Program

Program of the conference is available here

Short program of the conference is available here

Detailed program

May 3

10:00-10:45 A.Bakushinsky

A posteriori error analysis for nonlinear inverse problems

The possibility of a posteriori error estimates for nonlinear ill-posed operator equations are considered. For these estimates we utilize  in place of original infinite model  F(x)=0 its finite dimentional approximation Ф(w)=0. We estimate the distance between z: ||Ф(z)||<=delta and  some x*: F(x*)=0

11:00-11:45 A.Yagola

Multidimensional ill-posed problems

It is very important now to develop methods of solving multidimensional ill-posed problems using regularization procedures and parallel computers.  The main purpose of the talk is to show how 2D and 3D Fredholm integral equations of the 1st kind can be effectively solved.We will consider ill-posed problems on compact sets of convex functions and functions convex along lines parallel to coordinate axes. Recovery of magnetic target parameters from magnetic sensor measurements has attracted wide interests and found many practical applications. However, difficulties present in identifying the permanent magnetization due to the complications of magnetization distributions over the ship body, and errors and noises of measurement data degrade the accuracy and quality of the parameter identification. In this paper, we use a two step sequential solutions to solve the inversion problem. In the first step, a numerical model is built and used to determine the induced magnetization of the ship. In the second step, we solve a type of continuous magnetization inversion problem by solving 2D and 3D Fredholm integral equations of the 1st kind. We use parallel computing which allows solve the inverse problem with high accuracy. Tikhonov regularization has been applied in solving the inversion problems. The proposed methods have been validated using simulation data with added noises. 2D and 3D inverse problems also could be found in tomography and electron microscopy. We will demonstrate examples of applied problems and discuss methods of numerical solving.

12:00-12:20 M. Yudytskiy

Wavelet-based methods in atmospheric tomography and adaptive optics

The problem of atmospheric tomography arises in ground-based telescope imaging with adaptive optics, where one aims to compensate in real-time for the rapidly changing optical distortions in the atmosphere. The mathematical formulation of the problem resembles limited angle tomography. The recent developments of wavelet-based reconstructions methods in atmospheric tomography are discussed. In this talk we give a short introduction to the topic, discuss the theoretical results, and present a few numerical examples.

12:30-13:30 Lunch

13:30-14:15 G. Bocharov

Mathematical models and parameter estimation for cell proliferation studies

Proliferative response of cells to antigenic stimulation underlies the functioning of the immune system. The lymphocytes differ enormously in their proliferative potential. To assess the kinetics of cell divisions with time the flow cytometry analysis of heterogeneous cell populations labeled with a fluorescence marker is broadly used. The experimental technique allows one to generate histograms of cell distribution with respect to the fluorescence intensity at various times. The data can be used to estimate the kinetic parameters of cell proliferation and death. To this end various types of mathematical models have been proposed. We review the application of mathematical models formulated with ODEs, DDEs and hyperbolic PDEs to describe dynamics of the fluorescence marker- and division number structured cell populations and to estimate their turnover parameters. Regularization techniques and information–theoretic based model selection are essential for a robust maximum likelihood estimation of the parameters.

14:30-15:15 P. Elbay

Photoacoustic Sectional Imaging

    Photoacoustic Imaging is a non-invasive hybrid imaging method with the intention of determining the absorption density inside an object. This is done by measuring the ultrasonic pressure wave which is emitted from the object after an illumination with a short laser pulse.

    Focusing the laser pulse so that only one slice of the object gets illuminated, this technique also allows for cross-sectional imaging. The aim of this talk is to discuss some setups for this photoacoustic sectional imaging and provide explicit reconstructions formulas for the absorption density from the corresponding measurements.

    Moreover, it shall be shown how additional physical parameters, such as the slightly spatially varying speed of sound in the object, can be recovered by performing multiple sectional measurements.

