Infinite-dimensional integration on weighted Hilbert spaces (Q2840006)

The author studies the numerical integration problem for functions defined over the infinite-dimensional unit cube \([0,1]^{N}\). The functions belong to a reproducing kernel Hilbert space \({\mathcal H}_{\gamma}\). Its kernel is built up from weighted sums of products of the \(1\)-dimensional reproducing kernel \(K(x,y)=\min\{ x,y\}\). The role of the weights \(\gamma\) is to moderate the importance of different groups of variables. In fact, the infinite-dimensional integration problem consists of infinitely many finite-dimensional integration sub-problems of varying importance, and the importance of each sub-problem is proportional to the corresponding weight. The finite-dimensional integration sub-problems are intimately related to \(L_{2}\)-star discrepancy.NEWLINENEWLINEThe author studies the worst case \(\epsilon\)-complexity which is defined as the minimal cost among all algorithms whose worst case error over the Hilbert space unit ball is at most \(\epsilon\). The infinite-dimensional integration problem is said to be (polynomially) tractable if the \(\epsilon\)-complexity is bounded by a constant times a power of \(1/\epsilon\). The smallest such power is called the exponent of tractability.NEWLINENEWLINEThe author provides improved lower bounds for the exponent of tractability for general finite-order weights and, with the help of multilevel algorithms, improves upper bounds for three newly defined classes of finite-order weights. The newly defined finite-intersection weights model the situation where each group of variables interacts with at most \(\rho\) other groups of variables, \(\rho\) some fixed number. For these weights sharp upper bounds for any decay of the weights and any polynomial degree of the cost function are obtained. For the other two classes of finite-order weights these upper bounds are sharp if, e.g., the decay of the weights is fast or slow enough.