Elisabeth Larsson
Division of Scientific Computing
Department of Information Technology
Uppsala University
The use of radial basis function (RBF) approaches is motivated by the possibility of achieving high (spectral) accuracy with low complexity algorithms, for arbitrary choices of node locations, possibly in multi-dimensional spaces. An application area where this is of interest is multi-asset option pricing, involving the solution of the multi-dimensional Black-Scholes PDE. A common way of applying RBF methods to PDE problems is to choose an approximation space with a fixed RBF shape parameter and then use collocation to find the approximate solution. However, a typical situation when solving parabolic PDEs is that the initial data contains high frequencies or is of low regularity, whereas the solution for positive times is infinitely smooth and dominated by low frequencies. This indicates that an approximation space that works well for the initial data is bad for the final solution and vice versa. In this talk we show that by combining RBFs of different scales in a multi-level scheme and using least-squares approximation instead of collocation, we can derive a method that works well for the whole time-interval. Furthermore, for a given error tolerance at a fixed time, the proposed method consumes less time and memory than the usual collocation approach.