Intersection numbers with Witten's top Chern class

Witten’s top Chern class is a particular cohomology class on the moduli space of Riemann surfaces endowed with r-spin structures. It plays a key role in Witten’s conjecture relating to the intersection theory on these moduli spaces.
Our first goal is to compute the integral of Witten’s class over the so-called double ramification cycles in genus 1. We obtain a simple closed formula for these integrals.
This allows us, using the methods of the first author [Int. Math. Res. Not. 38 (2003) 2051-2094], to find an algorithm for computing the intersection numbers of the Witten class with powers of the ψ-classes over any moduli space of r-spin structures, in short, all numbers involved in Witten’s conjecture.