On the adaptive tensor product wavelet Galerkin method with applications in finance

Abstract

The adaptive tensor product wavelet Galerkin method is a well-known method for solving linear well-posed operator equations on product domains. For a wide class of operators this method can be shown to be of optimal computational complexity and to converge at the best possible rate w.r.t. the underlying tensor product wavelet basis. We focus on new domain types to which this method can be applied and on the improvement of quantitative aspects. As a first part, the adaptive tensor product wavelet Galerkin method is extended to linear operator problems on unbounded domains. In particular, it is shown that the method maintains its optimality w.r.t. computational complexity and convergence rate. As a second part, we present a fast algorithm for the application of finite dimensional system matrices arising from the tensor product wavelet discretization of the underlying linear operator. This algorithm is shown to be of linear complexity for a large class of local operators and, in particular, does not require compressions of the system matrix. Furthermore, a new efficient way of approximate residual evaluation within the adaptive tensor product wavelet Galerkin method is developed. The corresponding algorithm for the residual approximation is based on the aforementioned fast evaluation of system matrices and leads to a significant improvement of the performance of the adaptive tensor product wavelet Galerkin method. Theoretical findings are underlined by various numerical experiments. Moreover, the application of the adaptive tensor product wavelet Galerkin method to partial integrodifferential equations arising in Lévy models in finance is discussed using selected examples.