MATH 488/489: Dimitrios Mitsotakis' Projects

 

Project on numerical analysis

Numerical analysis is the mathematical field that deals with methods for approximating the solutions of various mathematical problems.The objective of this project is to study the theory and the implementation of a particular numerical method in the field of numerical analysis. This numerical method can be a method for the numerical solution of nonlinear equations or for the numerical solution of differential equations. Physical applications will also be considered. The project will combine methods of functions analysis, linear algebra, numerical analysis and scientific computations.  

Project on fluid mechanics

Equations of fluid mechanics can be very complicated and further simplifications are often required for the complete understanding of the properties of the fluids. The objective of this project is to study the derivation and the properties of equations from fluid mechanics and their applications. The derivation of the equations will be based on basic principles of physics. Then asymptotic analysis will be used for further understanding of the properties of the solutions of the equations. Applications can be related with water waves, blood flow or nonlinear optics.  

Project on applied analysis

Partial differential equations are characterised as good deterministic models when they have an established well-posedness theory. In other words, they are good models when they have a unique solution in appropriate spaces and the solution is stable under small perturbations of the initial data. This is equivalent to Newton's principle of determinacy that guarantees that an equation can predict the future state of a physical system given the current state only. The objective of this project is to study the theoretical properties of partial differential equations that describe certain physical phenomena of great importance. Theoretical properties such as existence, uniqueness and stability of their solutions will be studied in appropriate functional spaces. Certain techniques of mathematical analysis will be reviewed and extended.  

Project on machine learning

Modern numerical techniques evolved in such a way that solutions to problems can be predicted using neural network and optimization techniques. The focus of this project is on Physics Informed Neural Networks (PINNs) and on their applications to the approximation of solutions to nonlinear and dispersive water wave equations. An immediate application of this project involves forecast of tsunami waves, which are nonliner and dispersive.