Infinite-Dimensional Quadrature and Approximation of Distributions

Abstract

We study numerical integration of Lipschitz functionals on a Banach space by means of deterministic and randomized (Monte Carlo) algorithms. This quadrature problem is shown to be closely related to the problem of quantization and to the average Kolmogorov widths of the underlying probability measure. In addition to the general setting, we analyze, in particular, integration with respect to Gaussian measures and distributions of diffusion processes. We derive lower bounds for the worst case error of every algorithm in terms of its cost, and we present matching upper bounds, up to logarithms, and corresponding almost optimal algorithms. As Communicated by Arieh Iserles. 392 Found Comput Math (2009) 9: 391-429 auxiliary results, we determine the asymptotic behavior of quantization numbers and Kolmogorov widths for diffusion processes.

High resolution quantization and entropy coding of jump processes

“…Moreover, quadrature problems are linked to the quantization problem via estimates involving both quantities. Recent results in that direction can be found in [7] (see also [20] for earlier results).…”

Liberating the dimension

“…Since K (x j , y j ) ∈ [0, 1], we note that K γ is well defined due to (1). The reproducing kernel Hilbert space H γ :…”

“…1), and let λ satisfy (12). The FD algorithm Q FD given by (6), with d and n given by (9) and (14), and with the sample points t (i) being randomly shifted rank-1 lattice points with the generating vector obtained from the CBC algorithm, has error at most ε and cost bounded by…”

“…Other studies of tractability of the integration problem for functions with an infinite number of variables have been done in, e.g., [1,8,17,21,26], with the first and the third considering varying cost of function evaluations. More precisely, in [1] sample points are functions from increasing subspaces H j ⊂ H j+1 , j = 1, 2, .…”

“…More precisely, in [1] sample points are functions from increasing subspaces H j ⊂ H j+1 , j = 1, 2, . .…”