Thesis topic proposal
János Karátson
Finite elements, neural networks and iterative solvers for diffusion type PDEs


Institute: Eötvös Loránd University, Budapest
mathematics and computing
Doctoral School of Mathematics

Thesis supervisor: János Karátson
Location of studies (in Hungarian): ELTE TTK Math. Institute, ​Department of Applied Analysis ​
Abbreviation of location of studies: ELTE

Description of the research topic:

"This research plan focuses on some aspects of finite difference and finite element solution of diffusion type partial differential equations (PDEs).

Elliptic or parabolic PDEs arise in various models in physical and other applications. The spatial discretization of linear or nonlinear steady-state or time-dependent PDEs is a time-consuming step in the numerical solution. We should store a grid structure, apply transformations, compute Jacobians and also assemble a large matrix. The focus of the planned PhD work is to assist and accelerate this procedure using neural networks and thereby develop efficient 2 or 3 dimensional numerical simulations.

For nonlinear elliptic PDEs a quasi-Newton iterative approach based on spectral equivalence has been developed and applied to the finite element discretization of symmetric second-order problems. In the proposed research this approach should be extended to more general models, including equations and also systems of PDEs with nonsymmetric convection terms that appear in modelling transport phenomena, and for higher order PDEs arising in surface diffusion and crystal models. The arising nonlinear solvers can then be coupled with the previously described neural network assisted grid generation process.

Required language skills: English
Further requirements: 
A solid background in the theory of numerical methods for PDEs and in programming in Matlab and Python. A good level of English knowledge.

Number of students who can be accepted: 1

Deadline for application: 2023-05-31

All rights reserved © 2007, Hungarian Doctoral Council. Doctoral Council registration number at commissioner for data protection: 02003/0001. Program version: 2.2358 ( 2017. X. 31. )