18:30-21:00 Dinner

May 4

9:00-9.45 V. G. Romanov

2D and 3D Inverse Problems for viscoelasticity equations

Inverse problems for integro-differential equations of viscoelasticity are considered in 2D and 3D statements. In the 2D variant, the inverse problem consists in a determination of a density, shear modulus and a space part of a kernel of an integral operator of the equation. A series of the direct Cauchy problems with zero initial data and impulse plane sources are considered and solutions to these problems are supposed to be given on a boundary of a disc for a sufficiently large finite time interval [0, T]. Corresponding data are used to solve the inverse problem, i.e. to determine the unknown variable parameters of a medium inside of the disc. It is demonstrate that the original inverse problem can be reduce to 3 more simple inverse problems that can be solved one after another. Stability estimates of solutions are given. In the 3D statement, a similar problem is considered under the assumption that a density is given, so unknown functions are elastic moduli and space parts of two kernels of integral operators of viscoelasticity equations. In this case instead of impulse plane sources a series of impulse point sources are used. The solutions of corresponding direct problems are given on the boundary of a ball for a sufficiently large finite time interval [0, T]. This information is used to find uniquely unknown functions inside the ball.

10:00-10:45 B.T. Johansson

On the alternating method for Cauchy problems and its finite element discretisation

We consider the alternating method [2] for the stable reconstruction of the solution to the Cauchy problem for the stationary heat equation in a bounded Lipschitz domain. Using results from [1], we show that the alternating method can be equivalently formulated as the minimization of a certain gap functional and we prove some properties of this functional and its minimum. It is shown that the original alternating method can be interpreted as a method for the solution of the Euler-Lagrange first order optimality equations for the gap functional. Moreover,we show how to discretize this functional and equations via the Finite Element Method (FEM). The error between the minimum of the continuous functional and the discretised one is investigated, and an estimate is given between these minima in terms of the mesh size and the error level in the data. Numerical examples are included showing that accurate reconstructions can be obtained also with a non-constant heat conductivity. References 1. Baranger, T. N. and Andrieux, S., Constitutive law gap functionals for solving the Cauchy problem for linear elliptic PDE, Appl. Math. Comput. 218 (2011), 970–1989. 2. Kozlov, V. A. and Maz’ya, V. G., On iterative procedures for solving ill-posed boundary value problems that preserve differential equations, Algebra i Analiz 1 (1989), 144–170. English transl.: Leningrad Math. J. 1 (1990), 1207–1228.

11:00-11:45 P. Kuegler

Covariance fitting for parameter estimation in biochemical reaction networks

Based on the chemical master equation for stochastic biochemical reaction networks we consider rate parameter dependent ODE systems that describe the time evolution of the mean and the covariance of the number of species molecules involved. Cost functions that compare these ODE outputs with sample data obtained from repeated observations feature a landscape in parameter space with more pronounced curvatures at the solution. In numerical tests of parameter identification problems we observe advantages both with respect to accuracy and efficiency.

12:00-12:20 Y. Korolev

Error estimations for ill-posed problems in Banach lattices

We consider ill-posed inverse problems for linear operator equations Az= u in Banach lattices. We suppose a priori that the exact solution belongs to a compact set. With this a priori information, we construct a set of approximate solutions and provide an error estimate for an approximate solution. We also consider the properties of exact lower and upper bounds of the set of approximate solutions.

12:30-13:30 Lunch

13:30-14:15 G. Kuramshina

Determination of transferable force fields for biological molecules by stable numerical methods

The importance of large biological molecules stimulates the development of special approaches for describing their physicochemical properties such as molecule geometry, vibrational frequencies, and thermodynamic functions, etc. Fast growing computational resources and numerical methods leads to the great advantage of modern methods of quantum chemistry for the solving many problems of structural chemistry in application to the large molecular systems. But there are also existed obvious severe limitations of using purely ab initio methods for the analysis of molecular systems consisting of a few hundred atoms. In cases where such systems are organized from separate smaller size units the most successful approaches for their analysis are as a rule based on the joint use of theoretical results (e.g. ab initio or DFT data obtained for these chosen unites) with some empirical approach (e.g. molecular mechanics) and in many cases result in good descriptions of investigated systems. Solving such problems requires the knowledge of molecular force field which can be obtained as within purely empirical approach based on experimental frequencies so within the joint treatment of quantum mechanical and experimental data using some force field model. This is so-called inverse vibrational problem and it is known to be ill-posed. It means that this problem may have no solution, or may have non-unique solution, or/and these solutions may be unstable. Usually, non-existence and nonuniqueness can be overcome by searching some ”generalized” solutions, the last

