Preliminary list of abstracts

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A non linear approximation method for solving high dimensional partial differential equations: Application in Finance.
Keywords: Greedy algorithms, high-dimensional partial differential equations
Thu, 09:50--10:15
  • Infante Acevedo, Jos√© Arturo (CERMICS, Universit√© Paris-Est, France)
  • Lelièvre, Tony (CERMICS, Universit√© Paris Est, Ecole des Ponts - Paristech, FRANCE)

We study an algorithm which has been proposed in [1,4] and analyzed in [2,3] to solve high dimensional partial differential equations. The idea is to represent the solution as a sum of tensor products and to compute iteratively the terms of this sum. This algorithm is very much related to so called greedy algorithms, see [5]. Convergence results of this approach obtained in [2,3] will be presented.

Besides, we will also show the application of this non linear approximation method to the option pricing problem, an important subject of the mathematical finance domain. This leads us to consider two extensions of the standard algorithm, which applies to symmetric linear partial differential equations: (i) nonsymmetric linear problems to value European options, (ii) nonlinear variational problems to price American options. We will present theoretical and numerical difficulties arising in this context.

[1] A. Ammar, B. Mokdad, F. Chinesta, and R. Keunings. A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. J. Non-Newtonian Fluid Mech., 139:153–176, 2006.
[2] E. Cancès, V. Ehrlacher, and T. Lelièvre. Convergence of a greedy algorithm for high-dimensional convex nonlinear problems, 2010. To appear in Mathematical Models and Methods in Applied Sciences.
[3] C. Le Bris, T. Lelièvre, and Y. Maday. Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations. Constructive Approximation, 30(3):621–651, 2009.
[4] A. Nouy. A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations. Comput. Methods Appl. Mech. Engrg., 196:4521–4537, 2007.
[5] V.N. Temlyakov. Greedy approximation. Acta Numerica, 17:235–409, 2008.