Lecturers and topics covered



Helge Holden: Introduction to hyperbolic conservation laws (Crash course, Sunday)

Helge Holden is professor at Department of Mathematical Sciences, NTNU, Trondheim. He has written many influential papers on mathematical analysis and numerical methods for hyperbolic conservation laws. In particular, he is co-inventor of the Norwegian front-tracking algorithm, on which he also has written a book on Springer Verlag.

Plan of lectures

Based upon feedback from the participants, we offer a crash course on the intermediate level. We will cover the basic properties of scalar conservation laws (one and more space dimensions) as well as systems in one space dimension, in particular properties of the Riemann initial value problem.

Suggested reading

The lectures will be based on Chs. 1, 3, and 5 of H. Holden and N. H. Risebro: Front Tracking for Hyperbolic Conservation Laws, Springer, NY, 2002.



Eitan Tadmor: Modern numerical methods for nonlinear evolution PDEs (Monday and Tuesday)

Eitan Tadmor is professor at Department of Mathematics and Institute for Physical Sciences and Technology, University of Maryland College Park. He is also director for the Center for Scientific Computation and Mathematical Modeling (CSCAMM) at University of Maryland, College Park. Moreover, he was founding co-director of Institute for Pure and Applied Mathematics (IPAM).

Eitan Tadmor is co-inventor of non-oscillatory central difference schemes for conservation laws and Hamilton-Jacobi equation, the kinetic formulation of conservation laws, and the spectral viscosity method. He has also authored a large number of influential papers on stability of linear time-dependent problems, regularity and homogenization of convection-diffusion problems, and (numerical) approximation and mathematical analysis of nonlinear conservation laws. An overview is provided at his web-page.

Plan of lectures

The lectures will cover important classes of nonlinear evolution PDEs:

The unifying theme for the lectures will be entropy stability theory for discrete approximations of nonlinear conservation laws and related problems. Subtopics include:

Suggested reading

The following reference gives a good background for the topics covered in the lectures:

For those interested in learning more on a specific sub-topic, we can recommend the following collections of references:



Hongkai Zhao: Level set technology for front propagation problems with applications in CFD and image processing (Wednesday to Friday)

Hongkai Zhao is assistant professor of computational mathematics at Department of Mathematics, University of California, Irvine.

Hongkai Zhao's research interests include level-set methods, moving interfaces and free boundaries, image processing and computer graphics, time reversal for acoustics, and domain decomposition.

If you have Realplayer installed, you can watch Prof. Zhao's lecture "Can we hear the size of a target?" on streaming video from MSRI.

Plan of lectures

The lectures will provide an introduction to front propagation problems and in particular level-set methods. Moreover, some applications in CFD and image processing will be discussed.

Lecture 1-6, Basic formulation for the level set method and corresponding numerical algorithms: implicit representation and capturing of moving interfaces, general equation of motion, computing geometric quantities, motion by mean curvature, reinitialization, extension of normal velocity, fast marching method and fast sweeping method for Hamilton-Jacobi equations.

Lecture 7-11: Applications to multiphase problems: two phase fluids problem, elliptic problems with jump conditions at the interface, solving PDEs on moving interfaces.

Lecture 12-14 (16): Application to image processing and computer vision: Geometric PDEs for denoising and segmentation images, reconstruction and dynamic operations of implicit surfaces.

Suggested reading

Books:

The Level Set Method:

Moving Interface and Free Boundary:

Image Processing and Computer Graphics:

Fast Sweeping Method:

Miscellaneous: