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Journals and Conferences
We study polynomial interpolation on a d-dimensional cube, where d is large. We suggest to use the least solution at sparse grids with the extrema of the Chebyshev polynomials. The polynomial exactness of this method is almost optimal. Our error bounds show that the method is universal, i.e., almost optimal for many different function spaces. We report on… (More)
It is known that quantum computers yield a speed-up for certain discrete problems. Here we want to know whether quantum computers are useful for continuous problems. We study the computation of the integral of functions from the classical Hölder classes F k,α d on [0, 1] d and define γ by γ = (k + α)/d. The known optimal orders for the complexity of… (More)
We give a path integral prescription for the pair correlation function of Wilson loop observables on the worldvolume of a Dbrane in the bosonic string theory in flat spacetime. We determine the coefficient of the 1/R term in the static heavy quark potential from the static pair correlation function of Wilson lines at small spatial separation in the critical… (More)
We prove that L∞-approximation of C ∞-functions defined on [0, 1]d is intractable and suffers from the curse of dimensionality. This is done by showing that the minimal number of linear functionals needed to obtain an algorithm with worst case error at most ε ∈ (0, 1) is exponential in d. This holds despite the fact that the rate of convergence is infinite.
We mainly study multivariate (uniform or Gaussian) integration defined for integrand spaces Fd such as weighted Sobolev spaces of functions of d variables with smooth mixed derivatives. The weight #j moderates the behavior of functions with respect to the jth variable. For #j #1, we obtain the classical Sobolev spaces whereas for decreasing #j 's the… (More)
Many high dimensional problems are diicult to solve for any numerical method. This curse of dimension means that the computational cost must increase exponentially with the dimension of the problem. A high dimension , however, can be compensated by a high degree of smoothness. We study numerical integration and prove that such a compensation is possible by… (More)
We prove the curse of dimensionality for multivariate integration of Ck functions. The proofs are based on volume estimates for k = 1 together with smoothing by convolution. This allows us to obtain smooth fooling functions for k > 1. MSC: 65D30,65Y20,41A63,41A55
We consider the computation of the mean of sequences in the quantum model of computation. We determine the query complexity in the case of sequences which satisfy a p-summability condition for 1 ≤ p < 2. This settles a problem left open in Heinrich (2001).