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 models, uncertainty 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 (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.
- P. Çiloğlu and H. Yücel, Stochastic discontinuous Galerkin methods for robust deterministic control of convection diffusion equations with uncertain coefficients, Advances in Computational Mathematics, 49, 16, 2023.
- S. Can Toraman and H. Yücel, A stochastic gradient algorithm with momentum terms for optimal control problems governed by a convection-diffusion equation with random diffusivity, Journal of Computational and Applied Mathematics 422, 114919, 2023.
- S. Can Toraman (MSc Graduate): Stochastic momentum methods for optimal control problems governed by convection-diffusion equations with uncertain coefficients, January 2022.
- 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)
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.
- H. Yücel, Residual based a posteriori error estimation for Dirichlet boundary control problems, ESAIM: ProcS 71 185-195, 2021.
- H. Yücel, Goal–oriented a posteriori error estimation for Dirichlet boundary control problems, Journal of Computational and Applied Mathematics 381, 2020.
- H. Yücel, M. Stoll and P. Benner, Adaptive discontinuous Galerkin approximation of optimal control problems governed by transient convection-diffusion equations, Electronic Transactions on Numerical Analysis (ETNA) 48, 407–434, 2018.
- P. Benner and H. Yücel, Adaptive symmetric interior penalty Galerkin method for boundary control problems, SIAM Journal on Numerical Analysis 55(2), 1101-1133, 2017.
- M. Uzunca, T. Kucukseyhan, H. Yücel and B. Karasözen, Optimal control of convective FitzHugh-Nagumo equation, Computers & Mathematics with Applications 73(9), 2151-2169, 2017.
- H. Yücel, M. Stoll and P. Benner, A discontinuous Galerkin method for optimal control problems governed by a system of convection-diffusion PDEs with nonlinear reaction terms, Computers & Mathematics with Applications 70(10), 2414-2431, 2015.
- H. Yücel and P. Benner, Adaptive discontinuous Galerkin methods for state constrained optimal control problems governed by convection diffusion equations, Computational Optimization and Applications 62(1), 291-321, 2015.
- Z. K. Seymen, H. Yücel and B. Karasözen, Distributed optimal control of time-dependent diffusion-convection-reaction equations using space-time discretization, Journal of Computational and Applied Mathematics (261), 146-157, 2014.
- T. Akman, H. Yücel and B. Karasözen, A priori error analysis of the upwind symmetric interior penalty Galerkin (SIPG) method for the optimal control problems governed by unsteady convection diffusion equations, Computational Optimization and Applications (57), 703-729, 2014.
- H. Yücel and B. Karasözen, Adaptive symmetric interior penalty Galerkin (SIPG) method for optimal control of convection diffusion equations with control constraints, Optimization (63), 145-166, 2014.
- H. Yücel, and P. Benner, Distributed optimal control problems governed by coupled convection dominated PDEs with control constraints, in Assyr Abdulle, Simone Deparis, Daniel Kressner, Fabio Nobile, and Marco Picasso: Numerical Mathematics and Advanced Applications – ENUMATH 2013, Lecture Notes in Computational Science and Engineering, Springer International Publishing, 469-478, 2015.
- H. Yücel, and B. Karasözen, Optimal control of diffusion-convection-reaction equations using upwind symmetric interior penalty Galerkin (SIPG) method, in Stavros G. Stavrinides, Santo Banerjee, Suleyman Hikmet Caglar, Mehmet Ozer: Chaos and Complex Systems, Springer Berlin Heidelberg, 83-94, 2013.
- H. Yücel, M. Heikenschloss, and B. Karasözen, Distributed optimal control of diffusion-convection-reaction equations using discontinuous Galerkin methods, in Andrea Cangiani, Ruslan L. Davidchack, Emmanuil Georgoulis, Alexander N. Gorban, Jeremy Levesley, Michael V. Tretyakov: Numerical Mathematics and Advanced Applications 2011, Springer Berlin Heidelberg, 389-397, 2013.
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.
- M. Stoll and H. Yücel, Symmetric interior penalty Galerkin method for fractional-in-space phase-field equations, AIMS Mathematics 3(1), 66-95, 2018.
- B. Karasözen, M. Uzunca, A. Sariaydin Filibelioglu and H. Yücel, Energy stable discontinuous Galerkin finite element method for the Allen-Cahn equation, International Journal of Computational Methods 15(3), 1850013, 2018.