Ugur G. Abdulla

Educational Background

• Dr. rer. nat. habil., Saxon State Ministry of Science and Fine Arts, Dresden, Germany 2003
• Doctor of Physical & Mathematical Sciences, Baku State University, Azerbaijan 1994
• Ph.D., Soviet Union Academy of Sciences, USSR 1988

Professional Experience

• Professor, Florida Institute of Technology (2004 – present)
• Head of the Department of Mathematical Sciences (2013-2019)
• Visiting Professor, Massachusetts Institute of Technology (Fall 2012)
• Humboldt Visiting Professor, University of Bonn, Germany (Summer 2008)
• Professor, University of Leipzig, Germany (2002-2004)
• Max-Planck Research Fellow, Max-Planck Institute, Leipzig, Germany (2000-2002)
• Humboldt Research Fellow, University of Paderborn, Germany (1998-2000)
• Royal Society Research Fellow, University of Nottingham, United Kingdom (1996-1998)

Selected Publications

• U. G. Abdulla, Removability of the Logarithmic Singularity for the Elliptic PDEs with Measurable Coefficients and its Consequences, Calculus of Variations and Partial Differential Equations, 57, (6), (2018), 57-157.
• U. G. Abdulla, First Boundary Value Problem for the Diffusion Equation. I. Iterated Logarithm Test for the Boundary Regularity and Solvability, SIAM J. Math. Anal., 34(6), (2003), 1422–1434.
• U. G. Abdulla, Well-posedness of the Dirichlet Problem for the Nonlinear Diffusion Equation in Non-smooth Domains, Trans. Amer. Math. Soc., 357(1), (2005), 247–265.
• U. G. Abdulla, On the Dirichlet problem for the nonlinear diffusion equation in non-smooth domains, J. Math. Anal. Appl., 246, 2, 2001, 384-403.
• U. G. Abdulla, Wiener’s Criterion at ∞ for the Heat Equation, Advances in Differential Equations, 13(5-6), (2008), 457-488.
• U. G. Abdulla, Wiener’s Criterion for the Unique Solvability of the Dirichlet Problem in Arbitrary Open Sets with Non-Compact Boundaries, Nonlinear Analysis, 67(2), (2007), 563-578.
• U.G. Abdulla, Regularity of ∞ for Elliptic Equations with Measurable Coefficients and Its Consequences, Discrete and Continuous Dynamical Systems - Series A (DCDS-A), 32, 10(2012), 3379-3397.
• U. G. Abdulla, Reaction-diffusion in Irregular Domains, J. Differential Equations, 164(2), (2000), 321-354
• U. G. Abdulla and J. King, Interface Development and Local Solutions to Reaction-Diffusion Equations, SIAM J. Math. Anal., 32(2), (2000), 235-260.
• U. G. Abdulla, Exact local estimates for the supports of solutions in problems for nonlinear parabolic equations, Sbornik: Mathematics, Russian Academy of Sciences, 186(8), (1995), 1085-1106.

Recent Publications (2018-2020)

• Abdulla, Ugur G.; Removability of the logarithmic singularity for the elliptic PDEs with measurable coefficients and its consequences. Calculus of Variations and Partial Differential Equations 57 (2018), no. 6, 57:157.
• Abdulla, Ugur G.; Poggi, Bruno; Optimal Stefan Problem. Calculus of Variations and Partial Differential Equations 59, 61(2020). 
• Abdulla, Ugur G.; Cosgrove, Evan; Optimal Control of Multiphase Free Boundary Problems for Nonlinear Parabolic Equations, Applied Mathematics and Optimization (2020).
• Abdulla, Ugur G.; Poggi, Bruno; Optimal control of the multiphase Stefan problem. Applied Mathematics and Optimization. 80 (2019), no. 2, 479-513. 
• Abdulla, Ugur G.; Abuweden, Amna; Interface development for the nonlinear degenerate multidimensional reaction-diffusion equations. Nonlinear Differential Equations and Applications NoDEA, February 2020, 27:3.
• Abdulla, Ugur G.; Goldfarb, Jonathan; Hagverdiyev, Ali; Optimal Control of Coefficients in Parabolic Free Boundary Problems Modeling Laser Ablation, Journal of Computational and Applied Mathematics (2020). 
• Abdulla, Ugur G.; Bukshtynov, Vladislav; Hagverdiyev, Ali; Gradient method in Hilbert–Besov spaces for the optimal control of parabolic free boundary problems. Journal of Computational and Applied Mathematics, 346 (2019), 84–109.
• Abdulla, Ugur G.; Poteau, Roby; Identification of parameters in systems biology. Mathematical Biosciences, 305 (2018), 133–145. 
• Abdulla, Ugur G.; Du, Jian; Prinkey, Adam; Ondracek, Chloe; Parimoo, Suneil Evolution of interfaces for the nonlinear double degenerate parabolic equation of turbulent filtration with absorption. Mathematics and Computers in Simulation, 153 (2018), 59–82.
• Abdulla, Ugur G.; Jeli, Roqia Evolution of interfaces for the non-linear parabolic p -Laplacian type reaction-diffusion equations. II. Fast diffusion vs. strong absorption. European Journal of Applied Mathematics, 31 (2020), no. 3, 385-406.
• Abdulla, Ugur G.; Goldfarb, Jonathan Frechet Differentiability in Besov Spaces in the Optimal Control of Parabolic Free Boundary Problems, Inverse and Ill-posed Problems, Volume 26, Issue 2, 2018, 211-228.
• Abdulla, Ugur G.; Poteau, Roby, Identification of Parameters for Large-scale Kinetic Models, Journal of Computational Physics, 26 November 2020, 110026.

