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Nonlinear Partial Differential Models

Project

Project Details

Program
Applied Mathematics and Computer Science
Field of Study
Mathematics or related field
Division
Computer, Electrical and Mathematical Sciences and Engineering
Faculty Lab Link
https://www.kaust.edu.sa/en/study/faculty/diogo-gomes

Project Description

In this project we will study some nonlinear partial differential equations models that arise in applications ranging from population dynamics, mean-field games, quantum chemistry and mechanics, medicine, quasi-geostrophic flows, and water waves.​​​​​

About the Researcher

Diogo Gomes
Professor, Applied Mathematics and Computational Science
Computer, Electrical and Mathematical Science and Engineering Division

Affiliations

Education Profile

  • 2006, Habilitation in Mathematics, Universidade Tecnica de Lisboa, Portugal
  • 2000, Ph.D. in Mathematics, University of California at Berkeley, USA1996, M.Sc. in Mathematics, Instituto Superior Tecnico, Portugal1995, B.Sc. in Physics Engineering, Instituto Superior Tecnico, Portugal

Research Interests

Professor Gomesa's research interests are in partial differential equations (PDE), namely on viscosity solutions of elliptic, parabolic and Hamilton-Jacobi equations as well as in related mean-field models. This area includes a large class of PDEs and examples, ranging from classical linear equations to highly nonlinear PDEs, including the Monge-Ampere equation, geometric equations for image processing, non-linear elasticity equations and the porous media equation. His research is motivated directly or indirectly by concrete applications. These include population and crowd modeling, price formation and extended mean-field models, numerical analysis of infinite dimensional PDEs and computer vision. Before joining KAUST, Professor Gomes has been a Professor in the Mathematics Department at the Instituto Superior Tecnico since 2001. He did his Post-Doc at the Institute for Advanced Study in Princeton in 2000 and at the University of Texas at Austin in 2001.

Selected Publications

  • Gomes, Diogo; SA¡nchez Morgado, HA©ctor (2014); A stochastic Evans-Aronsson problem. Trans. Amer. Math. Soc. 366 (2), 903a-929.
  • Cagnetti, F., Gomes, D., & Tran, H. V. (2013). Adjoint methods for obstacle problems and weakly coupled systems of PDE. ESAIM - Control, Optimisation and Calculus of Variations, 19(3), 754-779.
  • Gomes, D. A., Mohr, J., & Souza, R. R. (2013). Continuous time finite state mean field games. Applied Mathematics and Optimization, 68 (1), 99-143.
  • Cagnetti, F., Gomes, D., & Tran, H. V. (2011). Aubry-mather measures in the nonconvex setting. SIAM Journal on Mathematical Analysis, 43 (6), 2601-2629.
  • Gomes, D. A., Mohr, J., & Souza, R. R. (2010). Discrete time, finite state space mean field games. Journal Des Mathematiques Pures Et Appliquees, 93(3), 308-328.

Desired Project Deliverables

The objective of the project is to study in detail modeling, analytical and numerical aspects of partial differential equations from concrete applications. A final report and presentation will be required. The work will be developed and the guidance of Prof. Diogo Gomes and the Research Scientist Dr. Saber Trabelsi.​
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