Partial Differential Equations

Peter Markowich Research Group

Professor Peter Markowich leads the so-called P-Markowich research group, which stands for Peter Markowich Research Group Research group areas of expertise and current scientific interests: numerical analysis, partial differential equations Led by Prof. Peter Markowich, the P-Markowich research group focuses on the mathematical and numerical analysis of partial differential equations (PDEs) and their applications in physics, biology, and engineering.

Numerical Mathematics Research Group

Professor David I. Ketcheson leads the NUMERICS research group, which stands for Numerical Mathematics Research group areas of expertise and current scientific interests: numerical methods, partial differential equations, nonlinear waves Research within our group focuses on the design, analysis, and implementation of numerical methods for ordinary and partial differential equations, as well as the application of numerical methods to problems in nonlinear wave propagation. Meet the Team Get to know NUMERICS Team Events Learn more about NUMERICS events Projects Learn more about NUMERICS projects

Mean-field Games and Nonlinear PDE Research Group

Professor Diogo Gomes leads the so-called MFG research group, which stands for Mean-field Games and Nonlinear PDE Research group areas of expertise and current scientific interests: partial differential equations, mean-field models, numerical analysis Professor Gomes’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

PDE with Random Coefficients As A Problem in High-Dimensional Numerical Integration By Prof. Ian H. Sloan (University of New South Wales, Australia)

B1 R4214

This talk is concerned with the use of quasi-Monte Carlo methods combined with finite-element methods to handle an elliptic PDE with a random field as a coefficient. A number of groups have considered such problems (under headings such as polynomial chaos, stochastic Galerkin and stochastic collocation) by reformulating them as deterministic problems in a high dimensional parameter space, where the dimensionality comes from the number of random variables needed to characterize the random field.

Discontinuous Galerkin approximation to the Vlasov-Poisson system By Dr. Blanca Ayuso de Dios (Centre de Recerca Matematica at Barcelona) ​

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B1 R4214

One of the simplest model problems in the kinetic theory of plasma--physics is the Vlasov-Poisson system with periodic boundary conditions. Such system describes the evolution of a plasma of charged particles (electrons and ions) under the effects of the transport and self-consistent electric field. In this talk, we present a family of discontinuous Galerkin (DG) methods for the approximation of the Vlasov-Poisson system.

Part 3: Variational structures and Fourier approximation techniques for mean-field game systems

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B1 L3 R3119

Mean-field game (MFG) systems of partial differential equations (PDE) arise in modeling huge populations of indistinguishable rational agents that play non-cooperative differential games. Mathematically, an MFG system comprises of a Hamilton-Jacobi-Bellman PDE coupled with a Kolmogorov-Fokker-Planck PDE in a highly nonlinear fashion. Hence, theoretical and numerical treatments of MFG systems are highly challenging problems.

Part 1: Variational structures and Fourier approximation techniques for mean-field game systems

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B1 L3 R3119

Mean-field game (MFG) systems of partial differential equations (PDE) arise in modeling huge populations of indistinguishable rational agents that play non-cooperative differential games. Mathematically, an MFG system comprises of a Hamilton-Jacobi-Bellman PDE coupled with a Kolmogorov-Fokker-Planck PDE in a highly nonlinear fashion. Hence, theoretical and numerical treatments of MFG systems are highly challenging problems. Day 1: I will show how to transform suitable mean-field game (MFG) systems into infinite-dimensional convex optimization problems. Furthermore, I will present Uzawa’s algorithm and augmented Lagrangian approach for solving convex optimization problems. Finally, I will demonstrate how to apply these methods to approximate solutions of corresponding MFG systems.

Estimation Problems for Partial and Fractional Differential Equations - Spring 2019

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B9 L2 H1

We will present some new methods for source and parameters estimation for partial and fractional differential equations and illustrate the results with some simulations and real applications.

