DE LUIGI , Christophe

We describe an adaptive algorithm to compute sparse polynomial approximations and the integral of a multivariate function over hyper-rectangular regions in medium dimensions. Numerical examples are given on functions taken from the Genz package and on basket options pricing in dimension up to 5 and on basket option pricing.

We improve an adaptive integration algorithm proposed by two of the authors by introducing a new splitting strategy based on a geometrical criterion. This algorithm is tested especially on the pricing of multidimensional vanilla options in the Black–Scholes framework which emphasizes the numerical problems of integrating non-smooth functions....

We describe how to use new reduced size polynomial approximations for the numerical solution of the Poisson equation over hypercubes. Our method is based on a non-standard Galerkin method which allows test functions which do not verify the boundary conditions. Numerical examples are given in dimensions up to 8.