.e., polynomials $Q_{n,j}$ of degree $n$ and such that $$ (Q_{n,0}+Q_{n,1}f+Q_{n,2}f^2)(z)=O(z^{-2n-2}), \quad z\to\infty. $$ For our analysis the so-called Nuttall's partition of a three-sheeted Riemann surface is crucial; see [Nut], [SuKo]. These results are obtained jointly with A. V. Komlov, N. G. Kruzhilin, and R. V. Palvelev and announced in [KKPS]. [Nut] J. Nuttall, Asymptotics of Diagonal Hermite--Pad\'e Polynomials, J. Approx. Theory 42 (1984), 299-386. [SuKo] R. K. Kovacheva, S. P. Suetin, Distribution of zeros of the Hermite--Pad\'e polynomials for a system of three functions, and the Nuttall condenser, Proc. Steklov Inst. Math. 284 (2014), 168-191. [KKPS] A. V. Komlov, N. G. Kruzhilin, R. V. Palvelev, S. P. Suetin, Convergence of Shafer quadratic approximations, Uspekhi Mat. Nauk, 71, Issue 2 (428) (2016), 205-206.

  • Taimanov, I.
    The Moutard transformation of two-dimensional Dirac operators and the Mobius geometry

    We demonstrate that the Moutard transformation for two-dimensional Dirac operators has a geometrical meaning in terms of the transformation of the data of the Weierstrass representation of a surface in the three-space under the Mobius inversion. We show how to apply that to constructing blowing-up solutions of the modified Novikov-Veselov equation.

  • Tumanov, A. E., Sukhov, A.
    Symplectic non-squeezing for the discrete nonlinear Schroedinger equation
    Abstract, Slides

    The celebrated Gromov's non-squeezing theorem of 1985 says that the unit ball in a symplectic space can be symplectically embedded in the circular cylinder only if the radius of the cylinder is at least 1. Hamiltonian differential equations provide examples of symplectic transformations in infinite dimension. Known results on the non-squeezing property in Hilbert spaces cover compact perturbations of linear symplectic transformations and several specific non-linear PDEs, including the periodic Korteweg - de Vries equation and the periodic cubic Schrödinger equation. We prove a new version of the non-squeezing theorem for Hilbert spaces. We apply the result to the discrete nonlinear Schrödinger equation.

  • Utkin, P., Lopato, A.
    Mathematical modeling of the long-time evolution of the pulsating detonation wave in the shock-attached frame
    Abstract, Slides

    Detonation is a hydrodynamic wave process of the supersonic propagation of an exothermic reaction through a substance. The detonation wave is a self-sustained shock-wave discontinuity behind the front of which a chemical reaction is continuously initiated due to heating caused by adiabatic compression. It is known from experimental and numerical studies that the propagation of the detonation wave in space is characterized by a complicated nonlinear oscillatory process. The paper is devoted to the numerical investigation of the pulsating detonation wave propagation in the shock-attached frame using special numerical algorithm with the second approximation order derived by the authors. The detailed features of very long-time detonation wave evolution behavior for four regimes – stable, weakly unstable, irregular and strongly unstable – will be presented including quantitative characteristics of the process such as Fourier analysis of the pressure signal.

  • Vabishchevich, P. N.
    Iterative computational identification of a spacewise dependent the source in a parabolic equations
    Abstract, Slides

    Coefficient inverse problems related to identifying the right-hand side of an equation with use of additional information is of interest among inverse problems for partial differential equations. When considering non-stationary problems, tasks of recovering the dependence of the right-hand side on time and spatial variables can be treated as independent. These tasks relate to a class of linear inverse problems, which sufficiently simplifies their study. This work is devoted to a finding the dependence of right-hand side of multidimensional parabolic equation on spatial variables using additional observations of the solution at the final point of time - the final overdetermination. More general problems are associated with some integral observation of the solution on time - the integral overdetermination. The first method of numerical solution of inverse problems is based on iterative solution of boundary value problem for time derivative with non-local acceleration. The second method is based on the known approach with iterative refinement of desired dependence of the right-hand side on spacial variables. Capabilities of proposed methods are illustrated by numerical examples for model two-dimensional problem of identifying the right-hand side of a parabolic equation. The standard finite-element approximation on space is used, the time discretization is based on fully implicit two-level schemes.

