Fred J. Hickernell

A generalized discrepancy and quadrature error bound

An error bound for multidimensional quadrature is derived that includes the Koksma-Hlawka inequality as a special case. This error bound takes the form of a product of two terms. One term, which depends only on the integrand, is defined as a generalized variation. The other term, which depends only on the quadrature rule, is defined as a generalized discrepancy. The generalized discrepancy is a figure of merit for quadrature rules and includes as special cases the L p -star discrepancy and P α that arises in the study of lattice rules.

Lattice Rules: How Well Do They Measure Up?

A simple, but often effective, way to approximate an integral over the s-dimensional unit cube is to take the average of the integrand over some set P of N points. Monte Carlo methods choose P randomly and typically obtain an error of 0(N-1/2). Quasi-Monte Carlo methods attempt to decrease the error by choosing P in a deterministic (or quasi-random) way so that the points are more uniformly spread over the integration domain.

Randomized Halton sequences

The Halton sequence is a well-known multi-dimensional low-discrepancy sequence. In this paper, we propose a new method for randomizing the Halton sequence. This randomization makes use of the description of Halton sequence using the von Neumann-Kakutani transformation. We randomize the starting point of the sequence. This method combines the potential accuracy advantage of Halton sequence in multi-dimensional integration with the practical error estimation advantage of Monte Carlo methods. Theoretically, using multiple randomized Halton sequences as a variance reduction technique we can obtain an efficiency improvement over standard Monte Carlo. Numerical results show that randomized Halton sequences have better performance not only than Monte Carlo, but also than randomly shifted Halton sequences. They have similar performance with the randomly digit-scrambled Halton sequences but require much less generating time.

Extensible Lattice Sequences for Quasi-Monte Carlo Quadrature

Integration lattices are one of the main types of low discrepancy sets used in quasi-Monte Carlo methods. However, they have the disadvantage of being of fixed size. This article describes the construction of an infinite sequence of points, the first bm of which forms a lattice for any nonnegative integer m. Thus, if the quadrature error using an initial lattice is too large, the lattice can be extended without discarding the original points. Generating vectors for extensible lattices are found by minimizing a loss function based on some measure of discrepancy or nonuniformity of the lattice. The spectral test used for finding pseudorandom number generators is one important example of such a discrepancy. The performance of the extensible lattices proposed here is compared to that of other methods for some practical quadrature problems.

Integration and Approximation in Arbitrary Dimensions

We study multivariate integration and approximation for various classes of functions of d variables with arbitrary d. We consider algorithms that use function evaluations as the information about the function. We are mainly interested in verifying when integration and approximation are tractable and strongly tractable. Tractability means that the minimal number of function evaluations needed to reduce the initial error by a factor of ɛ is bounded by C(d)ɛ−p for some exponent p independent of d and some function C(d). Strong tractability means that C(d) can be made independent of d. The ‐exponents of tractability and strong tractability are defined as the smallest powers of ɛ{-1} in these bounds.We prove that integration is strongly tractable for some weighted Korobov and Sobolev spaces as well as for the Hilbert space whose reproducing kernel corresponds to the covariance function of the isotropic Wiener measure. We obtain bounds on the ‐exponents, and for some cases we find their exact values. For some weighted Korobov and Sobolev spaces, the strong ‐exponent is the same as the ‐exponent for d=1, whereas for the third space it is 2.For approximation we also consider algorithms that use general evaluations given by arbitrary continuous linear functionals as the information about the function. Our main result is that the ‐exponents are the same for general and function evaluations. This holds under the assumption that the orthonormal eigenfunctions of the covariance operator have uniformly bounded L∞ norms. This assumption holds for spaces with shift-invariant kernels. Examples of such spaces include weighted Korobov spaces. For a space with non‐shift‐invariant kernel, we construct the corresponding space with shift-invariant kernel and show that integration and approximation for the non-shift-invariant kernel are no harder than the corresponding problems with the shift-invariant kernel. If we apply this construction to a weighted Sobolev space, whose kernel is non-shift-invariant, then we obtain the corresponding Korobov space. This enables us to derive the results for weighted Sobolev spaces.

The existence of good extensible rank-1 lattices

Extensible integration lattices have the attractive property that the number of points in the node set may be increased while retaining the existing points. It is shown here that there exist generating vectors, h, for extensible rank-1 lattices such that for n = b, b2, ... points and dimensions s = 1, 2, ... the figures of merit Rα, Pα and discrepancy are all small. The upper bounds obtained on these figures of merit for extensible lattices are some power of log n worse than the best upper bounds for lattices where h is allowed to vary with n and s.

Obtaining O(N- 2+∈) Convergence for Lattice Quadrature Rules

Good lattice quadrature rules are known to have O(N - 2+∈) convergence for periodic integrands with sufficient smoothness. Here it is shown that applying the bakers transformation to lattice rules gives O(N - 2+∈) convergence for nonperiodic integrands with sufficient smoothness. This approach is philosophically different than making a periodizing transformation of the integrand as it results in a different error analysis.

Time-dependent critical layers in shear flows on the beta-plane

The problem of a finite-amplitude free disturbance of an inviscid shear flow on the beta-plane is studied. Perturbation theory and matched asymptotics are used to derive an evolution equation for the amplitude of a singular neutral mode of the Kuo equation. The effects of time-dependence, nonlinearity and viscosity are included in the analysis of the critical-layer flow. Nonlinear effects inside the critical layer rather than outside the critical layer determine the evolution of the disturbance. The nonlinear term in the evolution equation is some type of convolution integral rather than a simple polynomial. This makes the evolution equation significantly different from those commonly encountered in fluid wave and stability problems.

Quadrature Error Bounds with Applications to Lattice Rules

Reproducing kernel Hilbert spaces are used to derive error bounds and worst-case integrands for a large family of quadrature rules. In the case of lattice rules applied to periodic integrands these error bounds resemble those previously derived in the literature. However, the theory developed here does not require periodicity and is not restricted to lattice rules. An analysis of variance (ANOVA) decomposition is employed in defining the inner product. It is shown that imbedded rules are superior when integrating functions with large high-order ANOVA effects.

The mean square discrepancy of randomized nets

One popular family of low dicrepancy sets is the (<italic>t, m, s</italic>)-nets. Recently a randomization of these nets that preserves their net property has been introduced. In this article a formula for the mean square <italic>L</italic><supscrpt>2</supscrpt>-discrepancy of (<italic>0, m, s</italic>)-nets in base <italic>b</italic> is derived. This formula has a computational complexity of only O(s log(<italic>N</italic>) + s<supscrpt>2</supscrpt>) for large <italic>N</italic> or s, where <italic>N = b<supscrpt>m</supscrpt></italic> is the number of points. Moreover, the root mean square <italic>L</italic><supscrpt>2</supscrpt>-discrepancy of (<italic>0, m, s</italic>)-nets is show to be O(<italic>N</italic><supscrpt>-1</supscrpt>[log(N)]<supscrpt>(s-1)/2</supscrpt>) as <italic>N</italic> tends to infinity, the same asymptotic order as the known lower bound for the <italic>L</italic><supscrpt>2</supscrpt>-discrepancy of an arbitrary set.