Part 2: Variational structures and Fourier approximation techniques for mean-field game systems

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B1 L3 R3119

Mean-field game (MFG) systems of partial differential equations (PDE) arise in modeling huge populations of indistinguishable rational agents that play non-cooperative differential games. Mathematically, an MFG system comprises of a Hamilton-Jacobi-Bellman PDE coupled with a Kolmogorov-Fokker-Planck PDE in a highly nonlinear fashion. Hence, theoretical and numerical treatments of MFG systems are highly challenging problems. Day 2: I will present new Fourier approximation techniques for nonlocal MFG systems and discuss how these techniques fit the variational framework presented on Day 1. Then I will introduce the Chambolle-Pock primal-dual hybrid gradient optimization method and apply a variant of this method to approximate nonlocal MFG models via Fourier approximation techniques above.

Part 3: The isoperimetric inequality. Stability and related topics

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B1 L4 R4102

The aim of the course is to give a self contained introduction, at the level of a graduate course, to the stability of the isoperimetric inequality and other related geometric and functional inequality such as the Sobolev inequality, the Faber-Krahn and the Brunn-Minkowski inequality. Lecture 3: I will present Fuglede’s proof of the quantitative isoperimetric inequality for convex and for nearly spherical sets

Part 1: The isoperimetric inequality. Stability and related topics

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B1 L4 R4102

The aim of the course is to give a self contained introduction, at the level of a graduate course, to the stability of the isoperimetric inequality and other related geometric and functional inequality such as the Sobolev inequality, the Faber-Krahn and the Brunn-Minkowski inequality. Lecture 1: I will introduce some basic notions of the theory of sets of ignite perimeter and of geometric measure theory.

Part 1: Boundary Control of Continuum Problems

Professor Mamadou L. Diagne, Rensselaer Polytechnic Institute

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B9 L3 R3131

Partial Differential Equations (PDEs) are often used to model various complex physical systems. Representative engineering applications such as heat exchangers, transmission lines, oil wells, road traffic, multiphase flow, melting phenomena, supply chains, collective dynamics, and even chemical processes governing the state of charge of Lithium-ion battery, extrusion, reactors to mention a few. Generally, key aspects of these processes operating mode are driven by convection phenomena with a spatiotemporal dynamic that cannot be approximated straightforwardly using a finite-dimensional representation. This course will explore the boundary control of several class of PDEs via the well-known backstepping method.

Part 2: The isoperimetric inequality. Stability and related topics

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B1 L4 R4102

The aim of the course is to give a self contained introduction, at the level of a graduate course, to the stability of the isoperimetric inequality and other related geometric and functional inequality such as the Sobolev inequality, the Faber-Krahn and the Brunn-Minkowski inequality. Lecture 2: I will present De Giorgi’s proof of the isoperimetric inequality and comment on other proofs of the isoperimetric inequality

Part 4: The isoperimetric inequality. Stability and related topics

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B1 L4 R4102

The aim of the course is to give a self contained introduction, at the level of a graduate course, to the stability of the isoperimetric inequality and other related geometric and functional inequality such as the Sobolev inequality, the Faber-Krahn and the Brunn-Minkowski inequality. Lecture 4: I will discuss the quantitative isoperimetric inequality in the general case and the quantitative form of related geometric and functional inequalities.

Part 2: Boundary Control of Continuum Problems

Professor Mamadou L. Diagne, Rensselaer Polytechnic Institute

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B9 L3 R3131

Partial Differential Equations (PDEs) are often used to model various complex physical systems. Representative engineering applications such as heat exchangers, transmission lines, oil wells, road traffic, multiphase flow, melting phenomena, supply chains, collective dynamics, and even chemical processes governing the state of charge of Lithium-ion battery, extrusion, reactors to mention a few. Generally, key aspects of these processes operating mode are driven by convection phenomena with a spatiotemporal dynamic that cannot be approximated straightforwardly using a finite-dimensional representation. This course will explore the boundary control of several class of PDEs via the well-known backstepping method.

Part 3: Boundary Control of Continuum Problems

Professor Mamadou L. Diagne, Rensselaer Polytechnic Institute

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B9 L3 R3131

Partial Differential Equations (PDEs) are often used to model various complex physical systems. Representative engineering applications such as heat exchangers, transmission lines, oil wells, road traffic, multiphase flow, melting phenomena, supply chains, collective dynamics, and even chemical processes governing the state of charge of Lithium-ion battery, extrusion, reactors to mention a few. Generally, key aspects of these processes operating mode are driven by convection phenomena with a spatiotemporal dynamic that cannot be approximated straightforwardly using a finite-dimensional representation. This course will explore the boundary control of several class of PDEs via the well-known backstepping method.

