Monte Carlo Complexity of Initial Value Problems and Indefinite Integration

Beschreibung

This thesis consists of two major parts. In the first one we study the complexity of randomized solution of initial value problems for systems of ordinary differential equations. We assume the input functions to be γ-smooth, where γ = r + ρ, which means that the r-th derivatives satisfy a ρ-Hölder condition. In the second major part we study the problem of indefinite integration for functions f Lp([0, 1]d). We show that for 1 p, the family of integrals [0,x] f(t)dt (x = (x1,.,xd) [0,1]d) 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 optimal order, the other up to logarithmic factors. We also prove lower bounds and discuss the dependence of the constants in the error estimates on the dimension.
This thesis consists of two major parts. In the first one we study the complexity of randomized solution of initial value problems for systems of ordinary differential equations. We assume the input functions to be γ-smooth, where γ = r + ρ, which means that the r-th derivatives satisfy a ρ-Hölder condition. In the second major part we study the problem of indefinite integration for functions f Lp([0, 1]d). We show that for 1 p, the family of integrals [0,x] f(t)dt (x = (x1,.,xd) [0,1]d) 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 optimal order, the other up to logarithmic factors. We also prove lower bounds and discuss the dependence of the constants in the error estimates on the dimension.

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Bernhard Milla

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