## 62D05 Sampling theory, sample surveys

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The aim of this thesis is to deepen our understand of how
IDA* heuristics influence the number of nodes expanded during
search. To this end, we develop Korf's formula for the number
of expanded nodes into a heuristic quality η which
expresses the quality of a heuristic function as a constant factor
on the number of expanded nodes, independent of a particular problem
instance.
We proceed to show how to compute η for some common kinds of
heuristics and how to estimate η by means of a random sample for
arbitrary heuristics. Using the value of η for some concrete
examples, we then inspect for which parts of the search space the
values of h(v) are particularly critical to the performance of the
heuristic, allowing us to build better heuristics for future problems.
This report originally appeared as a master thesis at Humboldt
University of Berlin.

Wigner transformation provides a one-to-one correspondence between functions on position space (wave functions) and functions on phase space (Wigner functions). Weighted integrals of Wigner functions yield quadratic quantities of wave functions like position and momentum densities or expectation values. For molecular quantum systems, suitably modified classical transport of Wigner functions provides an asymptotic approximation of the dynamics in the high energy regime. The article addresses the computation of Wigner functions by Monte Carlo quadrature. An ad aption of the Metropolis algorithm for the approximation of signed measures with disconnected support is systematically tested in combination with a surface hopping algorithm for non-adiabatic quantum dynamics. The numerical experiments give expectation values and level populations with an error of two to three percent, which agrees with the theoretically expected accuracy.