Estimation Problems for Partial and Fractional Differential Equations - Autumn 2019

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B9 L2 H1 R2322

I will present an overview of our activities around estimation problems for partial and fractional differential equations. I will present the methods and the algorithms we develop for the state, source and parameters estimation and illustrate the results with some simulations and real applications.

Part 1B | PDE Backstepping Boundary Control of Parabolic PDEs

Prof. Mamadou Diagne, Rensselaer Polytechnic Institute

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KAUST

PDE Backstepping Boundary Control of Parabolic PDEs Partial Differential Equations (PDEs) are often used to model various complex physical systems. Representative engineering applications such as heat exchangers, transmission lines, oil wells, road traffic, multiphase flow, melting phenomena, supply chains, collective dynamics, and even chemical processes governing the state of charge of Lithium-ion battery, extrusion, reactors to mention a few. This course will explore the boundary control of a class of parabolic PDE via the well-known backstepping method.

Part 1A | PDE Backstepping Boundary Control of Parabolic PDEs

Prof. Mamadou Diagne, Rensselaer Polytechnic Institute

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KAUST

Partial Differential Equations (PDEs) are often used to model various complex physical systems. Representative engineering applications such as heat exchangers, transmission lines, oil wells, road traffic, multiphase flow, melting phenomena, supply chains, collective dynamics, and even chemical processes governing the state of charge of Lithium-ion battery, extrusion, reactors to mention a few. This course will explore the boundary control of a class of parabolic PDE via the well-known backstepping method.

Part 2A | PDE Backstepping Boundary Control of Parabolic PDEs

Prof. Mamadou Diagne, Rensselaer Polytechnic Institute

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KAUST

Abstract Partial Differential Equations (PDEs) are often used to model various complex physical systems. Representative engineering applications such as heat exchangers, transmission lines, oil wells, road traffic, multiphase flow, melting phenomena, supply chains, collective dynamics, and even chemical processes governing the state of charge of Lithium-ion battery, extrusion, reactors to mention a few. Generally, key aspects of these processes operating mode are driven by convection phenomena with a spatiotemporal dynamic that cannot be approximated straightforwardly using a finite

Part 2B | PDE Backstepping Boundary Control of Parabolic PDEs

Prof. Mamadou Diagne, Rensselaer Polytechnic Institute

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KAUST

Partial Differential Equations (PDEs) are often used to model various complex physical systems. Representative engineering applications such as heat exchangers, transmission lines, oil wells, road traffic, multiphase flow, melting phenomena, supply chains, collective dynamics, and even chemical processes governing the state of charge of Lithium-ion battery, extrusion, reactors to mention a few. This course will explore the boundary control of a class of parabolic PDE via the well-known backstepping method.

On the collapse mechanism of Wave Maps

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B1 L4 R4214

The Wave Map system describes the evolution of waves constrained on a (Riemannian) manifold. For the 2 + 1 dimensional problem, when the target manifold is a sphere, the solution collapses in finite time. The Analysis is due to the pioneering work of Merle, Paphael and Rodnianski. Motivated by their work I will present a somewhat novel approach of the collapsing mechanism which is based on a view of the equations as a nonlinear gauge system. This is joint work with Dan Geba.

Aggregation-Diffusion and Kinetic PDEs for collective behavior: applications in the sciences

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B1 L3 R3119

In this seminar, I will present a survey of micro, meso and macroscopic models where repulsion and attraction effects are included through pairwise potentials. I will discuss their interesting mathematical features and applications in mathematical biology and engineering. Qualitative properties of local minimizers of the interaction energies are crucial in order to understand these complex behaviors. I will showcase the breadth of possible applications with three different phenomena in applications: segregation, phase transitions, and consensus.

Tensor-train Methods for Partial Differential Equations and its application to a Neutron Transport Problem

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B9 L4 R4225

Tensor network techniques are known for their ability to approximate low-rank structures and beat the curse of dimensionality. They are also increasingly acknowledged as fundamental mathematical tools for efficiently solving high-dimensional Partial Differential Equations (PDEs). In this talk, we present a novel method that incorporates the Tensor Train (TT) and Quantized Tensor Train (QTT) formats for the computational resolution of time-independent Boltzmann Neutron Transport equations (BNTEs) in Cartesian coordinates.

