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

Abstract

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-calledSmolyak 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.

Nontensorial Clenshaw–Curtis cubature

“…Even in the case of tensor-product domains, where tensor-product orthogonal expansions can be used (see, e.g., [24,42]), hyperinterpolation is intrinsically nontensorial and thus generates nontensorial cubature formulas. As known, there are also other ways of constructing useful nontensorial cubature formulas, like the so-called sparse grids introduced by Smolyak in the '60s (cf., e.g., [40,7,28,30,31] and references therein). In the present bidimensional context numerical tests and comparisons with available implementations (like, e.g., [8]), have shown that nontensorial formulas generated via (hyper)interpolation seem more effective.…”

Cubature formulas for symmetric measures in higher dimensions with few points

“…The n i are defined as follows: n 1 = 1, n 2 = n 3 = 3, n 4 = n 5 = n 6 = 7, n 7 = · · · = n 12 = 15, n 13 = · · · = n 24 = 31 and so on. Some of these numbers are larger than 2i − 1 and hence we can modify those n i , used by Petras (2003), tõ…”

“…We do not claim that the results of Table 3 are optimal for fully symmetric (or Smolyak) rules. It was proved by Petras (2003), however, that only minor improvements are possible if one uses Smolyak formulas. The same also holds for the more general fully symmetric formulas.…”

“…There are many other papers on the Smolyak method. The papers Gerstner and Griebel (1998), , and Petras (2003) study the polynomial exactness of A (q, d). See also Novak, Ritter, Schmitt and Steinbauer (1999) and the recent survey on sparse grids by Bungartz and Griebel (2004).…”

Fast Construction of the Fejér and Clenshaw–Curtis Quadrature Rules

“…Bungartz and Griebel [1], K. Petras [10], [11]. Comprehensive lists of references on sparse grids and their applications are given in [1], [11].…”