Research

I am an expert on Partial Differential Equations and its applications. My research interest and contributions can be classified into four groups.

• Partial Differential Equations
• Optimal Control and Inverse Problems for PDEs
• Dynamical Systems and Ergodic Theory
• Quantum Biology and Mathematical Biosciences

1. Partial Differential Equations (PDEs)

1.1. Potential Theory & PDEs

Norbert Wiener's celebrated result on the boundary regularity of harmonic functions is one of the most beautiful and delicate results of 20^th century mathematics. It has shaped the boundary regularity theory for elliptic and parabolic PDEs, and has become a central result in the development of potential theory at the intersection of functional analysis, PDE, and measure theories. My recent developments precisely characterize the regularity of the point at ∞ for second order elliptic and parabolic PDEs and broadly extend the role of the Wiener test in classical analysis. The Wiener test at ∞ arises as a global characterization of uniqueness in boundary value problems for arbitrary unbounded open sets. From a topological point of view, the Wiener test at ∞ arises as a thinness criteria at ∞ in fine topology. In a probabilistic context, the Wiener test at ∞ characterizes asymptotic laws for the characteristic Markov processes whose generator is the given differential operator. The counterpart of the new Wiener test at a finite boundary point leads to uniqueness in the Dirichlet problem for a class of unbounded functions growing at a certain rate near the boundary point; a criterion for the removability of singularities and/or for unique continuation at the finite boundary point. Recent related publications are listed below:

• Removability of the Logarithmic Singularity for the Elliptic PDEs with Measurable Coefficients and its Consequences, Calculus of Variations and Partial Differential Equations Volume 57:157, December 2018.
• Regularity of ∞ for Elliptic Equations with Measurable Coefficients and Its Consequences, Discrete and Continuous Dynamical Systems - Series A (DCDS-A), 32, 10(2012), 3379-3397.
• Wiener's Criterion at ∞ for the Heat equation and its Measure-theoretical Counterpart, Electronic Research Announcements in Mathematical Sciences, 15 (2008), 44-51.
• Wiener’s Criterion at ∞ for the Heat Equation, Advances in Differential Equations, 13, 5-6(2008), 457-488.
• Wiener’s Criterion for the Unique Solvability of the Dirichlet Problem in Arbitrary Open Sets with Non-Compact Boundaries, Nonlinear Analysis, 67, 2(2007), 563-578.

1.2. Geometric Boundary Regularity Tests for PDEs and Asymptotic Laws for Wiener Processes

One of the oldest problems in the theory of PDEs is the problem of finding geometric conditions on the boundary manifold for the regularity of the solution of the elliptic and parabolic PDEs. There is a deep connection between this problem and the delicate problem of asymptotics of the corresponding Wiener processes. In my paper

• First Boundary Value Problem for the Diffusion Equation. I. Iterated Logarithm Test for the Boundary Regularity and Solvability, SIAM J. Math. Anal., 34, 6, 2003, 1422-1434.

 I proved a geometric iterated logarithm test for the boundary regularity of the solution to the heat equation. In addition, I proved an exterior hyperbolic paraboloid condition for the boundary regularity, which is the parabolic analogy of the exterior cone condition for the Laplace equation. In fact, for the characteristic top boundary point of the symmetric rotational boundary surfaces, the necessary and sufficient condition for the regularity coincides with the well-known Kolmogorov-Petrovsky test for the local asymptotics of the multi-dimensional Brownian motion trajectories:

• Multidimensional Kolmogorov-Petrovsky test for the boundary regularity and irregularity of solutions to the heat equation, Boundary Value Problems, 2, 2005, 181-199.

