We present sharp bounds on the minimal errors of linear estimators for multivariate integration and L 2 -approximation. This is done for a random field whose covariance kernel is a tensor product of one dimensional kernels that satisfy the Sacks-Ylvisaker regularity conditions. 1. Introduction We study multivariate integration and L 2 -approximation for random fields Y which are defined on the d dimensional unit cube, D = [0; 1] d , and which have mean zero and known covariance kernel K. We assume that K is at least continuous, and hence we may assume that Y is a measurable random field whose realizations are in L 2 (D) with probability one. For integration we want to estimate the integral R D Y (t) dt, whereas for L 2 -approximation we want to estimate the values Y (t) for all t and we study the distance of the estimate and the realization of the field in the space L 2 (D). For both problems we mainly consider linear estimators that use n observations of the random field. These e...

. We study linear estimators for the weighted integral of a stochastic process. The process may only be observed on a finite sampling design. The error is defined in mean square sense, and the process is assumed to satisfy Sacks-Ylvisaker regularity conditions of order r 2 N 0 . We show that sampling at the quantiles of a particular density already yields asymptotically optimal estimators. Hereby we extend results by Sacks and Ylvisaker for regularity r = 0 or 1, and we confirm a conjecture by Eubank, Smith, and Smith. 1. Introduction Let X(t), t 2 [0; 1], be a centered stochastic process which is at least continuous in quadratic mean. For a known function ae 2 L 2 ([0; 1]) we want to estimate the weighted integral Int ae (X) = Z 1 0 X(t) \Delta ae(t) dt: We consider linear estimators I n which are based on n observations of X. Hence I n (X) = n X i=1 X(t i ) \Delta a i with sampling points 0 t 1 ! \Delta \Delta \Delta ! t n 1 and coefficients a i 2 R. The error of I n is de...

We study weighted approximation and integration of Gaussian stochastic processes X defined over R+ whose rth derivatives satisfy a Holder condition with exponent fi in the quadratic mean. We assume that the algorithms use samples of X at a finite number of points. We study the average case (information) complexity, i.e., the minimal number of samples that are sufficient to approximate/integrate X with the expected error not exceeding ". We provide sufficient conditions in terms of the weight and the parameters r and fi for the weighted approximation and weighted integration problems to have finite complexity. For approximation, these conditions are necessary as well. We also provide sufficient conditions for these complexities to be proportional to the complexities of the corresponding problems defined over [0; 1], i.e., proportional to " \Gamma1=ff where ff = r + fi for the approximation and ff = r + fi +1=2 for the integration.

Introduction We study probabilistic linear (n; ffi)-widths and average linear n-widths for L1 -approximation of functions that are distributed according to the r-fold Wiener measure. As the classical n- widths, see e.g., [9], probabilistic and average widths quantify the error of best approximating operators. However, in the classical approach, the errors are defined by their worst case with respect to a given class (typically a unit ball of the underlying space). In the probabilistic approach, the errors are defined by the worst case performance on a subset of measure at least 1 \Gamma ffi, and in the average case approach, they are defined by their expectations, both with respect to a given probability measure. The study of probabilistic and average widths has been suggested only recently, see, e.g., [8, 13] and relatively few results have be

Abstract. We study the average case complexity of a linear multivariate problem (LMP) defined on functions of d variables. We consider two classes of information. The first Λ std consists of function values and the second Λ all of all continuous linear functionals. Tractability of LMP means that the average case complexity is O((1/ε) p) with p independent of d. We prove that tractability of an LMP in Λ std is equivalent to tractability in Λ all, although the proof is not constructive. We provide a simple condition to check tractability in Λ all. We also address the optimal design problem for an LMP by using a relation to the worst case setting. We find the order of the average case complexity and optimal sample points for multivariate function approximation. The theoretical results are illustrated for the folded Wiener sheet measure. 1.