List of available projects for Master's Theses
If You are interested to do Master's thesis in the field of applied mathematics or applied mathematics combined with computer graphics, choose one of the available projects and sent your application by e-mail to:
List of available Master's projects in applied mathematics:
- Finite element method in quantitative elastic imaging
- Adaptive finite element method in quantitative imaging technique
- Optimization technique for approximate cloaking
- Adaptive finite element method for solution of elastodynamics system with anisotropic coefficients
- Domain decomposition finite element/finite difference method for elastodynamics system with anisotropic coefficients
List of available Master's projects in computer graphics
Open source applications developed in all these projects are thought to be used in the course on the Master's program ``Numerical Linear Algebra''.
- Real-time visualization of acoustic wave equation solver implemented in WavES.
Aim of this project is developing (designing and programming) of an open source application to load and visualize simulated solution of the time-dependent acoustic wave equation in isotropic media in two and three dimensions using OpenGL. Computed solutions are obtained by C++ software package WavES
using domain decomposition finite element/finite difference method.
- Real-time visualization of Maxwell's system solver implemented in WavES.
Aim of this project is developing (designing and programming) of an open source application to load and visualize simulated solutions of the time-dependent Maxwell's system to visualize behavior of the electric field in two and three dimensions using OpenGL. Computed solutions are obtained by C++ software package WavES
using domain decomposition finite element/finite difference method.
- Real-time visualization of elastodynamics system solver implemented in WavES.
Aim of this project is developing (designing and programming) of an open source application to load and visualize simulated solutions of the time-dependent elastodynamics system with isotropic coefficients in two and three dimensions using OpenGL. Computed solutions are obtained by C++ software package
using domain decomposition finite element/finite difference method.
List of available Master's theses in numerical linear algebra
- Theoretical and numerical comparison of different algorithms for the solution of symmetric eigenproblems.
Algorithms for the solution of eigenproblems arise in many different fields of science like computational fluid dynamics, solid mechanics, electrical networks, signal analysis, and optimisation. In this project we will study numerical methods for the solution of eigenvalue problems which are based on different transformation techniques for symmetric matrices. We are going to study following algorithms for the symmetric eigenproblem: tridiagonal QR iteration. Rayleigh quotient iteration, Divide-and-conquer, bisection and inverse iteration, Jacobi method.
We will discover convergence for all these algorithms and compare their performance with respect to applicability, reliability, accuracy, and efficiency. Programs
written in Matlab will demonstrate performance for every algorithm on the solution of practical problems.
- Theoretical and numerical investigation of the solution of Linear Least Squares Problems
In this project we will study three different methods for the solution of linear least square problems: the method of normal equations, the QR decomposition and the SVD decomposition. Such problems arise in curve fitting, statistical modeling with noisy data as well as in image compression. We will study why minimization is possible and discover the solution algorithms for all three cases. Programs written in Matlab will demonstrate performance for every algorithm on the solution of practical problems.
Scientific results in 2003-2015
My main research interests are in the solution of Coefficient Inverse Problems (CIPs) in 2D and 3D using domain decomposition Adaptive Finite Element/Finite Difference Methods and Approximate Globally Convergent Method. Domain decomposition Adaptive Finite Element/Finite Difference Method was developed in my PhD Thesis in 2003 with applications to the solution of time-dependent acoustic and elastic wave equations. Later in 2005 and 2011 I elaborated similar method for the solution of time-dependen Maxwell equations in 2D and 3D. Approximate globally convergent method was theoretically developed (approximate globally convergence theorem was proved) and numerically verified at the first time in 2008 (together with Professor M.V.Klibanov, University of North Carolina at Charlotte, USA). Results of my recent research on Adaptivity and Approximate Global Convergence are published in a book by Springer ( written jointly with M.V.Klibanov).
Applications of the above named methods are huge, we will mention some of them where I have direct experience: in medicine (detection of breast cancer or determining of elastic properties of bones), geological profiling (oil prospecting), military applications and airport security (determining of dielectric constants of explosives), reconstruction of blurred images in nanomaterials using electron microscopy, detection of defects in photonic crystals.
