Smolyak cubature of given polynomial degree with few nodes for increasing dimension

Summary. Some recent investigations (see e.g., Gerstner and Griebel [5], Novak and Ritter [9] and [10], Novak, Ritter and Steinbauer [11], Wasilkowski and Woźniakowski [18] or Petras [13]) show that the so-called Smolyak algorithm applied to a cubature problem on the d-dimensional cube seems to be particularly useful for smooth integrands. The problem is still that the numbers of nodes grow (polynomially but) fast for increasing dimensions. We therefore investigate how to obtain Smolyak cubature formulae with a given degree of polynomial exactness and the asymptotically minimal number of nodes for increasing dimension d and obtain their characterization for a subset of Smolyak formulae. Error bounds and numerical examples show their good behaviour for smooth integrands. A modification can be applied successfully to problems of mathematical finance as indicated by a further numerical example.