Numerical Analysis, Scientific Computing and Mathematical Finance

Project ID: 2531ad1556

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UCL Lead department: Mathematics

Project Summary:

One strand of this research is the design, implementation and analysis of computational methods for the approximation of solutions to partial differential equations. A second is stochastic analysis and the theory of stochastic processes. A wide range of applications are considered, including continuum mechanics, acoustics, electromagnetics, stochastic optimal control, inverse problems, and mathematical finance. Specific research areas include:

Computational methods in continuum mechanics - finite element methods for fluid and solid mechanics including multiphysics coupling, such as fluid-structure interaction or contact, and fictitious domain methods.
Analysis and computational methods for wave scattering - boundary integral equations, wavenumber dependent stability, fractal domains, boundary element methods, fast solvers.
Inverse problems - robust and accurate finite element methods with Carleman estimate based analysis, medical imaging. An example of a multidisciplinary research project in this context is the computational optimisation of treatment plans for high intensity focused ultrasound (HIFU) cancer treatment.
Numerical methods for nonlinear partial differential equations in control theory and game theory, such as Hamilton-Jacobi-Bellman equations and mean field games. Analysis and computation of adaptive discretizations with a posteriori error control.
Fully nonlinear differential equations emerging in stochastic optimal control and optimal transport, encompassing both numerical analysis (finite element methods, semi-Lagrangian schemes, Monte Carlo techniques, reinforcement learning) and applied analysis (comparison principles, coupling with nonlinear boundary conditions, etc.). Applications span industrial and scientific domains, including energy storage, micromagnetism, and mass transfer.
Mathematical finance, including asset pricing and hedging (fixed-income, equity, credit, commodities & emissions markets, insurance, etc.), financial risk management, computational methods for finance and insurance, algorithmic finance, filtrations and information modelling, rough path theory, statistical inference and machine learning, and research on heavy-tailed processes.