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:

• hyperbolic conservation laws
• Hamilton-Jacobi equations
• Navier-Stokes equations

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

• high-resolution non-oscillatory central schemes
• piecewise regularity and convergence rate estimates for nonlinear transport equations
• spectral viscosity methods
• critical threshold phenomena in Eulerian dynamics

Suggested reading

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

• E. Tadmor: Approximate solutions of nonlinear conservation laws. In "Advanced Numerical Approximations of Nonlinear Hyperbolic Equations Cetraro, Italy, 1997", (A. Quarteroni, ed.), Lecture Notes in Mathematics, Vol. 1697, Springer, 1998.

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

• High-resolution non-oscillatory central schemes
• Piecewise regularity and convergence rate estimates for nonlinear transport equations
• High-frequencies wave-dependent methods for time-dependent problems with large gradients
• Critical threshold phenomena in Eulerian dynamics

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:

• S. Osher and R. Fedkiw: Level Set Methods and Dynamic Implicit Surfaces, Springer Verlag. See also book listing at Amazon.
• James Sethian: Level set methods and fast marching methods, Cambridge, 1999.

The Level Set Method:

• A Variational Level Set Approach to Multiphase Motion, H.K. Zhao, T.F. Chan, B. Merriman, S.Osher. J. Comp. Phys. Vol. 127, 1996, pp 179-195. (ps file, pdf file)
• Capturing the Behavior of Bubbles and Drops Using the Variational Level Set Approach, H.K. Zhao, B. Merriman, S. Osher, L. Wang. J. Comp. Phys. Vol. 143, 1998, pp 495-518. (ps file, pdf file)
• A PDE Based Fast Local Level Set Method, D. Peng, B. Merriman, S. Osher, H.K. Zhao, M. Kang. J. Comp. Phys. Vol. 155, 1999, pp 410-438. (ps file, pdf file)
• An Eulerian Formulation for Solving Partial Differential Equations Along a Moving Interface, J. Xu, H.K. Zhao, to appear in J. Sci. Comp. (ps file, pdf file)

Moving Interface and Free Boundary:

• A Hybrid Method for Moving Interface Problems with Application to the Hele-Shaw Flow, T.Y. Hou, Z.L. Li, S. Osher, H.K. Zhao. J. Comp. Phys. Vol. 134, 1997, pp 236-252. (ps file, pdf file)
• A Numerical Study of Electro-migration Voiding by Evolving Level Set Functions on a Fixed Cartesian Grid, Z. Li, H.K. Zhao, H. Gao. J. Comp. Phys. Vol. 152, 1999, pp 281-304. (ps file, pdf file)
• Reactive Autophobic Spreading of Drops, J.K. Hunter, Z. Li and H.K. Zhao. J. Comp. Phys. Vol. 183, No. 2, pp. 335-366, 2002. (ps file,pdf file)

Image Processing and Computer Graphics:

• Visualization, Analysis and Shape Reconstruction of Unorganized Data Sets, H. Zhao and S. Osher, contributed chapter in S. Osher and N. Paragios, editors, Geometric Level Set Methods in Imaging, Vision and Graphics, Springer, 2002. (ps file, pdf file)
• Implicit and Non-parametric Shape Reconstruction from Unorganized Points Using Variational Level Set Method, H.K. Zhao, S. Osher, B. Merriman, M. Kang. Computer Vision and Image Understanding. Vol. 80, 2000, pp 295-319. (ps file, pdf file)
• Fast Surface Reconstruction Using the Level Set Method, H.K.Zhao, S. Osher, R. Fedkiw. Proceedings of IEEE Workshop on Variational and Level Set Methods in Computer Vision (VLSM 2001), July, 2001. (ps file, pdf file)
• Analysis and Visualization of Large Set of Unorganized Data Points Using the Distance Function, H.K. Zhao, preprint (ps file, pdf file)

Fast Sweeping Method:

• Fast Sweeping Method for Eikonal Equations I, H.K. Zhao, submitted to SIAM Numerical Analysis. (ps file, pdf file)
• Fast Sweeping Algorithms for a Class of Hamilton-Jacobi Equations,Y.R. Tsai, L.T. Cheng, S. Osher, H.K. Zhao, to appear in SINUM (ps file, pdf file).

Miscellaneous: 

• References on immersed interface method can be found at the homepage of Dr. Zhilin Li at http://www4.ncsu.edu/~zhilin/
• Coupling of the level set method with volume of fluid method and some level set codes for two phase flow can be found at the homepage of Dr. Mark Sussman at http://web.math.fsu.edu/~sussman/
• For applications in image processing go to website http://www.math.ucla.edu/~imagers/