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# Department of Mathematical Physics

Head of the department: Denisov Alexander, Academician of RANS, Professor, Dr.Sc.

The Department was established and headed for a long time by Academician of RAS A.N. Tikhonov, twice honoured with the title of the Hero of the Socialist Labour, was the Lenin Prize and State Prize laureate.

The Department trains specialists who can use applied mathematics and computer science methods for scientific, technical and economic activities. The training of students focuses on the combination of fundamental mathematical education and practising in modern programming technologies.

The main areas of scientific research are the following:

- mathematical models of various processes of natural science, engineering, medicine;
- inverse problems for mathematical physics equations and their applications in geophysics, electrodynamics, optics, medicine;
- mathematical methods of image processing and visualization.

Scientific problems are tackled using mathematical methods and computer technologies. The Department cooperates with scientific associations of USA, Great Britain, France, Italy, China, Japan and other countries.

The Department supervises the following laboratories:

- the Lab of Mathematical Physics,
- the Lab of Computational Electrodynamics,
- the Lab of Heat and Mass Transfer Processes Simulations,
- the Lab of Inverse Problems,
- the Lab of Mathematical Methods of Image Processing.

### Staff members:

- Dmitriev Vladimir, Academician of RANS, Professor, Dr.Sc., Head of the Laboratory of Mathematical Physics, Deputy Dean
- Zakharov Evgeny, Corresponding Member of RANS, Professor, Dr.Sc., Vice Dean for graduate studies
- Ilyinsky Anatoly, Corresponding Member of RANS, Professor, Dr.Sc., Head of the Laboratory of Computational Electrodynamics
- Baev Andrey, Professor, Dr.Sc.
- Krylov Andrey, Professor, Dr.Sc., Head of the Laboratory of Mathematical Methods of Image Processing
- Razgulin Alexander, Professor, Dr.Sc., Vice Dean for International Relations
- Tikhonov Ivan, Professor, Dr.Sc.
- Berezin Sergey, Associated Professor, PhD, Head of the Research Laboratory of Computational Modeling Tools
- Grigoryev Yevgeny, Associated Professor, PhD, Vice Dean, Secretary of the Faculty Council
- Dmitrieva Irina, Associated Professor, PhD, Academic Secretary of the Department
- Morozova Valentina, Associated Professor, PhD
- Orlik Sergey, Associated Professor, PhD
- Razborov Aleksey, Associated Professor, PhD
- Solovieva Svetlana, Associated Professor, PhD
- Tuykina Svetlana, Associated Professor, PhD
- Fedotov Mikhail, Associated Professor, PhD, Vice Dean for academic work
- Shcheglov Aleksey, Associated Professor, PhD
- Golovina Svetlana, Senior Lecturer, PhD
- Nikitina Ekaterina, Research Fellow, PhD
- Pavelyeva Elena, Assistant Professor
- Romanenko Tatiana, Assistant Professor
- Gavrilov Sergey, Researcher, PhD
- Pavelchak Ivan, Mathematician
- Karysheva Lyudmila, Engineer

### Regular courses:

- Differential equations by Prof. Denisov, Prof. Dmitriev, Prof. Razgulin, 64 lecture hours and 64 seminar hours, 3rd and 4th semesters.
- Equations of mathematical physics, 64 lecture hours and 64 seminar hours, 5th and 6th semesters.
- Equations of mathematical physics by Prof. Denisov and Prof. Zakharov, 32 lecture hours and 32 seminar hours, 5th semester.
- Integral equations by Assoc. Prof. Orlik, 32 lecture hours, 5th semester.
- Mathematical models of natural science by Prof. Ilyinsky, 64 lecture hours, 7th and 8th semesters.
- Mathematical methods of image processing by Prof. Krylov, 32 lecture hours, 6th semester.
- Mathematical models of medicine and biology by Prof. Zakharov, 28 lecture hours.
- Inverse problems by Prof. Baev, 32 lecture hours, 7th semester.
- Network technologies by Assoc. Prof. Tuykina, 32 lecture hours, 6th semester.
- Object-oriented programming: C++ language by Dr. Berezina, 32 lecture hours, 5th semester.
- Object-oriented programming: user interface development by Dr. Berezina, 32 lecture hours, 6th semester.
- Object-oriented programming: NET technology by Assoc. Prof. S.Berezin, 32 lecture hours, 7th semester.
- Object-oriented programming: WEB technologies by Assoc. Prof. S.Berezin, 32 lecture hours, 8th semester.

