A fast probabilistic component-by-component construction of exactly integrating rank-1 lattices and applications
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Several more and more efficient component–by–component (CBC) constructions for suitable rank-1 lattices were developed during the last decades. On the one hand, there exist constructions that are based on minimizing some error functional. On the other hand, there is the possibility to construct rank-1 lattices whose corresponding cubature rule exactly integrates all elements within a space of multivariate trigonometric polynomials. In this paper, we focus on the second approach, i.e., the exactness of rank-1 lattice rules. The main contribution is the analysis of a probabilistic version of an already known algorithm that realizes a CBC construction of such rank-1 lattices. It turns out that the computational effort of the known deterministic algorithm can be considerably improved in average by means of a simple randomization. Moreover, we give a detailed analysis of the computational costs with respect to a certain failure probability, which then leads to the development of a probabilistic CBC algorithm. In particular, the presented approach will be highly beneficial for the construction of so-called reconstructing rank-1 lattices, that are practically relevant for function approximation. Subsequent to the rigorous analysis of the presented CBC algorithms, we present an algorithm that determines reconstructing rank-1 lattices of reasonable lattice sizes with high probability. We provide estimates on the resulting lattice sizes and bounds on the occurring failure probability. Furthermore, we discuss the computational complexity of the presented algorithm. Various numerical tests illustrate the efficiency of the presented algorithms. Among others, we demonstrate how to exploit the efficiency of our algorithm even for the construction of exactly integrating rank-1 lattices, provided that a certain property of the treated space of trigonometric polynomials is known.
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