Tractability of multivariate problems for standard and linear information in the worst case setting: part II

We study QPT (quasi-polynomial tractability) in the worst case setting for linear tensor product problems defined over Hilbert spaces. We assume that the domain space is a reproducing kernel Hilbert space so that function values are well defined. We prove QPT for algorithms that use only function values under the three assumptions: 1) the minimal errors for the univariate case decay polynomially fast to zero, 2) the largest singular value for the univariate case is simple and 3) the eigenfunction corresponding to the largest singular value is a multiple of the function value at some point. The first two assumptions are necessary for QPT. The third assumption is necessary for QPT for some Hilbert spaces

Smolyak's algorithm: A powerful black box for the acceleration of scientific computations

We provide a general discussion of Smolyak's algorithm for the acceleration of scientific computations. The algorithm first appeared in Smolyak's work on multidimensional integration and interpolation. Since then, it has been generalized in multiple directions and has been associated with the keywords: sparse grids, hyperbolic cross approximation, combination technique, and multilevel methods. Variants of Smolyak's algorithm have been employed in the computation of high-dimensional integrals in finance, chemistry, and physics, in the numerical solution of partial and stochastic differential equations, and in uncertainty quantification. Motivated by this broad and ever-increasing range of applications, we describe a general framework that summarizes fundamental results and assumptions in a concise application-independent manner

Hot new directions for quasi-Monte Carlo research in step with applications

This article provides an overview of some interfaces between the theory of quasi-Monte Carlo (QMC) methods and applications. We summarize three QMC theoretical settings: first order QMC methods in the unit cube $[0,1]^s$ and in $\mathbb{R}^s$, and higher order QMC methods in the unit cube. One important feature is that their error bounds can be independent of the dimension $s$ under appropriate conditions on the function spaces. Another important feature is that good parameters for these QMC methods can be obtained by fast efficient algorithms even when $s$ is large. We outline three different applications and explain how they can tap into the different QMC theory. We also discuss three cost saving strategies that can be combined with QMC in these applications. Many of these recent QMC theory and methods are developed not in isolation, but in close connection with applications

Tractability of Approximation and Integration for Weighted Tensor Product Problems over Unbounded Domains

We study tractability and strong tractability of multivariate approximation and integration in the worst case deterministic setting. Tractability means that the number of functional evaluations needed to compute an "-approximation of the multivariate problem with d variables is polynomially bounded in " \Gamma1 and d. Strong tractability means that this minimal number is bounded independently of d by a polynomial in " \Gamma1 . Both problems are considered for certain Sobolev spaces of functions defined over the whole space IR d . These spaces are characterized by a number of parameters: r is the smoothness of functions, fl d;k is a space weight which measures the relative importance of the kth variable for d-variate functions, and a weight function / that monitors the behavior of the functions at infinity. The approximation and integration problems are defined in a weighted sense with respect to a probability density ! and variances oe d;k . We find conditions on the weights ! and / such that the approximation and integration are well defined. For the approximation problem, we consider two classes of functional evaluations: all consisting of all linear continuous functionals and std consisting of function evaluations. Of course, for integration we only consider std . Under natural assumptions on the weight functions ! and /, we prove that strong tractability holds iff sup d1 P d k=1 (fl d;k oe 2r\Gamma1 d;k ) b ! 1, and tractability holds iff sup d1 P d k=1 (fl d;k oe 2r\Gamma1 d;k ) b = ln(d + 1) ! 1: Here b can be any positive number for approximation in all , and b = 1 for approximation and integration in std .