PDE

Part 1: Introduction to homogenisation and Gamma-Convergence

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B2 B3 A0215

We will start by introducing the notion of Gamma-convergence and its main properties. We will then show how this can be applied to the study of homogenization of integral functionals. This includes, as a particular case, the homogenization of uniformly elliptic PDEs. Lecture 1: (1) The homogenisation problem; (2) Gamma-convergence - definition, main properties, examples, comparison with pointwise limit; (3) Lower semicontinuity, lower semicontinuous envelope; (4) Coercivity

Part 2: Introduction to homogenisation and Gamma-Convergence

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B1 L2 R2202

We will start by introducing the notion of Gamma-convergence and its main properties. We will then show how this can be applied to the study of homogenization of integral functionals. This includes, as a particular case, the homogenization of uniformly elliptic PDEs. Lecture 2: (1) Integral functionals with p growth, 1 p 1; The localisation method of Gamma-convergence; Integral representation.

Part 3: Introduction to homogenisation and Gamma-Convergence

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B1 L4 R4214

We will start by introducing the notion of Gamma-convergence and its main properties. We will then show how this can be applied to the study of homogenization of integral functionals. This includes, as a particular case, the homogenization of uniformly elliptic PDEs. Lecture 3: (1) Compactness; (2) Application of the compactness result; 93) The homogenisation theorem.

A very short introduction to regularity theory for linear elliptic equations - Part 1

Professor Jose Urbano, Department of Mathematics at University of Coimbra, Portugal

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B1 L3 R3119

Mini Course Part 1 of 4. The course is a very short introduction to regularity for linear elliptic pdes of second order. We start with equations with regular coefficients and the difference quotient method of Nirenberg. We then treat the case of coefficients that are merely measurable and bounded, putting forward the basics of De Giorgi-Nash-Moser theory. If time permits, we present some characterizations of Hölder spaces which are very useful in regularity theory.

A very short introduction to regularity theory for linear elliptic equations - Part 3

Professor Jose Urbano, Department of Mathematics at University of Coimbra, Portugal

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B1 L3 R3119

Mini Course Part 3 of 4. The course is a very short introduction to regularity for linear elliptic pdes of second order. We start with equations with regular coefficients and the difference quotient method of Nirenberg. We then treat the case of coefficients that are merely measurable and bounded, putting forward the basics of De Giorgi-Nash-Moser theory. If time permits, we present some characterizations of Hölder spaces which are very useful in regularity theory.

A very short introduction to regularity theory for linear elliptic equations - Part 4

Professor Jose Urbano, Department of Mathematics at University of Coimbra, Portugal

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B1 L3 R3119

Mini Course Part 4 of 4. The course is a very short introduction to regularity for linear elliptic pdes of second order. We start with equations with regular coefficients and the difference quotient method of Nirenberg. We then treat the case of coefficients that are merely measurable and bounded, putting forward the basics of De Giorgi-Nash-Moser theory. If time permits, we present some characterizations of Hölder spaces which are very useful in regularity theory.

A very short introduction to regularity theory for linear elliptic equations - Part 2

Professor Jose Urbano, Department of Mathematics at University of Coimbra, Portugal

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B1 L2 R2202

Mini Course Part 2 of 4. The course is a very short introduction to regularity for linear elliptic pdes of second order. We start with equations with regular coefficients and the difference quotient method of Nirenberg. We then treat the case of coefficients that are merely measurable and bounded, putting forward the basics of De Giorgi-Nash-Moser theory. If time permits, we present some characterizations of Hölder spaces which are very useful in regularity theory.

Unveiling the potential of energy/entropy stable numerical methods for hyperbolic/mixed hyperbolic-parabolic PDEs

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B9 L2 H1

Abstract The demand for increasingly multidisciplinary reliable simulations, for both analysis and design optimization purposes, requires transformational advances in individual components of future PDEs solvers. At the algorithmic level, hardware compatibility and efficiency are of paramount importance in determining viability on future hardware. However, equally important (if not more so) is provable algorithmic robustness which becomes progressively more challenging to achieve as problem size and physics complexity increase. We show that rigorously designed adaptive semi- and fully-discrete

Deterministic particle approximation for nonlinear transport equations

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Building L3 R3119

Approximating the solution to an evolutionary partial differential equation by a set of "moving particles" has several advantages. It validates the use of a continuity equation in an "individuals-based" modeling setting, it provides a link between Lagrangian and Eulerian description, and it defines a "natural" numerical approach to those equations. I will describe recent rigorous results in that context. The main one deals with one-dimensional scalar conservation laws with nonnegative initial data, for which we prove that the a suitably designed "follow-the-leader" particle scheme approximates entropy solutions in the sense of Kruzkov in the many particle limit. Said result represents a new way to solve scalar conservation laws with bounded and integrable initial data. The same method applies to second order traffic flow models, to nonlocal transport equations, and to the Hughes model for pedestrian movements.

WAN Approximation of PDEs: Best Approximation, Stabilization and Boundary Conditions

Prof. Silvia Bertoluzza

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B2 L5 R5209

We present a theoretical analysis of the Weak Adversarial Networks (WAN) method, recently proposed in [1, 2], as a method for approximating the solution of partial differential equations in high dimensions and tested in the framework of inverse problems. In a very general abstract framework.

Virtual Element Approximation of Eigenvalue Problems

Prof. Francesca Gardini, Università di Pavia

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B1 L3 R3119

We will discuss the solution of eigenvalue problems associated with partial differential equations (PDE)s that can be written in the generalised form Ax = λMx, where the matrices A and/or M may depend on a scalar parameter. Parameter dependent matrices occur frequently when stabilised formulations are used for the numerical approximation of PDEs. With the help of classical numerical examples we will show that the presence of one (or both) parameters can produce unexpected results.

Myoungkeun Shin

PDE UWOC

Myoungkeun Shin is a MS/Ph.D student in the Communication Theory Lab (CTL) under the supervision of Professor Mohamed-Slim Alouini at King Abdullah University of Science and Technology (KAUST). Education and Early Career Myoungkeun obtained his bachelor degree in mathematics from Seoul National University (SNU) in 2011. Before joining KAUST, he taught mathematics to high school students in Korea. Research Interest Myoungkeun’s primary research interests are in PDE and UWOC. Education Profile M.S. in Applied Mathematics and Computational Science, Kind Abdullah University of Science and

David Ketcheson

Professor Ketcheson is an expert in numerical analysis and computational science, specializing in the study of numerical methods for differential equations and the modeling and analysis of time-dependent nonlinear wave phenomena.