Numerical schemes for Hamilton-Jacobi equations (Reference ESR5) ***this position has been filled***

• Title: Numerical schemes for Hamilton-Jacobi equations
• Location: Università degli Studi di Roma - La Sapienza -

• Secondment: Université Pierre & Marie Curie, Paris 6

• Contact: Maurizio Falcone
• Reference: ESR5

• Deadline for application:   May 15, 2011
• Key Words: Numerical analysis, scientific computing, adaptatif schemes

Description of the subject: A major difficulty in solving Hamilton-Jacobi equations is the high computational cost involved, when the state dimension is large. Hamilton-Jacobi approaches are seldom used in the practical applications, even though this method has the great advantage of providing global optima. When the value function is continuous, several numerical schemes are available for the solution of Hamilton-Jacobi equations, including Semi-Lagrangian, Markov Chain, ENO and WENO schemes. These schemes are known to be inefficient for the approximation of discontinuous solutions, however; there are only a few proposed techniques for dealing with numerical schemes in this case. The aim of the project area is to bring together four teams in the network (Rome, UBT, UPMC and Inria) to share their expertise concerning the numerical solution of Hamilton-Jacobi equations. The teams will develop algorithms, and supporting analysis, for problems where the solution is possibly discontinuous. They will also synthesize the latest advances in scientific calculus, to improving existing numerical methods for the continuous case.

• Semi-Lagrangian methods (SL) / Fast Marching Methods (FMM)

SL schemes are based on the coupling between an integration method for ordinary differential equations and an interpolation technique which makes it possible to compute the value at the foot of the up-wind characteristic. In order to deal with high-order methods and discontinuous solutions, we plan to use advanced interpolation techniques (ENO, WENO, ...) balancing the relative errors in the above two steps of the discretization. Also, an analysis of time-space adaptive schemes for first and second order Hamilton-Jacobi equations and the corresponding a-posteriori estimates will help to reduce the number of unknowns in the discrete problem, while retaining adequate accuracy of approximation.

•   Antidiffusive methods.

Classical monotone schemes have convergence properties but their drawback is that they are diffusive. They are therefore not suitable for computing discontinuous solutions or for application to problems involving long-time evolutions. Recently, some antidiffusive methods have been introduced, using ideas coming from schemes for conservation laws. Very sharp antidiffusive behavior has been established numerically. A rigorous convergence proof is available, however, only in very particular cases. These methods are still in an early stage of development and carrying out a general and rigorous analysis is a challenging research area.

• Requirements:  MSc degree in mathematics or a related subject, numerical analysis
  

Applications including a full curriculum vitae, and the names and addresses of at least two referees should be sent to Contacts.