Dr. Kirill Golubnichiy

Research Interests

Inverse Problems for Partial Differential Equations; Ill-Posed Problems; Mathematical Physics; Imaging and Image Reconstruction; Carleman Estimates; Scientific Computing; Machine Learning; Deep Learning; Mathematical Finance; Option Pricing; Volatility Modeling; Quantum Theory; Spectral Theory; Nonlinear Partial Differential Equations; Numerical Optimization; Artificial Intelligence

Biography

My research interests lie primarily in inverse problems for partial differential equations, ill-posed problems, mathematical physics, imaging, financial mathematics, and quantum theory. I am particularly interested in developing rigorous analytical and computational methods for recovering unknown coefficients, structures, or physical parameters in PDE models from indirect or incomplete data.

A central part of my research focuses on Carleman estimates, convexification methods, and globally convergent numerical algorithms for nonlinear inverse problems. I also work on combining these analytic approaches with deep learning and machine learning techniques, especially for applications such as electrical impedance tomography, coefficient inverse problems, and image reconstruction.

In addition, I am interested in mathematical finance, including inverse problems for financial models, option pricing equations such as the Black–Scholes and Heston models, volatility reconstruction, and statistical methods such as GARCH-type models.

Another direction of my research is connected with mathematical physics and quantum theory, including analytical methods for studying models arising in quantum mechanics, field theory, and related nonlinear PDEs. I am especially interested in the interaction between inverse problems, spectral theory, and quantum models, as well as possible applications of data-driven and computational methods to problems motivated by quantum theory.

My current and future research directions include the development of hybrid methods that combine PDE-based analysis, regularization theory, numerical optimization, artificial intelligence, and quantum-theoretic models to solve challenging inverse and ill-posed problems arising in science, engineering, finance, and mathematical physics.