# Research

My research interests and experiences focus on applied and computational mathematics, with  particular emphasis on  PDE-constrained optimization, discontinuous Galerkin methods, adaptive finite element methods, phase field modelsuncertainty quantification, and deep learning techniques for PDEs. These are usually large-scale ill-posed problems where solution involves combined techniques in computational science, engineering and mathematics. The aim is to develop robust and accurate methods that enable the solution reasonable time. My research can be categorized as follows:

Deep Neural Network for Partial Differential Equations

Partial differential equations (PDEs) are fundamental to model different phenomena in science and engineering mathematically. Solving them is a crucial step towards a precise knowledge of the behavior of natural and engineered systems. The fundamental idea to solve a PDE is to approximate the solution of the PDE by means of functions specially built to have some desirable properties. However, conventional methods such as  finite difference method, finite volume method, and finite element method have some shortcomings such as mesh dependence and the curse of dimensionality problem for especially high dimensional problems. Nowadays,  deep neural network can be considered as an alternative to conventional methods in order to investigate the numerical solution of PDEs.

Uncertainty Quantification

Uncertainty quantification (UQ) is a modern inter-disciplinary science that cuts across traditional research groups and combines statistics, numerical analysis and computational applied mathematics. When we attempt to simulate complex real-world phenomena, e.g., fluid dynamics, climate science, chemically reacting systems, oil field research, the price of stock pricing, using mathematical and computer models, there  almost always exits uncertainty in our predictions due to the lack of knowledge or inherent variability in the model parameters. Therefore, the idea of uncertainty quantification (UQ), i.e. quantifying the effects of uncertainty on the result of a computation, has attracted much interest in the last few years.

PDE-Constrained Optimization

Optimization problem with constraints given by PDEs arise in many real-life applications. The numerical solution to PDE-constrained optimization problems involves a series of theoretical and practical challenges: solving a PDE-constrained optimization problem not only requires a solution to state equations but  adjoint equations as well; the structural interaction between the optimization issue and the underlying PDE, and the impact of the discretization processes have to be taken into account; the numerical approaches typically can lead to large-scale nonlinear programming problems. With regard to algorithmic complexity, their numerical solutions require the use of efficient iterative schemes such as multilevel techniques or preconditioning to solve the saddle matrix system. Therefore, the appropriate mathematical treatment of PDE-constrained optimization problems requires the integrated use of advanced methodologies from the theory of optimization and optimal control in functional analytic setting, the theory of PDEs as well as the development and implementation of powerful algorithmic tools from numerical mathematics and scientific computing.

Phase Field Models

Phase-field models are mathematical models for solving interfacial problems emerged in material science, image processing, or chemistry. The reliability of solder joints connecting different components is becoming very popular in electronic devices. As voids and cracks typically develop at phase boundaries, a well-known source of failure is thermomechanically induced phase separation in solder alloys, called coarsening. Allen-Cahn or Cahn-Hilliard equations describe phase separation of a immiscible binary mixture at constant temperature in the presence of dissipation of free energy. They are stiff, nonlinear parabolic partial differential equations, which may serve as a prototype phase-field problems as intermediate step toward models that take other or additional phenomena into account, e.g., mechanical stress effect, elastic interactions.