. We propose isotropic probability measures defined on classes of smooth multivariate functions. These provide a natural extension of the classical isotropic Wiener measure to multivariate functions from C 2r . We show that in the corresponding average case setting the minimal errors of algorithms that use n function values are \\Theta \\Gamma n \\Gamma(d+4r+1)=2d \\Delta and \\Theta \\Gamma n \\Gamma(4r+1)=2d \\Delta for the integration and L 2 -approximation problems respectively. Here d is the number of variables of the corresponding class of functions. This means that the minimal average errors depend essentially on the number d of variables. In particular, for d large relative to r, the L 2 -approximation problem is intractable. The integration and L 2 -approximation problems have been recently studied with measures whose covariance kernels are tensor products. The results for these measures and for isotropic measures differ significantly. 1 Introduction We study the integratio...
We study integration and L2-approximation of functions of infinitely many variables in the following setting: The underlying function space is the countably infinite tensor product of univariate Hermite spaces and the probability measure is the corresponding product of the standard normal distribution. The maximal domain of the functions from this tensor product space is necessarily a proper subset of the sequence space RN. We establish upper and lower bounds for the minimal worst case errors under general assumptions; these bounds do match for tensor products of well-studied Hermite spaces of functions with finite or with infinite smoothness. In the proofs we employ embedding results, and the upper bounds are attained constructively with the help of multivariate decomposition methods.
We study a Monte Carlo algorithm that is based on a specific (randomly shifted and dilated) lattice point set. The main result of this paper is that the mean squared error for a given compactly supported, square-integrable function is bounded by
n−1/2 times the
L2-norm of the Fourier transform outside a region around the origin, where
n is the expected number of function evaluations. As corollaries we obtain the order of convergence for the Sobolev spaces
Hps with isotropic, anisotropic or mixed smoothness for all values of the parameters. This proves, in particular, that the optimal order of convergence in the latter case is
n−s−1/2 for
p≥2, which is, in contrast to the case of deterministic algorithms, independent of the dimension. This shows that Monte Carlo algorithms can improve the order by more than
n−1/2 for a whole class of practically important function classes. All results carry over to functions defined on the unit cube without boundary conditions.
We study lower bounds on the worst-case error of numerical integration in tensor product spaces. As reference we use the
N-th minimal error of linear rules that use
N function values. The information complexity is the minimal number
N of function evaluations that is necessary such that the
N-th minimal error is less than a factor
ε times the initial error. We are interested to which extent the information complexity depends on the number
d of variables of the integrands. If the information complexity grows exponentially fast in
d, then the integration problem is said to suffer from the curse of dimensionality. Under the assumption of the existence of a worst-case function for the uni-variate problem we present two methods for providing good lower bounds on the information complexity. The first method is based on a suitable decomposition of the worst-case function. This method can be seen as a generalization of the method of decomposable reproducing kernels, that is often successfully applied when integration in Hilbert spaces with a reproducing kernel is studied. The second method, although only applicable for positive quadrature rules, has the advantage, that it does not require a suitable decomposition of the worst-case function. Rather, it is based on a spline approximation of the worst-case function and can be used for analytic functions. The methods presented can be applied to problems beyond the Hilbert space setting. For demonstration purposes we apply them to several examples, notably to uniform integration over the unit-cube, weighted integration over the whole space, and integration of infinitely smooth functions over the cube. Some of these results have interesting consequences in discrepancy theory.
This paper reformulates the problems of approximation of imbedding and integration from anisotropic Sobolev classes B(Wrp([0, 1]d)) in the deterministic, randomized and average case settings, and obtains the exact orders of the n-th minimal error of these problems in all three settings. The results show that in the case of B(Wrp([0, 1]d)) being not imbedded into the space of continuous functions C([0, 1]d), the randomized and average case error are essential smaller than the deterministic ones. Quantitatively, this maximal gain for imbedding problem amounts to the factor n-1+ε for any ε > 0, and for integration problem the gain is up to factor n-1 which is also the maximal speedup over the deterministic algorithms observed so far in natural numerical problems.