is left to be unstable. So for solving such problems is necessary to use the special methods - regularizing algorithms. For overcoming the ill-posedness of inverse vibrational problem we have proposed to use the quantum mechanical data as an additional information and as a stabilizer within the regularization procedure while looking for the Regularized Quantum Mechanical Force Field (RQMFF). Quantum mechanical data also serve as a priori information on molecular force field model. At the present one of the most popular force field models is based on the introducing the scaling factors. Otherwise it is also possible to use some modification of harmonic force constant matrix. While the problem of finding the force constants or scaling factors is formulated as fitting the experimental

frequencies it leads to the solving the inverse vibrational problem, which belongs to a class of nonlinear ill-posed problems. As the result we have RQMFF or regularized scaled quantum mechanical force field (RSQMFF). Within this model the new algorithm for determination of scaling factors in Cartesian coordinates has been proposed and applied to biological molecules such as primary nucleobases (cytozine, adenine, guanine, thymine, uracil) and some imidazole derivatives.

14:30-15:15 L. Nazarova

Inverse problems of geomechanics and its applications in mining

We will discuss statements of Inverse problems for geomechanical objects of different scale level. Particularly we will consider following questions:

* Evaluation of natural stress field components by seismotectonical deformations.

* Determination of focal parameters of forthcoming earthquake based on strain components variation in epicentral zone on daylight surface.

* Estimation of deformation and strength properties for filling mass by measurements of displacements at mined-out void boundaries during mining gently sloping seams.

18:30-21:00 Dinner

May 5

9:00-9.45 M. Asadzadeh

 On inverse kinetic model and FEMs for Vlasov-Poisson-Fokker-Planck system

We construct and analyze a finite element procedure for an in- verse multidimensional kinetic model for the interaction of charged Coulomb plasma particles described by the Vlasov-Poisson-Fokker-Planck system. We employ numerical approximation in the realm of the hp based finite elements, which may be viewed as a generalized adaptivity procedure. To this approach we combined mixed finite volume and finite element (FV/FE) methods for the Poisson equation in spatial domain and the streamline diffusion (SD) and discontinuous Galerkin (DG) finite element methods in time, phase-space variables for he Vlasov-Fokker-Planck equation. Here for an inverse model problem we describe appropriate regularity re- quirements on physical data, in order to obtain

a classical solution. As for the weak formulations we derive stability and coercivity estimates and prove that the convergence rates for the combined schemes are optimal due to the maximal available regularity of the exact solutions.

10:00-10:45 L. Beilina

Approximate global convergence in imaging of land mines from backscattered data

We present approximate globally convergent method in the most challenging case of the backscattered data. In this case data for the coefficient inverse problem are given only at the backscattered side of the medium which should be reconstructed. We demonstrate efficiency and robustness of the proposed technique on the numerical solution of the coefficient inverse problem in two dimensions with the time-dependent backscattered data. Goal of our tests is to reconstruct dielectrics in land mines which is the special case of interest in military applications. Our tests show that refractive indices, locations and shapes/sizes of dielectric abnormalities are accurately imaged.

11:00-11:45 E. E. Tyrtyshnikov

New generation of numerical algorithms via tensor-train representations of data

 In the 20th century tensors were used chiefly as a language or a descriptive means.  In the 21st century they get to play a new role as a base for numerical algorithms for the data in really many (as well as in few) dimensions.