When we solve CIPs we face two challenges: nonlinearity and ill-posedness of CIPs. Since in many applications (military and airport security) we are working with a single measurement signal (data are generated by either a single location of the point source or by a single direction of the incident plane wave) we have third complicating factor - minimal amount of available information. A very important feature of Approximate Globally Convergent Method is that this method does not requires any knowledge of medium inside the domain of interest nor of any point in a small neighborhood of the exact solution. The main feature of an Adaptive Finite Element Method is that it takes a solution obtained in a globally convergent algorithm and then refines it. To do that we minimize corresponding Tikhonov functional on a sequence of locally adaptivelly refined meshes and improve the resulting solution.
In 2009-2012 my scientific work was concentrated on the combination of the approximately globally convergent numerical method for the solution of the hyperbolic coefficient inverse problem (CIP) with adaptive finite element method, or simply adaptivity technique. Adaptive finite element method for the solution of the hyperbolic coefficient inverse problem was developed in my PhD thesis together with follow-up publications , , , , , . Approximate global convergence, or simply globally convergence, means that there is a rigorous guarantee that the method provides a good approximation to the exact solution regardless on a availability of a good first guess for this solution. This method was developed together with prof. M.V.Klibanov from University of North Carolina at Charlotte, USA, with the first publication .
In 2009-2011 two-stage numerical procedure was developed (together with prof. M.V.Klibanov) where at the first stage the globally convergent method gives good first approximation to the solution of CIP, and on the second stage this approximation is used as a starting point for the locally convergent method. We use adaptive finite element method developed in my PhD thesis as the second stage to refine the solution obtained on the first stage.
In 2010 I applied the adaptive finite element method directly inside the globally convergent method. A posteriori error estimates for the nonlinear integral-differential equation was proven and corresponding adaptive globally convergent algorithm was tested numerically  (together with M.Asadzadeh). A posteriori error estimates for the Tikhonov functional and framework of the functional analysis for the adaptivity technique was developed in  , (together with prof.Klibanov).
Globally convergent method as also two-stage numerical procedure where verified in a picosecond scale experiment on blind experimental data, references , . Paper  became a featured article in J.Inverse Problems in 2010. In 2011 the globally convergent method was verified on the blind experimental data in the field collected by the Forward Looking Radar of the US Army Research Laboratory. A posteriori error analysis has revealed that computed and tabulated values of dielectric constants are in a good agreement, see preprint which is accepted in IEEE Transactions in Geoscience and Remote Sensing.
In 2010 in  the local strong convexity of the general Tikhonov functional was proven in the presence of the solution of the globally convergent method. This proof requires existence only the first Frechet derivative of this functional. Relaxation property for the adaptive mesh refinements for the general ill-posed problem was also developed in  (together with M.Klibanov and M.Kokurin). As result of the above described activities our research was published in different refereed journals, see references  - ,  - .
I also continued work on new numerical methods for the solution of the time-dependent Maxwell equations. New hybrid FEM/FDM method for the numerical solution of Maxwell's system was developed in . This method was applied to the solution of an 3D coefficient inverse problem with back-scattered data in  at the first time. I derived Energy estimates and numerical verified the stabilized Domain Decomposition Finite Element/Finite Difference approach for the Maxwell's system in time domain in 2D and 3D. Results of this research are presented in the preprint and in the paper "Energy estimates and numerical verification of the stabilized Domain Decomposition Finite Element/Finite Difference approach for time-dependent Maxwell's system", Cent. Eur. J. Math., 2013, 11(4), 702-733
DOI: 10.2478/s11533-013-0202-3, 2013.