### Special courses:

- Inverse problems of wave propagation and methods of theirs solutions by Prof. Baev, 32 lecture hours, 8th semester.
- Iterative methods of solving inverse problems by Assoc. Prof. Shcheglov, 32 lecture hours, 7th semester.
- Integral transformations in image processing by Prof. Krylov, 32 lecture hours, 5th semester. Computational methods of image processing by Prof. Krylov, 32 lecture hours, 8th semester.
- Inverse problems of mathematical physics by Prof. Dmitriev, 32 lecture hours, 8th semester.
- Integral equations method of solving wave diffraction problems by Assoc. Prof. Zakharov, 32 lecture hours, 7th semester.
- Variational methods of mathematical physics by Prof. Ilyinsky, 32 lecture hours, 8th semester.
- Nonlinear models of optical synergetics by Prof. Razgulin and Assoc. Prof. Fedotov, 32 lecture hours, 5th semester.
- Pattern formation models in optical synergetics by Prof. Razgulin, 32 lecture hours, 6th semester.
- Projection-difference method of controlled Fourier filtering and other problems of mathematical physics by Prof. Razgulin, 64 lecture hours, 7th and 8th semesters.
- Fractioned Sobolev spaces by Prof. Razgulin, 64 lecture hours, 7th and 8th semesters.

### Special scientific seminars:

- Inverse and ill-posed problems by Prof. Denisov, Prof. Baev, Assoc. Prof. Shcheglov.
- Image processing and computer simulations by Prof. Zakharov, Prof. Krylov.
- Mathematical models of electromagnetic processes by Prof. Dmitriev, Prof. Zakharov, Prof. Ilyinsky.
- Supercomputer simulations by Prof. Paskonov, Prof. Lebedev, Assoc. Prof. S.Berezin.
- Mathematical modeling of nonlinear phenomena by Prof. Razgulin and Assoc. Prof. Fedotov.

## Main Scientific Directions:

### Direct and inverse problems in medicine

(Professors A.M. Denisov and E.V. Zakharov)

The research group studies a heart activity in cooperation with Bakulev Scientific centre of cardiovascular surgery (Moscow, Russia). The inverse electrocardiography problem in the form of potentials consists in reconstructing the potential on the outer surface of the heart from the potential measured on the chest surface (see Fig.1). The study of this problem is motivated by the clinical implementation of new treatment methods for cardiac arrhythmia. Algorithms for solving the inverse electrocardiography problem for model geometry of the torso and the heart have been described by many authors. The research group considers the inverse electrocardiography problem for realistic geometry of the torso and the heart with admission of the internal inhomogeneity of the chest. A numerical method is proposed for solving an inverse electrocardiography problem for a medium with a piecewise constant electrical conductivity. The method is based on the method of boundary integral equations and the Tikhonov regularization. Refs:

- S.V. Gavrilov and A.M. Denisov, Numerical method for determining the inhomogeneous boundary in a boundary value problem for the Laplace equation in a piecewise homogeneous medium // Comput. Math. Math. Phys., vol. 51, pp. 1476–1489, 2011 (Pleiades Publishing).
- A.M. Denisov, E. V. Zakharov, A. V. Kalinin, and V. V. Kalinin, Numerical solution of an inverse electrocardiography problem for a medium with piecewise constant electrical conductivity // Comput. Math. Math. Phys., vol. 50, pp. 1172–1177, 2010 (Pleiades Publishing).
- A.M. Denisov, E. V. Zakharov, A. V. Kalinin, and V. V. Kalinin, Numerical solution of the inverse electrocardiography problem with the use of the Tikhonov regularization method // Moscow Univ. Comput. Math. Cybern., vol. 32, pp. 61–68. 2008.
- A.M. Denisov, E. V. Zakharov, A. V. Kalinin, and V. V. Kalinin, Numerical methods for some inverse problems of heart electrophysiology // Differ. Equations, vol. 45, pp. 1034–1043, 2009.
- E.V. Zakharov and A.V. Kalinin, Method of boundary integral equations as applied to the numerical solution of the three-dimensional Dirichlet problem for the Laplace equation in a piecewise homogeneous medium // Comput. Math. Math. Phys., vol. 49, pp. 1141–1150, 2009 (Pleiades Publishing).
- E.V. Zakharov and A.V. Kalinin, Algorithms and numerical analysis of DC fields in a piecewise-homogeneous medium by the boundary integral equation method // Comput. Math. and Modeling., vol. 20, no. 3, pp. 247-257, 2009.

