No abstract

In Kuznetsov et al. [28] a new Monte Carlo simulation technique was introduced for a large family of Lévy processes that is based on the Wiener-Hopf decomposition. We pursue this idea further by combining their technique with the recently introduced multilevel Monte Carlo methodology. Moreover, we provide here for the first time a theoretical analysis of the new Monte Carlo simulation technique in [28] and of its multilevel variant for computing expectations of functions depending on the historical trajectory of a Lévy process. We derive rates of convergence for both methods and show that they are uniform with respect to the "jump activity" (e.g. characterised by the Blumenthal-Getoor index). We also present a modified version of the algorithm in Kuznetsov et al. [28] which combined with the multilevel methodology obtains the optimal rate of convergence for general Lévy processes and Lipschitz functionals. This final result is only a theoretical one at present, since it requires independent sampling from a triple of distributions which is currently only possible for a limited number of processes.

We develop algorithms for multivariate integration and approximation in the weighted half-period cosine space of smooth non-periodic functions. We use specially constructed tent-transformed rank-1 lattice points as cubature nodes for integration and as sampling points for approximation. For both integration and approximation, we study the connection between the worst-case errors of our algorithms in the cosine space and the worst-case errors of some related algorithms in the well-known weighted Korobov space of smooth periodic functions. By exploiting this connection, we are able to obtain constructive worst-case error bounds with good convergence rates for the cosine space.

Recent research has seen several forecasting methods being applied for heat load forecasting of district heating networks. This paper presents two methods that gain significant improvements compared to the previous works. First, an automated way of handling non-linear dependencies in linear models is presented. In this context, the paper implements a new method for feature selection based on [1], resulting in computationally efficient models with higher accuracies. The three main models used here are linear, ridge, and lasso regression. In the second approach, a deep learning method is presented. Although computationally more intensive, the deep learning model provides higher accuracy than the linear models with automated feature selection. Finally, we compare and contrast the proposed methods with earlier work for day-ahead forecasting of heat load in two different district heating networks.