KAUST mathematician named Fellow to European Academy of Sciences

1 min read ·
KAUST Distinguished Professor of Applied Mathematics and Computational Science, Dr. Peter Markowich, is named a 2020 Fellow to the European Academy of Sciences (EurASc) in recognition of his outstanding scholarship in the field of mathematical and numerical analysis of partial differential equations.

Meet KAUST prospective student: Stefanos Georgiadis

1 min read ·
Stefanos Georgiadis is an applied mathematics graduate who will join KAUST from the School of Applied Mathematical and Physical Sciences, Athens, Greece. Stefanos will join KAUST this fall as a Ph.D. candidate and member of the Applied Partial Differential Equations (AppliedPDE) research group under the supervision of Professor Athanasios Tzavaras.

Stefanos Georgiadis

Stefanos Georgiadis is a Ph.D. candidate in a co-tutelle between KAUST and TU Wien, under the supervision of Professor Athanasios Tzavaras and Ansgar Jüngel. Before joining KAUST, Stefanos obtained his bachelor and master's degree in applied mathematics from the School of Applied Mathematical and Physical Sciences of the National Technical University of Athens in Greece. Research Interests Stefanos' research interests include the analysis of nonlinear partial differential equations as they apply to the modeling of multicomponent systems of gases.

Nuno J. Alves

Short Bio Mr. Nuno Alves is a Ph.D student in the AppliedPDE group of the Applied Mathematics and Computational Science program (AMCS) of CEMSE division at the King Abdullah University of Science and Technology, Saudi Arabia. He has got bachelor and master's degrees from Instituto Superior Tecnico, University of Lisbon, Portugal. Mr. Nuno had been a visiting student at KAUST from March 2014 to July 2014. Moreover, during the academic year 2017/2018 he was working as an invited teaching assistant at Polytechnic Institute of Beja, Portugal. His work is essentially related with non-linear partial

Julio Enrique Castrillon Candas

Julio Enrique Castrillon Candas worked as a Research Scientist at Professor Raul F. Tempone's Stochastic Numerics Research Group at King Abdullah University of Science and Technology (KAUST). Research Interests Julio's research focused on the development of fast and accurate numerical methods for the efficient solution of deterministic and stochastic computational models. In particular, his interests included, but not limited, to Stochastic PDEs, Radial Basis Function Interpolation, Regression, Hierarchical Basis, and Probability Theory and Stochastic Processes Selected Publications Julio E

David Evangelista

David Evangelista is a Ph.D. Candidate at King Abdullah University of Science and Technology (KAUST), working in the areas of partial differential equations, mean-field games (MFG), and market microstructure. He is expect to receive a Ph.D. from KAUST in June 2019, under the direction of Professor Diogo A. Gomes. David's research interests are in optimal control theory, nonlinear partial differential equations, MFG, and market microstructure. MFG is a framework to study interactions of a large number of indistinguishable players that play differential games. MFG has become an active research

David Keyes

Senior Associate to the President, King Abdullah University of Science and Technology
David Keyes works at the interface of computational science and engineering and high performance computing, which he helped edit into the DNA of KAUST as a founding Dean in 2009 and then as director of the Extreme Computing Research Center. He currently serves as Senior Associate to the President, Chair of the KAUST RDIA Economies of the Future committee, and delegate to the Saudi Center for the Fourth Industrial Revolution.

Carlos Andrés Muñoz Moncayo

Carlos Andrés Muñoz Moncayo is an M.S./Ph.d. candidate in applied mathematics and computational sciences program and a member of The Numerical Mathematics Group under the supervision of Professor David Ketcheson at King Abdullah University of Science and Technology (KAUST). Research Interests Carlos' research interests include partial differential equations, numerical analysis, and scientific computing.

Athanasios Tzavaras

KAUST Professor of Applied Mathematics and Computational Science Athanasios (Thanos) Tzavaras specializes in mathematical modelling, analysis and computation of partial differential equations pertaining to fluid mechanics or modeling of materials.