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:

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 is almost always uncertainty in our predictions. 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.

  • S. Can Toraman (MSc Graduate): Stochastic momentum methods for optimal control problems governed by convection-diffusion equations with uncertain coefficients, January 2022.
  • S. Can Toraman and H. Yücel, Stochastic momentum methods for optimal control problems containing uncertain inputs, Submitted 2021.
  • P. Çiloğlu and H. Yücel, Stochastic discontinuous Galerkin methods for robust deterministic control of convection diffusion equations with uncertain coefficients, Submitted 2021.
  • P. Çiloğlu and H. Yücel, Stochastic discontinuous Galerkin methods with low–rank solvers for convection diffusion equations, Applied Numerical Mathematics 172, 157-185, 2022.
  • Mehmet Alp Üreten (MSc Graduate): Numerical studies of Korteweg-de Vries equation with random input data, September 2018. (jointly supervised with Ömür Uğur, IAM)

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 solution requires 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.