  • Vedenyapin, V., Negmatov, M. A. and Fimin, N. N.
    Hydrodynamics and Kinetics of Vlasov and Liouville equations
    Abstract, Slides

    Hydrodynamics and Kinetics of Vlasov and Liouville equations. 1.Hydrodinamic anzats for Liouville equation and Hamilton-Jacoby equation. Hamilton-Jacoby method in nonhamiltonian situation. 2. Hydrodynamic anzats for Vlasov equation and Godunov double divergent form of magnetohidrodynamics. 3. On Arnold-Kozlov Topology of SteadyState Solutions of Hydrodynamic and Vortex Consequences of the Vlasov Equation. 1. A.A. Власов, Статистические функции распределения, Наука, M., 1966. 2. B.B. Козлов, Общая теория вихрей, Изд-во Удмуртского гос. ун-та, Ижевск, 1998. 3. В.В. Веденяпин, Н.Н. Фимин, `Уравнение Лиувилля, гидродинамическая подстановка и уравнение Гамильтона–Якоби', Доклады РАН, 446:2 (2012), 142--144. 4. B.B. Веденяпин В.В., M.A. Негматов O топологии гидродинамических и вихревых следствий уравнения Власова и метод Гамильтона--Якоби, Доклады РАН, 449:5 (2013), 521--526. 5. V. V. Vedenyapin and M. A. Negmatov. DERIVATION AND CLASSIFICATION OF VLASOV-TYPE AND MAGNETOHYDRODYNAMICS EQUATIONS: LAGRANGE IDENTITY AND GODUNOV’S FORM. Theoretical and Mathematical Physics, 170(3): 395–406 (2012)

  • Vakulenko, S. P., Volosov, K., Volosova, N. K.
    The modulational instability in models of Rayleigh beam lying on the elastic basis
    Abstract, Slides

    The dynamic behavior of Bernoulli‐Euler, Rayleigh and Timoshenko beam models lying on an elastic foundation is considered. A comparative analysis of their dispersion curves is given. The behaviour of the Rayleigh beam dispersion curve has been found to coincide qualitatively with the behaviour of the lower branch of the Timoshenko beam dispersion curve. Accounting for the cubic nonlinearity of the elastic foundation in these models leads to the generation of higher harmonics, which do not, however, practically interact (due to strong dispersion). The stability and instability domains of quasi‐harmonic flexural waves have been found. It is shown that, in contrast to the Bernoulli‐Euler model, the Rayleigh model may be used in studying low‐frequency flexural waves. Keywords: flexural vibrations of beams, elastic foundation, wave, dispersion, modulation instability.

  • Vorontsov, K.
    Ill-posed problems of non-negative matrix factorization with applications to text analysis
    Abstract, Slides

    Non-negative matrix factorizations are widely used in science and engineering to provide compressed low-rank representations of data. In text mining applications, this approach helps to reveal a hidden topical semantic representation of large text document collections. Topic model represents the matrix of term-document frequencies by a factorization into a term-topic matrix and a topic-document matrix. Topic modeling is an ill-posed problem because of non-uniqueness of this representation. We propose to use additive combinations of regularization criteria to meet multiple linguistic and subject-oriented requirements as well as multimodal metadata of heterogeneous data collections. The additive regularization allows us to immerse many well-known models into a single framework in order to build combined models with desired properties. This paradigm has been implemented in parallel open-source framework BigARTM. Now it is used in many text mining applications. In this report, we consider some convergence issues for the regularized Expectation-Maximization algorithm. Also, we propose some strategies for regularization trajectory optimization.

  • Voroshchuk, D.
    Application of Discontinuous Galerkin Method in direct 3D seismic xploration problems

    A discontinuous Galerkin method on unstructured grids is adapted and implemented for simulation of wave response of subvertical fractured systems in carbonate rocks for numerical solution of direct problems of seismic exploration.

  • Voynov, O., Golubev, V. I., Zhdanov, M. S.
    Migration imaging in elastic media using Born approximation
    Abstract, Slides

    One of the main methods of key fossil fuels discovery and engineering surveying of geological structure of Earth's crust is seismic tomography. Migration imaging is one of its techniques, which solves the problem of locating interfaces in the medium under investigation. There is a lot of methods solving this problem in acoustic case.
    In this work we discuss a method of migration imaging for elastic quasi-homogeneous media based on Born approximation. It is compared to analogous method for acoustic media in application to simple models and is shown to have better resolution of steep interfaces.