The key is a new tensor decomposition called the tensor-train (TT) decomposition and basic TT algorithms appeared only in 2009:   TT rounding, TT cross interpolation, and TT   wavelet transorms.  We discuss these new findings and the related perspectives for the development of numerical analysis

12:00-12:20 N. Koshev

A posteriori error estimates for the Fredholm integral equation of the first kind

We consider an adaptive finite element method for the solution of the Fredholm integral equation of the first kind and derive a posteriori error estimates both in the Tikhonov functional and in the regularized solution of this functional. We formulate an adaptive algorithm and present experimental verification of our adaptive technique on the backscattered data measured in microtomography.

12:30-13:30 Lunch

13:30-14:15 Y. Podlipenko

Minimax approach to solving some inverse problems in electromagnetics

Optimization under incomplete data is a relatively new part of the theory of optimal processes related to the study of control systems functioning under uncertainty conditions. It is motivated by the necessity of solving applied problems connected with interpretation of the results of experiments and data processing in gas dynamics, economics, ecology and other fields of science and technology. As a rule, control systems are described by certain boundary value problems (BVPs) for ordinary or partial differential equations whose parameters contain unknown quantities, and the additional data (observations) y = Cφ + η are needed in order to determine them. Here φ is an unknown solution of the considered BVP, C is an operator that specifies the method of measuring, and η is the measurement (observations) error. Depending on assumptions regarding unknown perturbations of parameters and data, there is a variety of approaches to their optimal determination, in particular, the minimax approach. Although numerous studies have been performed in this direction, this approach has not been developed for the important case of the internal and external BVPs for Maxwell equations. We will investigate optimal reconstruction (estimation) of solutions and right-hand sides of Maxwell equations (or, more generally, estimation of values of some functionals on their solutions or right-hand sides) under incomplete data. These problems play an important role in applied electromagnetics and acoustics. Depending on a character of a priori information, stochastic or deterministic approaches are possible. Their choice is determined by the nature of parameters, which can be random or not. Moreover, the optimality of estimations depends on a criterion with respect to which a given value is evaluated. In this work, we assume that (i) the right-hand sides of Maxwell equations are unknown and belong to the given bounded subsets of the space of square integrable functions in the considered domain; and (ii) observation errors (noises) are realizations of the stochastic processes with unknown moment functions of the second order also belong to certain given subsets. Our approach is as follows. We are looking for linear with respect to observations optimal estimates of solutions and right-hand sides of Maxwell equations from the condition of minimum of maximal mean square error of estimation taken over the above subsets. Such estimates are called minimax or guaranteed estimates. We consider constructive methods for obtaining these estimates expressed in terms of solutions to special integral or integro-differential equations. It should be noted that beyond purely theoretical interest, the mentioned problems find many other valuable applications, e.g. in automatized measurement data processing systems and interpretation of electromagnetic observations.

14:30-15:15 Y. Smirnov

Permittivity reconstruction of layered dielectrics in a rectangular waveguide from the transmission coefficients at different frequencies.

Determination of electromagnetic parameters of dielectric bodies of complicated structure is an urgent problem. However, as a rule, these parameters cannot be directly measured (because of composite character of the material

and small size of samples), which leads to the necessity of applying methods of mathematical modeling and numerical solution of the corresponding forward and inverse electromagnetic problems. It is especially important to develop the solution techniques when the inverse problem for bodies of complicated shape are considered in the resonance frequency range. In this paper we develop a method of solution to the inverse problem of reconstructing (complex) permittivity of layered dielectrics in the form of diaphragms in a waveguide of rectangular cross section from the transmission coefficients measured at different frequencies. The method enables in particular obtaining solutions in a closed form in the case of one-sectional diaphragm. In the case of an n-sectional diaphragm we solve the inverse problem using numerical solution of a nonlinear equation system of n complex variables. Solvability and uniqueness of the system are studied and convergence of the method is discussed. Numerical results of calculating (complex) permittivity of the layers are presented. The case of metamaterials is also considered. The results of solution to the inverse problem can be applied in nanotechnology, optics, and design of microwave devices.