Abstract. We reformulate the original component-by-component algorithm for rank-1 lattices in a matrix-vector notation so as to highlight its structural properties. For function spaces similar to a weighted Korobov space, we derive a technique which has construction cost O(sn log(n)), in contrast with the original algorithm which has construction cost O(sn 2 ). Herein s is the number of dimensions and n the number of points (taken prime). In contrast to other approaches to speed up construction, our fast algorithm computes exactly the same quantity as the original algorithm. The presented algorithm can also be used to construct randomly shifted lattice rules in weighted Sobolev spaces.

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Lattice rules are a family of equal-weight cubature formulae for approximating highdimensional integrals. By now it is well established that good generating vectors for lattice rules having n points can be constructed component-by-component for integrands belonging to certain weighted function spaces, and that they can achieve the optimal rate of convergence. Although the lattice rules constructed this way are extensible in dimension, they are not extensible in n; thus when n is changed the generating vector needs to be constructed anew. In this paper we introduce a new algorithm for constructing good generating vectors for embedded lattice rules which can be used for a range of n while still being extensible in dimension. By using an adaptation of the fast component-by-component construction algorithm (which makes use of fast Fourier transforms), we are able to obtain good generating vectors for thousands of dimensions and millions of points, under both product weight and order-dependent weight settings, at the cost of O(dn(log(n)) 2 ) operations. With a sufficiently large number of points and good overall quality, these embedded lattice rules can be used for practical purposes in the same way as a low-discrepancy sequence. We show for a range of weight settings in an unanchored Sobolev space that our embedded lattice rules achieve the same (optimal) rate of convergence O(n −1+δ ), δ > 0, as those constructed for a fixed number of points, and that the implied constant gains only a factor of 1.30 to 1.55.

In this paper we present a general, theoretical foundation for the construction of cubature formulae to approximate multivariate integrals. The focus is on cubature formulae that are exact for certain vector spaces of polynomials. Our main quality criteria are the algebraic and trigonometric degrees. The constructions using ideal theory and invariant theory are outlined. The known lower bounds for the number of points are surveyed and characterizations of minimal cubature formulae are given. We include references to all known minimal cubature formulae. Finally, some methods to construct cubature formulae illustrate the previously introduced concepts and theorems.