I am also three-years (2010-2013) grant holder and PI for the Project ``Adaptive finite element methods for solutions of inverse problems'' supported by the Swedish Institute, Visby Program. This is the collaborative project between Chalmers University of Technology an GU, and leading Universities in Russia (Moscow Lomonosov State University; International Institute of Earthquake Prediction Theory and Mathematical Geophysics, Moscow; Institute of Computational Mathematics and Mathematical Geophysics of Siberian Branch of the Russian Academy of Sciences, Novosibirsk; Institute for System Analysis of The Russian Academy of Science, Moscow; Penza State University of Architecture and Building, Saratov State University, Siberian State University of Telecommunications and Information Sciences, Novosibirsk, Russia)). Project includes two-sided scientific exchange, PhD student supervision and yearly workshop organization. Main idea of the project is development of new mathematical idea - adaptivity technique - to the solution of CIP in imaging using electromagnetic waves as well as in signal reconstruction in scanning electron tomography. Under my co-supervision participant of the project N.Koshev defended his PhD thesis in October 2012 at Moscow State University. Together with N.Koshev we developed and Adaptive Finite Element Method for solution of Fredholm Integral Equation of the First Kind and verified it on experimental data in signal reconstruction in scanning electron microscopy. One publication on this method is accepted for publication in Springer Conference Proceedings in Mathematics, 2012, and one more paper "An adaptive finite element method for Fredholm integral equations of the first kind and its verification on experimental data" is published in the Topical Issue ”Numerical Methods for Large Scale Scientific Computing” of CEJM, 11(8), 1489-1509, 2013.
I am one of co-workers on the Project "Globally Convergent Numerical Methods for Inverse Problems of Imaging of Buried Targets" supported by the USA Army Research Laboratory grant W911NF-11-1-0399, 2011-2014. This is a collaborative project with Prof. Michael V. Klibanov, University of North Carolina at Charlotte, USA, and Prof. Michael Fiddy, Optical Center of the University of North Carolina at Charlotte, USA. In this project I continue my work on the topic of approximately globally convergent numerical methods and adaptivity for Coefficient Inverse Problems. Both analytical and numerical issues will be studied. The focus will be on extension of the current Beilina-Klibanov technique to Coefficient Inverse Problems for the time dependent Maxwell equations.
I am grant holder and the leader of the project "Global convergence and adaptivity for coefficient inverse problems for Maxwell equations" supported by the Swedish Research Council, VR, Sweden, 2012-2015. Project includes development of the new mathematical idea - adaptive finite element method- to the solution of coefficient inverse problems with applications in subsurface imaging and dielectrics reconstruction using electromagnetic waves. One PhD student at GU is hired now and works under my supervision to achieve goals of this project. We have close collaboration with Optical Center of the University of North Carolina at Charlotte, USA. We are planning to test our new reconstruction methods on experimental data provided by this center.
In December 2011 I started the new scientific computing project WavES (Wave Equations Solutions) (together with V.Timonov, Siberian State University of Telecommunications and Information Sciences and from Novosibirsk State University , Novosibirsk, Russia). WavES supported by the Swedish Institute and Swedish Research Council. Project WavES is a combined theoretical and practical tool for the numerical solution of different types of time-dependent Wave Equations (acoustic, elastic and electromagnetic). Theoretical tool consists of published books, papers, courses and presentations where new efficient numerical methods and strategies will be presented for the solution of time-dependent wave equations. Practical tool is represented by the efficient C++ program library WavES with using PETSc in parallell infrastructure for the computational solution of time-dependent wave equations (acoustic, elastic and electromagnetic) using three different methods: Finite Element Method (FEM), Finite Difference Method (FDM) and domain decomposition Hybrid FEM/FDM method. Program Library WavES is already used in all my publications and in publications of my co-workers at UNCC in USA. Particularly, WavES with implemented Approximate Globally Convergent Algorithm is currently used by Optical Center of the University of North Carolina at Charlotte, USA for qualitative reconstruction of dielectrics in imaging of explosives. More precisely, WavES is used for the numerical testing of a globally convergent method in producing of prototype of working equipment which will use this method in reality. These tests are conducted in collaboration with ARO (Army Research Laboratory), USA.