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Recently, quasi-Monte Carlo algorithms have been successfully used for multivariate integration of high dimension d, 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 large d. It is done by identifying classes of functions for which the effect of the dimension d is negligible. These are weighted classes in which the behavior in the successive dimensions is moderated by a sequence of weights. We prove that the minimal worst case error of quasi-Monte Carlo algorithms does not depend on the dimension d iff 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 ε is bounded by Cε −p , where the exponent p ∈ [1, 2], and C depends 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 the worst case setting is of order ε −p with the exponent p = 2. The deterministic construction of strongly tractable quasi-Monte Carlo algorithms as well as the minimal exponent p is open.

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Our problem is to compute an approximation to the largest eigenvalue of an n x n large symmetric positive definite matrix with relative error at most c. We consider only algorithms that use Krylov inforrpation [b, Ab,. .. , Akb] consisting of k matrix-vector multiplications for some unit vector b. If the vector b is chosen deterministically then the problem cannot be solved no matter how many matrix-vector multiplications are performed and what algorithm is used. If, however, the vector b is chosen randomly with respect to the uniform distribution over the unit sphere, then the problem can be solved on the average and probabilistically. More precisely, for a randomly chosen vector b we study the power and Lanczos algorithms. For the power algorithm (method) we prove sharp bounds on the average relative error and on the probabilistic relative failure. For the Lanczos algorithm we present only upper bounds. In particular, In(n)Jk characterizes the average relative error of the power algorithm, whereas O((ln(n)jk)l) is an upper bound on the average relative error of the Lanczos algorithm. In the probabilistic case, the algorithm is characterized by its probabilistic relative failure which is defined as the measure of the set of vectors b for which the algorithm fails. We show that the probabilistic relative •Supported in part by the National Science Foundation under Grant DCR-86-03674. failure goes to zero roughly as vIn(l-c)k for the power algorithm and at most as ..;n e-(2k-1)y'i for the Lanczos algorithm. These bounds are for a worst case distribution of eigenvalues which may depend on k. vVe also study the behavior in the average and probabilistic cases of the two algorithms for a fixed matrix A as the number of matrix-vector multiplications k increases. The bounds for the power algorithm depend then on the ratio of the two largest eigenvalues and their multiplicities. The bounds for the Lanczos algorithm depend on the ratio between the difference of the two largest eigenvalues and the difference of the largest and the smallest eigenvalues.

We study multivariate tensor product problems in the worst case and average case settings. They are de ned on functions of d variables. For arbitrary d, w e p r o vide explicit upper bounds on the costs of algorithms which compute an "-approximation to the solution. The cost bounds are of the form (c(d) + 2) 1 2 + 3 ln 1=" d ; 1 4 (d;1) 1 " 5 : Here c(d) is the cost of one function evaluation (or one linear functional evaluation), and i 's do not depend on d they are determined by the properties of the problem for d = 1. F or certain tensor product problems, these cost bounds do not exceed c(d) K " ;p for some numbers K and p, both independent o f d. We apply these general estimates to certain integration and approximation problems in the worst and average case settings. We also obtain an upper bound, which is independent o f d, for the number, n(" d), of points for which discrepancy (with unequal weights) is at most ", n(" d)

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.

The inverse of the star-discrepancy depends linearly on the dimension

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 c 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 e in the worst-case setting is bounded by a polynomial in d and e −1 . Strong tractability means that the bound does not depend on d and depends polynomially on e −1 . We prove that strong tractability holds iff ;. j=1 c j < ., and tractability holds iff lim sup d Q . ; d j=1 c j /log d < .. Furthermore, strong tractability or tractability results are achieved by the relatively small class of lattice rules. We also prove that if ;. j=1 c 1/a j < ., where a measures the decay of Fourier coefficients in the weighted Korobov class, then for d \ 1, n prime and d > 0 there exist lattice rules that satisfy an error bound independent of d and of order n −a/2+d . This is almost the best possible result, since the order n −a/2 cannot be improved upon even for d=1. A corresponding result is deduced for weighted non-periodic Sobolev spaces: if ; there exist shifted lattice rules that satisfy an error bound independent of d and of order n −1+d . We also check how the randomized error of the (classical