Matias Vestberg's Research Page

Affiliation

I am an Assistant Professor at Uppsala University, Department of Mathematics. I am part of the group Analysis and PDEs led by Professor Kaj Nyström. My research is suported by WASP, The Wallenberg AI, Autonomous Systems and Software Program.

Research interests

My research is concerned with exploring various aspects of PDEs and their connection to Deep Learning. Originally my research focused on the regularity and existence theory of nonlinear diffusion equations, but I have later broadened my interests to include other equations such as the Monge-Ampère equation and approximation of solutions using neural networks.

Publications

      Mathematics

S. Ciani, V. Vespri and M. Vestberg: Boundedness, Ultracontractive Bounds and Optimal Evolution of the Support for Doubly Nonlinear Diffusion, Submitted to a journal, 2023. Available on arXiv.

K. Nyström and M. Vestberg: Solving the Dirichlet problem for the Monge-Ampère equation using Neural Networks, Journal of Computational Mathematics and Data Science, Vol 8, 2023.

C. Fjellström, K. Nyström and M. Vestberg: Tug-of-war with Kolmogorov, Journal of Differential Equations, Vol 342, 501-558, 2022.

V. Bögelein, N. Dietrich and M. Vestberg: Existence of solutions to a diffusive shallow medium equation, Journal of Evolution Equations, Vol. 21, pages 845–889, 2020.

V. Vespri and M. Vestberg: An Extensive Study of the Regularity of Solutions to Doubly Singular equations, Advances in Calculus of Variations, 2020.

T. Singer, M. Vestberg: Local Hölder continuity of weak solutions to a diffusive shallow medium equation. Nonlinear Analysis, Vol. 185, 306-335, 2019. doi: 10.1016/j.na.2019.03.013

T. Singer, M. Vestberg: Local boundedness of weak solutions to the diffusive wave approximation of the shallow water equations, Journal of Differential Equations, Vol. 266 no. 6, 3014-3033, 2019. doi: 10.1016/j.jde.2018.08.051

PhD Thesis: Doubly Nonlinear Parabolic equations, 2020.

Master's Thesis: The Wave Front Set and Oscillatory Integrals, 2015.

      Physics

Sinha, Heikkinen, Vestberg, Mether, Nordlund and Lahtinen: Adsorption of maleic anhydride on Pt(111) Surface Science, 620, 9-16, 2014.

Norris, Samela, Vestberg, Nordlund and Aziz: Crater functions for compound materials: A route to parameter estimation in coupled-PDE models of ion bombardment, Nuclear instruments & methods in physics research section b -Beam interactions with materials and atoms 318. 245-252, 2014.

See also

The Analysis and PDE group website