 In the case when symmetric rotational boundary surfaces extend to t=-∞, the regularity of the point at ∞ in my sense precisely characterize the uniqueness of the bounded solutions. I proved a geometric necessary and sufficient condition for the regularity of ∞. In the probabilistic context the result coincides with the Kolmogorov-Petrovsky test for the asymptotics of the multi-dimensional Brownian motion trajectories at infinity. My related publications are listed below.

• Necessary and sufficient condition for the existence of a unique solution to the first boundary value problem for the diffusion equation in unbounded domains, Nonlinear Analysis, 64, 5(2006), 1012-1017.
• Kolmogorov problem for the heat equation and its probabilistic counterpart, Nonlinear Analysis, 63, 5-7, 2005, 712-724.

1.3. Qualitative Theory of Nonlinear PDEs 

PDEs arising in a majority of real-world applications are nonlinear. Despite its complexity, in the theory of nonlinear PDEs, one can observe key equations or systems which are essential for the development of the theory for a particular class of equations. One of those key equations is the nonlinear degenerate/singular parabolic equation, so called nonlinear diffusion equation. In a series of publications, I developed a general theory of the nonlinear degenerate and/or singular parabolic equations in general non-smooth domains under the minimal regularity assumptions on the boundary. This development was motivated with numerous applications, including flow in porous media, heat conduction in a plasma, free boundary problems with singularities, etc. For example, primarily by applying this theory, I solved Barenblatt's problem about the short-time behavior of interfaces for the reaction-diffusion equations. Related publications are listed below.

• Well-posedness of the Dirichlet Problem for the Nonlinear Diffusion Equation in Non-smooth Domains, Transactions of Amer. Math. Soc., 357, 1, 2005, 247-265.
• Reaction-diffusion in nonsmooth and closed domains, Boundary Value Problems, Special issue: Harnack estimates, Positivity and Local Behavior of Degenerate and Singular Parabolic Equations, Vol. 2007 (2007). 
• On the Dirichlet problem for the nonlinear diffusion equation in non-smooth domains, J. Math. Anal. Appl., 246, 2, 2001, 384-403.
•  Reaction-diffusion in Irregular Domains, J. Differential Equations, 164, 2000, 321-354.

• Reaction-diffusion in a closed domain formed by irregular curves, J. Math. Anal. Appl., 246, 2000, 480-492. 

• Interface Development and Local Solutions to Reaction-Diffusion Equations (together with J.R.King), SIAM J. Math. Anal., 32, 2, 2000, 235-260.
• Evolution of interfaces and explicit asymptotics at infinity for the fast diffusion equation with absorption, Nonlinear Analysis, 50, 4, 2002, 541-560.
• Local structure of solutions of the Dirichlet problem of N-dimensional reaction-diffusion equations in bounded domains,  Advance in Differential Equations, Volume 4, Number 2, 1999, 197-224.
• Evolution of interfaces for the nonlinear parabolic p-Laplacian type reaction-diffusion equations (together with R.Jeli), European Journal of Applied Mathematics, Volume 28, Issue 5, 2017, 827-853.
• Evolution of Interfaces for the Nonlinear Double Degenerate Parabolic Equation of Turbulent Filtration with Absorption (together with J.Du, A.Prinkey, Ch.Ondracek, S.Parimoo), Mathematics and Computers in Simulation, 153 (2018), 59-82.
• Evolution of interfaces for the non-linear parabolic p -Laplacian type reaction-diffusion equations. II. Fast diffusion vs. strong absorption. European Journal of Applied Mathematics, March 2019
• Evolution of Interfaces for the Nonlinear Double Degenerate Parabolic Equation of Turbulent Filtration with Absorption. II. Fast diffusion case (together with A. Prinkey, M. Avery), submitted, March 2019, arXiv#1903.08155
• Interface development for the nonlinear degenerate multidimensional reaction-diffusion equations (together with Amna Abuweden), Nonlinear Differential Equations and Applications NoDEA, February 2020, 27:3.
• Instantaneous shrinking of the support of a solution of a nonlinear degenerate parabolic equation. (Russian) Mat. Zametki 63(1998), no.3,323-331; translation in Math. Notes 63(1998), no.3-4,285-292.

