Model Order Reduction in Contour Integral Methods for Parametric PDEs

In this paper we discuss a projection model order reduction method for a class of parametric linear evolution PDEs, which is based on the application of the Laplace transform. The main advantage of this approach is that the Laplace transform allows us to compute the solution directly at a given instant. This can be done efficiently by approximating the contour integral associated to the inverse Laplace transform by a suitable quadrature formula. This feature turns out to be very important in terms of classical model order reduction methodologies. In fact, differently from time stepping integrators (like Runge–Kutta methods), the use of the Laplace transform determines a significant reduction of the size of the reduced space. In this article we propose two reduction methods; the first one constructs a unique reduced space, based on the vectors representing the Laplace transform at the chosen quadrature points for different values of the parameters, and the second one instead constructs several reduced spaces, one at each quadrature node. This last approach is particularly well suited to a parallel computation approach, allowing a faster computation. We show the effectiveness of the method by some illustrative parabolic PDEs arising from finance.