### Direct and inverse problems of geophysics and electrodynamics

(Professors V.I. Dmitriev, A.S. Ilyinsky, A.M. Denisov, and A.V. Baev)

Methods of solving direct and inverse problems of geophysics and electrodynamics are studied. The iterative method of finding the conductive heterogeneous field under the ground which based on the known magnetic field on the earth surface is worked out and implemented to solve applied problems of geological prospecting. To solve 3D problems of geophysical prospecting minerals the method of integral equations and its software support are worked out (see Fig.2). Refs:

- M.S.Kruglyakov, Estimating the influence zone of a vertical magnetic dipole in aerial prospecting // Comput. Math. and Modeling., vol. 21, no. 1, pp. 30-40, 2010.
- V.I.Dmitriev, E.V.Zakharov, and E.V.Nikitina, Tensor Green’s functions for axisymmetrical models of magnetotelluric marine sounding of locally nonhomogeneous layered media // Comput. Math. and Modeling., vol. 20, no. 3, pp. 219-230, 2009.
- V.I.Dmitriev and M.N.Berdichevsky, Models and Methods of Magnetotellurics. Berlin: 563 p., Springer, 2008.
- A.S. Ilyinsky and T.N. Galishnikova, The integral equation method in problems of diffraction by a finite impedance section of a medium interface // Mosc. Univ. Comput. Math. Cybern., no. 4, pp. 187-193, 2008.
- V.I.Dmitriev and E.V.Zakharov, Three-dimensional models of marine magnetotelluric sounding of nonhomogeneous media // Comput. Math. and Modeling., vol. 19, no. 4, pp. 359-364, 2008.
- V.I.Dmitriev and M.S.Kruglyakov, Fast computation of the field of a vertical magnetic dipole above a layered medium // Comput. Math. and Modeling., vol. 19, no. 3, pp. 263-270, 2008.
- M.C.Zhdanov, V.I.Dmitriev, A.V.Gribenko and V.Burtman, Anisotropy of induced polarization in the context of the generalized effective-medium theory // Proc. of Annual Meeting SEG-2008. Las Vegas, USA: Society of Exploration Geophysicists, pp. 20-25, 2008.
- E.V.Zakharov and E.V.Nikitina, Axisymmetrical marine magnetotelluric models for locally nonhomogeneous layered media // Comput. Math. and Modeling., vol. 19, no. 4, pp. 365-374, 2008.
- M.C.Zhdanov, V.I.Dmitriev and A.V.Gribenko, Integral electric current method in 3D electromagnetic modeling for large conductivity contrast // IEEE Trans. on Geoscience and Remote Sensing., vol. 45, no. 5. pp. 1282-1290, 2007.
- A.S.Ilyinsky, Yu.N.Vasilenko, and Yu.Ya.Kharlanov, Cross-shaped and four-ridge waveguide-horn radiators: internal and external characteristics // J. Commun. Technology and Electronics., vol. 51, no. 1, p. 2, 2006.
- A.S.Ilyinsky, Simulation of corrugated horns and exciters for circular corrugated waveguides // J. Commun. Technology and Electronics., vol. 51, no. 9, p. 1036, 2006.

### Diffusive filtration problems

(Professors A.M. Denisov and A.S. Krylov)

Methods of diffusive filtration and of increasing the image sharpness in computer image processing are worked out on the base of solving direct and inverse problems for the 2D diffusion equation containing a special-type coefficient and source term (see Fig.3). Refs:

- A.M.Denisov, A.S.Krylov and V.Yu.Medvedeva, Edge detection using the reaction-diffusion equation with a variable diffusion coefficient // “Graphicon’2010”, 20th Intern. Conf. on Computer Graphics and Vision, pp. 129-132, 2010.
- G.V.Borisenko, A.M.Denisov, and A.S.Krylov, Image diffusion methods with an integral-based diffusion coefficient // Proc. of 17th Intern. Conf. on Comp. Graphics and Vision "GraphiCon'2007", pp. 182-185, 2007.
- A.M.Denisov, A.S.Krylov, and V.N.Tsibanov, Second-order Tikhonov regularization method for image filtering // Proc. of 16th Intern. Conf. on Comp. Graphics and Appl. "GraphiCon'2006", pp. 54-58, 2006.