•  On sharp local estimates for the support of solutions in problems for nonlinear parabolic equations. (Russian) Mat. Sbornik 186(1995), no.8,3-24; translation in Sb. Math. 186(1995), no.8,1085-1106.

2. Optimal Control and Inverse Problems for PDEs

I developed a new variational formulation of the inverse Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundary. This research project is motivated by the bioengineering problem on the laser ablation of biological tissues. I employed optimal control framework, where boundary heat flux and free boundary are components of the control vector, and optimality criteria consists of the minimization of the sum of L2-norm declinations from the available measurement of the temperature on the fixed boundary and available information on the phase transition temperature on the free boundary. This approach allows one to tackle situations when the phase transition temperature is not known explicitly, and is available through measurement with possible error. It also allows for the development of iterative numerical methods of least computational cost due to the fact that for every given control vector, the parabolic PDE is solved in a fixed region instead of full free boundary problem. Below are recent papers published by my research group on Optimal Control and Inverse Problems:

• U. G. Abdulla On the Optimal Control of the Free Boundary Problems for the Second Order Parabolic Equations. I. Well-posedness and Convergence of the Method of Lines, Inverse Problems and Imaging, Volume 7, Number 2(2013), 307-340.
• U. G. Abdulla On the Optimal Control of the Free Boundary Problems for the Second Order Parabolic Equations: II. Convergence of the Method of Finite Differences, Inverse Problems and Imaging, Volume 10, Number 4(2016), 869-898. 
• U. G. Abdulla and B. Poggi Optimal control of the multiphase Stefan problem, Applied Mathematics & Optimization, 80, 2(2019), 479-513.
• U.G. Abdulla and J. Goldfarb Frechet Differentiability in Besov Spaces in the Optimal Control of Parabolic Free Boundary Problems, Inverse and Ill-posed Problems, Volume 26, Issue 2, 2018, 211-228.
• U. G. Abdulla, E. Cosgrove, J. Goldfarb On the Frechet Differentiability in Optimal Control of Coefficients in Parabolic Free Boundary Problems, Evolution Equations and Control Theory, Volume 6, Issue 3, 2017, 319-344.
• U. G. Abdulla, V. Bukshtynov, A. Hagverdiyev Gradient Method in Hilbert-Besov Spaces for the Optimal Control of Parabolic Free Boundary Problems, Journal of Computational and Applied Mathematics, Volume 346, January 2019, 84-109.
• U. G. Abdulla and B. Poggi Optimal Stefan Problem, Calculus of Variations and Partial Differential Equations, 59, 61(2020).
• U. G. Abdulla and E. Cosgrove, Optimal Control of Multiphase Free Boundary Problems for Nonlinear Parabolic Equations, Applied Mathematics and Optimization (2020).
• U. G. Abdulla, J. Goldfarb and A. Hagverdiyev Optimal Control of Coefficients in Parabolic Free Boundary Problems Modeling Laser Ablation, Journal of Computational and Applied Mathematics (2020).

3. Dynamical Systems and Ergodic Theory

This project relates to two topics of Dynamical Systems and Chaos Theory: 1. Classification of periodic orbits for the continuous endomorphisms in the interval, and 2. Asymptotic behavior of parameter dependent continuous maps, chaos phenomena and universal transition from periodic to chaotic behavior for nonlinear maps. The first topic originates from the celebrated result by Sharkovski (1964) on the coexistence of periodic orbits of continuous maps on the interval, which presents a surprisingly simple and elegant hierarchy of distribution of periodic orbits according to their periods. This second topic originates from the pioneering work of Feigenbaum (1978) on the universal transition route from periodic to chaotic behavior through period doubling bifurcations for the logistic type unimodal maps. Recently my Dynamical Systems and Ergodic Theory research team solved an open problem on the classification of second minimal odd orbits of the continuous endomorphisms on the interval. It is proved that there are 4k−3 types of second minimal (2k+1)-orbits, each characterized with unique cyclic permutation and directed graph of transitions with accuracy up to inverses. We then revealed a fascinating universal law of distribution of periodic orbits in chaotic regime for one-parameter family of unimodal continuous maps on the interval, and very deep connection between Sharkovski ordering and universality in chaos. Recent papers are listed below:

