Polynomial approximation methods for solving Hamilton-Jacobi-Bellman PDEs in nonlinear model reduction.

Many physical applications involve modeling the state of complicated system as a high-dimensional system of ordinary differential equations (ODEs). These finite dimensional models of dynamical systems appear in many applications, including the control of multi-agent systems, mechanical systems, and systems of electrical signals. The high dimensionality of these models introduces significant computational challenges in trying to analyse the behaviour of the system. This is often known as the "curse of dimensionality", which often arises in attempting to analyse systems of high dimension or compute their control, for example in multi-agent systems where the number of dimensions in the state-space can be in the thousands or millions.
My PhD thesis is based on numerical approximation techniques for reduced-order modeling. This presents a particularly interesting challenge due to the range of different types of reduced-order modeling that are possible, and the computational and practical challenges that come with dealing with nonlinear and high-dimensional problems. Reduced-order modeling finds application in the analysis and design of systems in many fields, for example mechanical and electrical systems, and weather forecasting, and is also useful for finding optimal control solutions in these fields. In my project, I introduce novel computational methods using the Galerkin residual method with polynomial basis functions to solve specific types of model-order reduction problems.
The main research contributions that I expect to achieve through my thesis involve finding approximate polynomial solutions to various partial differential equations (PDEs) to aid in constructing reduced-order models. So far, in collaboration with the Electrical and Electronic Engineering department at Imperial College London, I have developed and implemented a Galerkin spectral method to find approximate solutions to the invariance PDE associated with the steady-state response of...