### Direct and inverse control problems in nonlinear optics

(Prof. A.V. Razgulin and Ass. Prof. M.V. Fedotov)

The distortion suppression and construction of light waves which have prescribed properties form an important problem for image processing and data transmitting through optical channels. Novel approaches to solve these problems utilize both nonlinear optical systems with controlling feedback loop and efficient optimal control methods for spatio-temporal coordinate transforms and Fourier filtering (Fig. 4). Corresponding models are governed by parabolic functional-differential equations (FDE). Methods of bifurcation theory and attractors are elaborated for a qualitative analysis of nonlinear dynamics for these FDE. In this direction the research succeeded in an analytical description of wide range of pattern formation phenomena: the Turing patterns, 1-D and 2-D rotating multi-petal waves, optical spirals, etc. Modern technique is elaborated for constructing and analyzing projection difference schemes (PDS) for finite-dimensional approximations of FDE. It allows to obtain new convergence rates of PDS for non-smooth solutions of FDE and better approximations of optimal control problems as well. An adjacent direction of research studies stable methods for solving the dual zone observability and controllability problems for linear and nonlinear Schrödinger type equations. Refs:

- A.V. Razgulin, Projection–difference method for controlled Fourier filtering // Computat. Mathem. and Modeling, Springer, NY, vol. 23, no. 1, pp. 56-71, 2012.
- V.A. Trofimov, M.V. Fedotov, et al., Performance of multicore and multiprocessor computers for some 3D problems of nonlinear optics and gaseous dynamics // Workshop on large-scale modeling, May 1-6, 2012, Sunne, Sweden, pp.12-13, 2012.
- A.V. Razgulin and T.E. Romanenko, An approach to the description of rotating waves in parabolic functional-differential equations with rotation of spatial arguments and time delay // The Sixth International Conference on Differential and Functional Differential Equations, (DFDE-2011), Moscow, Russia, pp. 56-57, 2011.
- S.Yu. Nikitin, A.E. Lugovtsov, A.V. Priezzhev, and V.D. Ustinov // Quantum Electronics, vol. 41, no. 9, pp. 843-846, 2011.
- V.A. Grebennikov and A.V. Razgulin, Weighted estimate for the convergence rate of a projection difference scheme for a quasilinear parabolic equation // Comput. Math. and Math. Physics, Pleiades Publishing, Ltd., vol. 51, no. 7, pp. 1208–1221, 2011.
- F.P. Vasil’ev, M.A. Kurzhanski, M.M. Potapov, and A.V. Razgulin, Approximate solving of dual control and observation problems. 384 p., M.: MAX-Press, 2010, (In Russian).
- V.A. Trofimov, M.V. Fedotov, et al., Computer simulation and observation of anomalous light emission from nonlinear photonic crystal with various geometry of its elements // Laser Physics., vol. 20, no. 5, pp. 1137-1143, 2010.
- A.V. Razgulin, Nonlinear models of optical synergetics, 204 p., M.: MAX-Press, 2008 (In Russian).

### Supercomputer simulation

(Prof. V.M. Paskonov, Ass. Professors B.I. Berezin and S.B. Berezin)

The Department conducts research on methods and algorithms for solving computation flow dynamics problems based on the finite-difference approximation of the Navier-Stokes equations on high performance computer systems of various architectures (Fig. 5). In particular, computational algorithms for gas and fluid flows in 3D areas of complex shapes are developed for NVIDIA Tesla graphics processors. A source code of the GPU-based solver is published in Internet for a wide community use. Further application of this solver includes simulation of sea currents. Supercomputers of traditional architectures don’t stay apart: a numerical algorithm for simulation of two fluid’s parting that makes use of the MPI application program interface for computational nodes coordination is implemented on IBM BlueGene/P. Refs:

- V.M. Paskonov and A.V. Morozov, Supercomputer simulation of viscous noncompressible flow past bodies of complex shape // Computat. Mathem. and Modeling, January 2011, vol. 22, Issue 1, pp. 15-29, 2011.
- V.M. Paskonov, S.B. Berezin, and E.S. Korukhova, A dynamic visualization system for multiprocessor computers with the common memory and its application for numerical modeling of turbulent flows of viscous fluids // Moscow University Computat. Mathem. and Cybernetics, vol. 31, Issue 4, pp. 133-142, 2007.
- S. Berezin, V. Paskonov, and D. Voitsekhovky, Implementing a unified access to scientific data from .NET platform // .NET Technologies 2006. Full Papers Proceedings. ISBN 80-86943-10-0, pp. 63-70, 2006.
- S.B. Berezin and V.M. Paskonov. Nonclassical solutions of the classical problem of the viscous incompressible flow in a plane channel // Computat. Mathem. and Modeling, vol. 16, no. 3, pp. 199-213, 2005.