• A. U. Abdulla, R.U. Abdulla, U.G. Abdulla, On the Minimal 2(2k+1)-orbits of the Continuous Endomorphisms on the Real Line with Application in Chaos Theory, Journal of Difference Equations and Applications, Volume 19, 9(2013), 1395-1416
• U. G. Abdulla, R.U. Abdulla, M.U. Abdulla and N.H. Iqbal, Second minimal orbits, Sharkovski ordering and universality in chaos., Volume 27, Number 5, May 2017, Arxiv:1610.00814.
• U. G. Abdulla, R.U. Abdulla, M.U. Abdulla, N.H. Iqbal, Classification of the Second Minimal Odd Periodic Orbits in the Sharkovskii Ordering, 2017, ArXiv:1701.02695

4. Quantum Biology and Mathematical Biosciences

4.1. Identification of Parameters in Systems Biology

Systems Biology is an actively emerging interdisciplinary area between biology and applied mathematics, based on the idea of treating biological systems as a whole entity which is more than the sum of its interrelated components. These systems are networks with emerging properties generated by complex interaction of a large number of cells and organisms. One of the major goals of systems biology is to reveal, understand, and predict such properties through the development of mathematical models based on experimental data. In many cases, predictive models of systems biology are described by large systems of nonlinear differential equations. Quantitative identification of such systems requires the solution of inverse problems on the identification of parameters of the system. In a recent project we consider the inverse problem for the identification of parameters for systems of nonlinear ODEs arising in systems biology. A new numerical method which combines Pontryagin optimization, Bellman's quazilinearization with sensitivity analysis and Tikhonov's regularization is implemented. The method is applied to various biological models such as Lotka-Volterra system, bistable switch model in genetic regulatory networks, gene regulation and repressilator models from synthetic biology. The numerical results and application to real data demonstrate that the method is very well adapted to canonical models of systems biology with moderate size parameter sets and has a quadratic convergence. Software package qlopt is developed to implement the method and posted in GitHub under the GNU General Public License v3.0. One recent paper is listed below.

• U. G. Abdulla and R. Poteau, Identification of Parameters in Systems Biology, Mathematical Biosciences, Volume 305, November 2018, 133-145.

To address adaptation and the scalability of the method to inverse problems with significantly larger size of parameter sets we developed a modification of the method by embedding a method of staggered corrector for sensitivity analysis and by enhancing multi-objective optimization which enables application of the method to large-scale models with practically non-identifiable parameters based on multiple data sets, possibly with partial and noisy measurements. Application of the modified method to benchmark model of a three-step pathway modeled by 8 nonlinear ODEs with 36 unknown parameters demonstrate geometric convergence with a minimum of five data sets and with minimum measurements per data set. MATLAB package AMIGO2 is used to demonstrate the advantage of qlopt over most popular methods/software such as lsqnonlin, fmincon and nl2sol. Here is a current preprint:

• U. G. Abdulla and R. Poteau, Identification of Parameters for Large-scale Kinetic Models, Journal of Computational Physics, 26 November 2020, 110026 

4.2. Cancer Detection through EIT and Optimal Control Theory

EIT is a non-invasive imaging technique recently gaining popularity in various medical applications including breast screening and cancer detection. Our recent project is on inverse EIT problem on recovering electrical conductivity tensor and potential in the body based on the measurement of the boundary voltages on the electrodes for a given electrode current. The inverse EIT problem presents an effective mathematical model of breast cancer detection based on the experimental fact that the electrical conductivity of malignant tumors of the breast may significantly differ from conductivity of the surrounding normal tissue. We analyze the inverse EIT problem in a PDE constrained optimal control framework in Besov space, where the electrical conductivity tensor and boundary voltages are control parameters, and the cost functional is the norm declinations of the boundary electrode current from the given current pattern and boundary electrode voltages from the measurements. The state vector is a solution of the second order elliptic PDE in divergence form with bounded measurable coefficients under mixed Neumann/Robin type boundary condition. To address the highly ill-posed nature of the inverse EIT problem, we develop a "variational formulation with additional data" which is well adapted to clinical situation. We prove existence of the optimal control problem and Frechet differentiability in the Besov space setting. Based on the Frechet gradient and optimality condition effective numerical method based on the projective gradient method in Besov spaces is developed. Below is the recent preprint.

• U. G. Abdulla, V. Bukshtynov, S. Seif Breast Cancer Detection through Electrical Impedance Tomography and Optimal Control Theory: Theoretical and Computatonal Analysis, ArXiv:1809.05936, September 16, 2018.
• U. G. Abdulla, S.Seif Discretization and Convergence of the EIT Optimal Control Problem in 2D and 3D Domains, to appear in Inverse and Ill-posed Problems, 2020.