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This article provides a strong law of large numbers for integration on digital nets randomized by a nested uniform scramble. The motivating problem is optimization over some variables of an integral over others, arising in Bayesian optimization. This ...

Algorithms that combat the curse of dimensionality take advantage of nonuniformity properties of the underlying functions, which may be rotational (e.g., grid alignment) or translational (e.g., near-singularities localized at certain points of the domain). ...

Subspace deformable body simulations can be very fast, but can behave unrealistically when behaviors outside the prescribed subspace such as novel external collisions, are encountered. We address this limitation by presenting a fast, flexible new method ...

We present an efficient new subspace method for simulating the self-contact of articulated deformable bodies, such as characters. Self-contact is highly structured in this setting, as the limited space of possible articulations produces a predictable set ...

We present a new subspace integration method that is capable of efficiently adding and subtracting dynamics from an existing high-resolution fluid simulation. We show how to analyze the results of an existing high-resolution simulation, discover an ...

Reasonable point set selection is of paramount importance to the accuracy of high-dimensional integrals that will be encountered in various disciplines. In the present paper, to improve the point selection and to overcome the computational complexity of ...

An optimization based method is proposed to generate moment matching scenarios for numerical integration and its use in stochastic programming. The main advantage of the method is its flexibility: it can generate scenarios matching any prescribed set of ...

A recent result in [B. T. Helenbrook, SIAM J. Numer. Anal., 47 (2009), pp. 1304-1318] on the nonexistence of Gauss-Lobatto cubature rules on the triangle is strengthened by establishing a lower bound for the number of nodes of such rules. A method of ...

We present a new algorithm for numerically computing quadrature formulas for arbitrary domains which exactly integrate a given polynomial space. An effective method for constructing quadrature formulas has been to numerically solve a nonlinear set of ...

This paper studies numerical integration (or cubature) over the unit sphere S² ⊂ R³ for functions in arbitrary Sobolev spaces H^s (S²), s > 1. We discuss sequences (Q_m(n))_n∈N of cubature rules, where (i) the rule Q_m(n) uses m(n) points and is assumed to ...

We show that the worst-case cubature error E(Q_m;H^s) of an m-point cubature rule Q_m for functions in the unit ball of the Sobolev space H^s = H^s (S²), s > 1, has the lower bound E(Q_m; H^s) ≥ c_sm^-s/2, where the constant c_s is independent of Q_m and m. This ...

Sequences of points with a low discrepancy are the basic building blocks for quasi-Monte Carlo methods. Traditionally these points are generated in a unit cube.To develop point sets on a simplex we will transform the low-discrepancy points from the unit ...

In this note cubature formulae of degree 5 are studied for n-dimensional integrals over the ball with constant weight function. We apply the method of reproducing kernel to show that the existence of such formulae attaining the best known lower bound is ...

We describe the results of a computer-based search for five and six-dimensional lattice rules of specified trigonometric degree. In this search only lattice rules that can be generated by a circulant or skew-circulant generator matrix are considered, ...

A globally adaptive algorithm for numerical cubature of a vector of functions over a collection of n-dimensional simplices is described. The algorithm is based on a subdivision strategy that chooses for subdivision at each stage the subregion (of the ...

CUBPACK aims to offer a collection of re-usable code for automatic n-dimensional (n ≥ 1) numerical integration of functions over a collection of regions, i.e., quadrature and cubature. The current version allows this region to consist of a union of n-...

r2d2lri is a non-adaptive algorithm implemented in C++ for performing automatic cubature over a wide variety of finite and non-finite two-dimensional domains. The core integrator uses a sixth-order Sidi transformation applied to a sequence of embedded ...