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- Stefan Heinrich, Bernhard Milla
- J. Complexity
- 2008

We study the complexity of randomized solution of initial value problems for systems of ordinary differential equations (ODE). The input data are assumed to be γ-smooth (γ = r+% : the r-th derivatives satisfy a %-Hölder condition). Recently, the following almost sharp estimate of the order of the n-th minimal error was given by Kacewicz (Almost optimal… (More)

- Stefan Heinrich, Bernhard Milla
- J. Complexity
- 2011

We show that for functions f ∈ Lp([0, 1] d), where 1 ≤ p ≤ ∞, the family of integrals ∫ [0,x] f(t)dt (x = (x1, . . . , xd) ∈ [0, 1] ) can be approximated by a randomized algorithm uniformly over x ∈ [0, 1]d with the same rate n−1+1/min(p,2) as the optimal rate for a single integral, where n is the number of samples. We present two algorithms, one being of… (More)

- Bernhard Milla
- 2011

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