Reduced basis methods for parabolic PDEs with parameter functions in high dimensions and applications in finance
Mayerhofer, Antonia Christine
FacultiesFakultät für Mathematik und Wirtschaftswissenschaften
InstitutionsInstitut für Finanzmathematik
Institut für Numerische Mathematik
The present thesis deals with variants of the space-time reduced basis method for parametrised parabolic partial differential equations. The reduced basis method is a well-known projection based model reduction technique for parameter dependent problems. We consider parabolic partial differential equations in space-time variational formulation. The formulation includes integration over space and time and a model reduction is achieved in both, space and time dimension. One objective of this thesis is the development of a reduced basis method to handle functions as parameters. A parameter in the reduced basis method is usually a vector of real numbers that describes the underlying model properties. The more general concept of a parameter space that is a subspace of a Hilbert space requires new reduced basis generation processes. In particular, the initial value of a parabolic partial differential equation is considered as a function parameter. A Two-Step Greedy Method is presented where two reduced bases are constructed, one for the approximation of the initial value and one for the evolution of the solution. The decomposition into two steps allows for a better approximation error control. A-priori as well as a-posteriori error bounds are provided. As the space-time reduced basis method deals with time as additional dimension, the reduced basis generation process is computationally expensive. We apply the H-Tucker low rank tensor format in the reduced basis offline phase to reduce the computational costs. The high dimensional linear system is decomposed in its tensor product components and can efficiently be solved applying the low rank tensor method. The resulting solution of the high dimensional problem implies an additional approximation error that has to be considered in the basis generation process. Error analysis is provided to combine both model reduction methods. For derivative pricing in finance, one needs to calculate the conditional expectation of the discounted payoff under a risk neutral measure. To this end one can equivalently solve the associated parabolic partial differential equation in diffusion based models. If we want to apply the reduced basis method as model reduction scheme, we usually have to construct a new reduced basis for each payoff function. To avoid this effort, the reduced basis method for parameter functions can be applied. Two standard option pricing models are considered, the Black-Scholes model and the Heston model. Both models provide a closed form solution such that the numerical results can easily be verified.
Graduiertenkolleg 110 "Modellierung, Analyse und Simulation in der Wirtschaftsmathematik"
Subject HeadingsParabolische Differentialgleichung [GND]
Differential equations, parabolic [LCSH]