Ian H. Sloan

When Are Quasi-Monte Carlo Algorithms Efficient for High Dimensional Integrals?

Recently, quasi-Monte Carlo algorithms have been successfully used for multivariate integration of high dimensiond, and were significantly more efficient than Monte Carlo algorithms. The existing theory of the worst case error bounds of quasi-Monte Carlo algorithms does not explain this phenomenon. This paper presents a partial answer to why quasi-Monte Carlo algorithms can work well for arbitrarily larged. It is done by identifying classes of functions for which the effect of the dimensiondis negligible. These areweightedclasses in which the behavior in the successive dimensions is moderated by a sequence of weights. We prove that the minimalworst caseerror of quasi-Monte Carlo algorithms does not depend on the dimensiondiff the sum of the weights is finite. We also prove that the minimal number of function values in the worst case setting needed to reduce the initial error by ? is bounded byC??p, where the exponentp? 1, 2], andCdepends exponentially on the sum of weights. Hence, the relatively small sum of the weights makes some quasi-Monte Carlo algorithms strongly tractable. We show in a nonconstructive way that many quasi-Monte Carlo algorithms are strongly tractable. Even random selection of sample points (done once for the whole weighted class of functions and then the worst case error is established for that particular selection, in contrast to Monte Carlo where random selection of sample points is carried out for a fixed function) leads to strong tractable quasi-Monte Carlo algorithms. In this case the minimal number of function values in theworst casesetting is of order ??pwith the exponentp= 2. The deterministic construction of strongly tractable quasi-Monte Carlo algorithms as well as the minimal exponentpis open.

High-dimensional integration: The quasi-Monte Carlo way

This paper is a contemporary review of QMC (‘quasi-Monte Carlo’) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1]s, where s may be large, or even infinite. After a general introduction, the paper surveys recent developments in lattice methods, digital nets, and related themes. Among those recent developments are methods of construction of both lattices and digital nets, to yield QMC rules that have a prescribed rate of convergence for sufficiently smooth functions, and ideally also guaranteed slow growth (or no growth) of the worst-case error as s increases. A crucial role is played by parameters called ‘weights’, since a careful use of the weight parameters is needed to ensure that the worst-case errors in an appropriately weighted function space are bounded, or grow only slowly, as the dimension s increases. Important tools for the analysis are weighted function spaces, reproducing kernel Hilbert spaces, and discrepancy, all of which are discussed with an appropriate level of detail.

Extremal systems of points and numerical integration on the sphere

This paper considers extremal systems of points on the unit sphere Sr⊂Rr+1, related problems of numerical integration and geometrical properties of extremal systems. Extremal systems are systems of dn=dim Pn points, where Pn is the space of spherical polynomials of degree at most n, which maximize the determinant of an interpolation matrix. Extremal systems for S2 of degrees up to 191 (36,864 points) provide well distributed points, and are found to yield interpolatory cubature rules with positive weights. We consider the worst case cubature error in a certain Hilbert space and its relation to a generalized discrepancy. We also consider geometrical properties such as the minimal geodesic distance between points and the mesh norm. The known theoretical properties fall well short of those suggested by the numerical experiments.

Quasi-Monte Carlo Finite Element Methods for a Class of Elliptic Partial Differential Equations with Random Coefficients

In this paper quasi-Monte Carlo (QMC) methods are applied to a class of elliptic partial differential equations (PDEs) with random coefficients, where the random coefficient is parametrized by a co...

Tractability of Multivariate Integration for Weighted Korobov Classes

We study the worst-case error of multivariate integration in weighted Korobov classes of periodic functions of d coordinates. This class is defined in terms of weights ?j which moderate the behavior of functions with respect to successive coordinates. We study two classes of quadrature rules. They are quasi-Monte Carlo rules which use n function values and in which all quadrature weights are 1/n and rules for which all quadrature weights are non-negative. Tractability for these two classes of quadrature rules means that the minimal number of function values needed to guarantee error ? in the worst-case setting is bounded by a polynomial in d and ??1. Strong tractability means that the bound does not depend on d and depends polynomially on ??1. We prove that strong tractability holds iff ?∞j=1?j 0 there exist lattice rules that satisfy an error bound independent of d and of order n??/2+?. This is almost the best possible result, since the order n??/2 cannot be improved upon even for d=1. A corresponding result is deduced for weighted non-periodic Sobolev spaces: if ?∞j=1?1/2j 0 there exist shifted lattice rules that satisfy an error bound independent of d and of order n?1+?. We also check how the randomized error of the (classical) Monte Carlo algorithm depends on d for weighted Korobov classes. It turns out that Monte Carlo is strongly tractable iff ?∞j=1log?j<∞ and tractable iff limsupd?∞?dj=1log?j/logd<∞. Hence, in particular, for ?j=1 we have the usual Korobov space in which integration is intractable for the two classes of quadrature rules in the worst-case setting, whereas Monte Carlo is strongly tractable in the randomized setting.

Component-by-component construction of good lattice rules

This paper provides a novel approach to the construction of good lattice rules for the integration of Korobov classes of periodic functions over the unit s-dimensional cube. Theorems are proved which justify the construction of good lattice rules one component at a time - that is the lattice rule for dimension s + 1 is obtained from the rule for dimension s by searching over all possible choices of the (s + 1)th component, while keeping all the existing components unchanged. The construction, which goes against accepted wisdom, is illustrated by numerical examples. The construction is particularly useful if the components of the integrand are ordered, in the sense that the first component is more important than the second, and so on.

Good Lattice Rules in Weighted Korobov Spaces with General Weights

We study the problem of multivariate integration and the construction of good lattice rules in weighted Korobov spaces with general weights. These spaces are not necessarily tensor products of spaces of univariate functions. Sufficient conditions for tractability and strong tractability of multivariate integration in such weighted function spaces are found. These conditions are also necessary if the weights are such that the reproducing kernel of the weighted Korobov space is pointwise non-negative. The existence of a lattice rule which achieves the nearly optimal convergence order is proven. A component-by-component (CBC) algorithm that constructs good lattice rules is presented. The resulting lattice rules achieve tractability or strong tractability error bounds and achieve nearly optimal convergence order for suitably decaying weights. We also study special weights such as finite-order and order-dependent weights. For these special weights, the cost of the CBC algorithm is polynomial. Numerical computations show that the lattice rules constructed by the CBC algorithm give much smaller worst-case errors than the mean worst-case errors over all quasi-Monte Carlo rules or over all lattice rules, and generally smaller worst-case errors than the best Korobov lattice rules in dimensions up to hundreds. Numerical results are provided to illustrate the efficiency of CBC lattice rules and Korobov lattice rules (with suitably chosen weights), in particular for high-dimensional finance problems.

Superconvergence in finite element methods and meshes that are locally symmetric with respect to a point

Consider a second-order elliptic boundary value problem in any number of space dimensions with locally smooth coefficients and solution. Consider also its numerical approximation by standard conforming finite element methods with, for example, fixed degree piecewise polynomials on a quasi-uniform mesh-family (the “h-method”). It will be shown that, if the finite element function spaces are locally symmetric about a point

Lattice methods for multiple integration: theory, error analysis and examples

x_0

How good can polynomial interpolation on the sphere be

with respect to the antipodal map