### Scientific data storage, manipulation and visualization

(Ass. Prof. S.B. Berezin)

Attention to the Big Data problem is constantly increasing during the last few years. Data sets generated by computational experiments and observations become extremely large and their processing using common tools becomes awkward. Multidimensional arrays are the main data structures for many problems of computational modeling and image processing. Efficient methods and algorithms for multidimensional arrays storage, manipulation and visualization are being researched and developed in the close collaboration with the Microsoft Research team. In particular, a highly scalable storage for multidimensional arrays of unlimited size is designed and implemented for the Windows Azure cloud environment. This storage is a central part of the FetchClimate computational service that provides aggregated information on climate parameters. Such information is required for building a computational model of global biological systems. Another good example of large data visualization tool developed in collaboration with the Microsoft Research is the Chronozoom project: a cloud-based application for the interactive visualization of the Big History – a large scale multidisciplinary database of historic events from the Big Bang up to nowadays. Refs:

- S.B. Berezin, D.V. Voitsekhovsky, et al., Videowalls for multiresolution visualization of natural environment // Scientific Visualization, 2009 Quarter 4, vol. 1, no. 1, pp. 100–107, 2009.
- M. Zhizhin, E. Kihn, V. Lyutsarev, S. Berezin, et al., Environmental scenario search and visualization // Proc. of the 15th ACM Intern. Symposium on Advances in Geographic Information Systems. New-York: ACM, pp. 117-126, 2007.
- Chronozoomproject.org

### Recent publications:

• 2013

- Baev A.V. Mathematical simulation of acoustic wave refraction near a caustic // Comput. Mathematics and Math. Phys. 2013. 53. N 7. P. 947-961.
- Berezin B.I., Blagodatskikh D.V. Sufficient conditions for the stability of a CABARET approximation of multidimensional convection-diffusion equations on orthogonal computational grids // Moscow University Comput. Mathematics and Cybern. 2013. 37. N 4. P. 155-161.
- Churbanov D.V. Numerical methods for reconstruction of dipole source parameters and examples of nonuniqueness in the determination of dipole coordinates // Computational Mathematics and Modeling. 2013. 24. N 4. P. 517-521.
- Churbanov D.V. Uniqueness of finding the coefficient of the derivative in a first order nonlinear equation // Moscow University Comput. Mathematics and Cybern.. 2013. 37. N 1. P. 8-13.
- Churbanov D.V., Shcheglov A. Yu. An iterative method for solving an inverse problem for a first-order nonlinear partial differential equation with estimates of guaranteed accuracy and the number of steps // Comput. Mathematics and Math. Phys. 2013. 53. N 2. P. 215-220.
- Denisov A.M. Asymptotic expansions of solutions to inverse problems for a hyperbolic equation with a small parameter multiplying the highest derivative // Comput. Mathematics and Math. Phys. 2013. 53. N 5. P. 580-587.
- Denisov A.M. Inverse problem for a hyperbolic equation with nonlocal boundary condition containing a delay argument // Proceedings of the Steklov Institute of Mathematics. 2013. 280. N 1. P. 80-87.
- Denisov A.M., Shirkova E.Yu. Inverse problem for a quasilinear hyperbolic equation with a nonlocal boundary condition containing a delay argument // Differential Equations. 2013. 49. N 9. P. 1053-1061.
- Denisov A.M., Solov'eva S.I. Inverse problem for the diffusion equation in the case of spherical symmetry // Comput. Mathematics and Math. Phys. 2013. 53. N 11. P. 1607-1613.
- Gavrilov S.V. Numerical solution of the electrical impedance tomography problem with piecewise-constant conductivity and one measurement on the boundary // Computational Mathematics and Modeling. 2013. 24. N 4. P. 498-504.
- Iroshnikov N.G., Larichev A.V., Potyagalova A.A., Razgulin A.V. Tikhonov-regularized bispectral variational method for optical signal reconstruction // Computational Mathematics and Modeling. 2013. 24. N 4. P. 505-516.
- Pavel'chak I.A., Tuikina S.R. Numerical solution of an inverse problem for the modified Aliev-Panfilov model // Computational Mathematics and Modeling. 2013. 24. N 1. P. 14-21.
- Razgulin A.V., Romanenko T. E. Rotating waves in parabolic functional differential equations with rotation of spatial argument and time delay // Comput. Mathematics and Math. Phys. 2013. 53. N 11. P. 1626-1643.
- Zakharov E.V., Kalinin A.V. Numerical solution of the three-dimensional Dirichlet problem for inhomogeneous media by the method of integral equations // Differential Equations. 2013. 49. N 9. P. 1160-1167.

• 2012

- Denisov A.M., Zakharov E.V, Kalinin A.V., Kalinin V.V. Numerical method for solving an inverse electrocardiography problem for a quasi stationary case // J. Inverse and Ill-Posed Problems. 2012. 20. N 4. P. 501-512.
- Razgulin A.V. Projection-difference method for controlled Fourier filtering // Computational Mathematics and Modeling. 2012. 23. N 1. P. 56-71.