Random weights, robust lattice rules and the geometry of the cbc$r$c algorithm

In this paper we study lattice rules which are cubature formulae to approximate integrands over the unit cube $[0,1]^s$ from a weighted reproducing kernel Hilbert space. We assume that the weights are independent random variables with a given mean and variance for two reasons stemming from practical applications: (i) It is usually not known in practice how to choose the weights. Thus by assuming that the weights are random variables, we obtain robust constructions (with respect to the weights) of lattice rules. This, to some extend, removes the necessity to carefully choose the weights. (ii) In practice it is convenient to use the same lattice rule for many different integrands. The best choice of weights for each integrand may vary to some degree, hence considering the weights random variables does justice to how lattice rules are used in applications. We also study a generalized version which uses $r$ constraints which we call the cbc$r$c (component-by-component with $r$ constraints) algorithm. We show that lattice rules generated by the cbc$r$c algorithm simultaneously work well for all weights in a subspace spanned by the chosen weights $\boldsymbolγ^{(1)},..., \boldsymbolγ^{(r)}$. Thus, in applications, instead of finding one set of weights, it is enough to find an $r$ dimensional convex polytope in which the optimal weights lie. The price for this method is a factor $r$ in the upper bound on the error and in the construction cost of the lattice rule. Thus the burden of determining one set of weights very precisely can be shifted to the construction of good lattice rules.