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Multivariate problems occur in many applications. These problems are defined on spaces of d-variate functions and d can be huge – in the hundreds or even in the thousands. Some high-dimensional problems can be solved efficiently to within ε, i.e., the cost increases polynomially in ε −1 and d. However, there are many multivariate problems for which even the(More)
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 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(More)
We mainly study multivariate (uniform or Gaussian) integration defined for integrand spaces F d 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)
We study the optimal approximation of the solution of an operator equation A(u) = f by linear mappings of rank n and compare this with the best n-term approximation with respect to an optimal Riesz basis. We consider worst case errors, where f is an element of the unit ball of a Hilbert space. We apply our results to boundary value problems for elliptic(More)
We study the approximation problem for C ∞ functions f : [0, 1] d → R with respect to a W m p-norm. Here, m = [m, m,. .. , m], d times, with the norm of the target space defined in terms of up to m partial derivatives with respect to all d variables. The optimal order of convergence is infinite, hence excellent, but the problem is still intractable and(More)