My main research interest is Computational Methods for Inverse Problems for Partial Differential Equations with applications to physics and biology. In particular, I am interested in the designation of convergent numerical schemes and mathematical analysis of their stability and convergence. I also use MATLAB and Python to verify their numerical performance.
I am very active in finding collaborations. If you want to share any ideas and to join my team, please feel free to contact me! My contact information, as well as further details, can be found in my CV.
My Erdős number is 4. Several sources of my public profile are, e.g., Google Scholar, ResearchGate and Mathscinet.
By the evolution of the inverse and ill-posed problems, there are now several types of coefficient inverse problems (CIPs) for different real-world applications. My early research focused on applications such as landmine and improvised explosive device detection in military actions, as well as initial state reconstruction for source localization problems in mathematical oncology. Representative publications include:
Convexification and experimental data for a 3D inverse scattering problem with the moving point source, Inverse Problems.
Convexification for a three-dimensional inverse scattering problem with the moving point source, SIAM Journal on Imaging Sciences.
Identification of the population density of a species model with nonlocal diffusion and nonlinear reaction, Inverse Problems.
Analysis of a quasi-reversibility method for a terminal value quasi-linear parabolic problem with measurements, SIAM Journal on Mathematical Analysis.
CIPs are typically highly nonlinear, severely ill-posed, and often involve incomplete noisy data, making them especially challenging. To address these difficulties, my research focuses on the development and analysis of convexification-based methods and variational quasi-reversibility techniques, with a strong emphasis on theoretical convergence analysis. Theoretical foundations of these techniques rely on Carleman weight functions, a powerful tool in inverse problems and widely used across various areas of applied mathematics.
Recently, I have been working with more advanced models related to the above-mentioned topics, showing promising progress across several challenging CIPs. In particular, my recent research has obtained great results for multiple-coefficient inverse problems (mCIPs) for the following PDEs:
Age- and size-structured population models:
Since 2015, I have nurtured this research direction with a focus on advanced source localization models aimed at understanding the growth and evolutionary dynamics of cancer. Cancer is an age- and size-structured disease, characterized by the uncontrolled proliferation of cell populations in specific regions of the body. In this context, cell age represents the progression through the cell cycle, while cell size reflects growth and division dynamics.
The mathematical modeling of such phenomena remains largely unresolved due to the involvement of multiple nonlocal and nonlinear interactions. To unravel the complexity of cancer progression, it is essential to develop and analyze numerical schemes for both the forward and inverse problems associated with these models. Including mCIPs, some studies of mine have been done in this research line:
Simultaneous reconstruction of birth condition and mortality rate in an age-structured tumor model, Inverse Problems.
An explicit Fourier-Klibanov method for an age-dependent tumor growth model of Gompertz type, Applied Numerical Mathematics.
Uniqueness result for an age-dependent reaction–diffusion problem, Applicable Analysis.
A finite difference scheme for nonlinear ultra-parabolic equations, Applied Mathematics Letters.
Schrödinger evolution equations:
TBA
Ziad Musslimani (Tallahassee, USA), Kbenesh W. Blayneh (Tallahassee, USA), Solomon Manukure (Tallahassee, USA), Michael Klibanov (Charlotte, USA), Adrian Muntean (Karlstad, Sweden), Daniel Lesnic (Leeds, UK), Loc Hoang Nguyen (Charlotte, USA), Nguyen Huy Tuan (HCMC, VN), Nguyen Thanh Long (HCMC, VN), Thanh Tran (New South Wales, Australia), Dang Duc Trong (HCMC, VN), Mai Thanh Nhat Truong (Dongguk, South Korea), Tran The Hung (Warsaw, Poland), Mach Nguyet Minh (Helsinki, Finland), Thieu Thi Kim Thoa (Edinburg, USA), Vo Van Au (HCMC, VN), Vasily Astratov (Charlotte, USA), Van Thinh Nguyen (Seoul, Korea)