  • Yagola, A.
    A priori and a posteriori error estimates for solutions of ill-posed problems

    In order to calculate a priori or a posteriori error estimates for solutions of an ill-posed operator equation with an injective operator we need to describe a set of approximate solutions that contains an exact solution. After that we have to calculate a diameter of this set or maximal distance from a fixed approximate solution to any element of this set. We will describe three approaches for constructing error estimates and also their practical applications. 1) Error estimates for quasisolutions on a given compact set [1, 2]. This method can be generalized for inverse problems on Banach lattices [3]. 2)A posteriori error estimates in the method of extending compacts [2, 4]. This method can be generalized for nonlinear ill-posed problems [5]. Using the Lagrange principle optimal a posteriori error estimates can be constructed [6, 7]. 3)Extra-optimal regularizing algorithms proposed by A.S. Leonov [8]. This paper was supported by the RFBR grants 14-01-00182-a and 16-01-00450. References 1. A. Yagola, V. Titarenko. Using a priori information about a solution of an ill-posed problem for constructing regularizing algorithms and their applications, Inverse Probl. in Sci. and Eng., 15, 3 – 17 (2007). 2. V. Titarenko, A. Yagola. Error estimation for ill-posed problems on piecewise convex functions and sourcewise represented sets, J. Inverse Ill-Posed Probl., 16, 625–638 (2008). 3. Y. Korolev, A. Yagola. Making use of a partial order in solving inverse problems, Inv. Probl., 29, (2013) doi:10.1088/0266-5611/29/9/095012 4. K. Yu. Dorofeev, V. N. Titarenko, and A. G. Yagola. Algorithms for constructing a posteriori errors of solutions to ill-posed problems, Comp. Math. Math. Phys., 43, 12-25 (2003). 5. K.Yu. Dorofeev, A.G. Yagola. The method of extending compacts and a posteriori error estimates for nonlinear ill-posed problems, J. Inverse Ill-Posed Probl., 12, 627-636 (2004). 6. A.V. Bayev, A.G. Yagola. Optimal recovery in problems of solving linear integral equations with a priori information, J. Inverse Ill-Posed Probl., 15, 569-586 (2007). 7. Y. Zhang, D.V. Lukyanenko, A.G. Yagola. Using Lagrange principle for solving two-dimensional integral equation with a positive kernel, Inverse Probl. in Sci. and Eng. (2015), DOI: 10.1080/17415977.2015.1077445. 8. A.S. Leonov. Extra-optimal methods for solving ill-posed problems, J. Inverse Ill-Posed Probl., 20, 637-665 (2012).

  • Yavich, N., Malovichko M., Zhdanov, M. S.
    Efficient Preconditioning in 3D Marine Electromagnetic Geophysical Modeling
    Abstract, Slides

    We discuss design and application of a contraction operator (CO) preconditioner to 3D marine controlled-source electromagnetic data modeling. Performance of this preconditioner is compared with the discrete Green’s function preconditioner. Analysis and numerical examples demonstrate that the CO preconditioner speeds up convergence of iterative solvers. This research was partially funded by RFBR, project No. 16-37-00018.

  • Zhang, S.
    Modeling and Computation of Energy Efficiency Management with Emission Permits Trading

    In this talk, we present an optimal feedback control model to deal with the problem of energy efficiency management. Especially, an emission permits trading scheme is considered in our model, in which the decision maker can trade the emission permits flexibly. We make use of optimal control theory to derive the Hamilton-Jacobi-Bellman (HJB) equation satisfied by the value function, and then propose an upwind finite difference method to solve it. The stability of this method is demonstrated and the accuracy, as well as the usefulness, are shown by the numerical examples. The optimal management strategies, which maximize the discounted stream of the net revenue, together with the value functions, are obtained. The effects of the emission permits price and other parameters in the established model on the results have been also examined. We find that the influences of emission permits price on net revenue for the economic agents with different initial quota are quite different. All results demonstrate that the emission permits trading scheme plays an important role in the energy efficiency management.

  • Zubov, V., Albu, A.
    The use of Fast Automatic Differentiation technique for solving coefficient inverse problems
    Abstract, Slides

    The work deals with the problem of determining dependence of the material thermal conductivity upon temperature. Consideration is based on the first boundary value problem for the one-dimensional non-stationary heat equation. The standard deviation of the temperature field and heat flux distribution on the left boundary upon the experimental data is chosen as an objective function. An analytical expression for the cost functional gradient is derived. An algorithm for the numerical solution of the problem based on modern fast automatic differentiation technique is proposed. Examples of solving the problem are given.