Ronald Cools

Monomial cubature rules since “Stroud”: a compilation

A bibliography of references to cubature rules which have appeared since the publication of Strouds book (1971) is presented. The standard regions that are treated in this paper are the n-cube, the n-simplex, the n-sphere and the entire space.

An encyclopaedia of cubature formulas

About 13 years ago we started collecting published cubature formulas for the approximation of multivariate integrals over some standard regions. In this paper we describe how we make this information available to a larger audience via the World Wide Web.

Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces

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(snlog(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.

Homotopies exploiting Newton polytopes for solving sparse polynomial systems

This paper is concerned with the problem of finding all isolated solutions of a polynomial system. The BKK bound, defined as the mixed volume of the Newton polytopes of the polynomials in the system, is a sharp upper bound for the number of isolated solutions in

Template-Based Continuous Speech Recognition

\mathbb{C}_0^n ,\mathbb{C}_0 = \mathbb{C} \backslash \{ 0\}

Constructing cubature formulae: the science behind the art

, of a polynomial system with a sparse monomial structure. First an algorithm is described for computing the BKK bound. Following the lines of Bernshtei˘n’s proof, the algorithmic construction of the cheater’s homotopy or the coefficient homotopy is obtained. The mixed homotopy methods can be combined with the random product start systems based on a generalized Bezout number. Applications illustrate the effectiveness of the new approach.

Monomial cubature rules since “Stroud”: a compilation—part 2

Despite their known weaknesses, hidden Markov models (HMMs) have been the dominant technique for acoustic modeling in speech recognition for over two decades. Still, the advances in the HMM framework have not solved its key problems: it discards information about time dependencies and is prone to overgeneralization. In this paper, we attempt to overcome these problems by relying on straightforward template matching. The basis for the recognizer is the well-known DTW algorithm. However, classical DTW continuous speech recognition results in an explosion of the search space. The traditional top-down search is therefore complemented with a data-driven selection of candidates for DTW alignment. We also extend the DTW framework with a flexible subword unit mechanism and a class sensitive distance measure-two components suggested by state-of-the-art HMM systems. The added flexibility of the unit selection in the template-based framework leads to new approaches to speaker and environment adaptation. The template matching system reaches a performance somewhat worse than the best published HMM results for the Resource Management benchmark, but thanks to complementarity of errors between the HMM and DTW systems, the combination of both leads to a decrease in word error rate with 17% compared to the HMM results

Constructing Embedded Lattice Rules for Multivariate Integration

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.

Fast component-by-component construction of rank-1 lattice rules with a non-prime number of points

More than 25 years ago, Stroud published his encyclopedic work on multiple numerical integration, Approximate Calculation of Multiple Integrals [98]. This book contains a listing of almost all multiple integration or cubature rules for a variety of regions known at that time. About 5 years ago, Monomial cubature rules since “Stroud ”: a compilation [104] was published. This paper was concerned with continuing the work of Stroud in one speci c area, namely the compilation of all the so-called monomial cubature rules which have appeared since the publication of [98] for most of the regions contained there plus some cubature rules which appeared earlier but were not included in [98] for some reason. The study of multiple numerical integration has continued during the past years, although we notice a slow-down. Nevertheless, we believe that this is the right time to come out with a small list of errata and addenda. We also use the occasion to present some cubature rules that are of theoretical importance but are not well known. For background material, a description of the material included and the tables we refer to Cools and Rabinowitz [104]. References given in [104] are not repeated in this paper. So for references from [1] to [101], the reader must look at [104]. The numbering of the new references starts at [102]. Readers with access to Russian literature were referred to the book by Mysovskikh [81]. Indeed, as a rule, we ignored the Russian literature except when it appeared in translation. Meanwhile a

Advances in multidimensional integration

Lattice rules are a family of equal-weight cubature formulae for approximating high-dimensional integrals. By now it is well established that good generating vectors for lattice rules having