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

Approximation of linear functionals on a banach space with a Gaussian measure

AbstractWe study approximation of linear functionals on separable Banach spaces equipped with a Gaussian measure. We study optimal information and optimal algorithms in average case, probabilistic, and asymptotic settings, for a general error criterion. We prove that adaptive information is not more powerful than nonadaptive information and that μ-spline algorithms, which are linear, are optimal in all three settings. Some of these results hold for approximation of linear operators. We specialize our results to the space of functions with continuous rth derivatives, equipped with a Wiener measure. In particular, we show that the natural splines of degree 2r + I yield the optimal algorithms. We apply the general results to the problem of integration

Infinite-dimensional integration and the multivariate decomposition method

We further develop the Multivariate Decomposition Method (MDM) for the Lebesgue integration of functions of infinitely many variables x_1, x_2, x_3, ... with respect to a corresponding product of a one dimensional probability measure. Although a number of concepts of infinite-dimensional integrals have been used in the literature, questions of uniqueness and compatibility have mostly not been studied. We show that, under appropriate convergence conditions, the Lebesgue integral equals the "anchored" integral, independently of the anchor. The MDM assumes that point values of f_u are available for important subsets u, at some known cost. In this paper we introduce a new setting, in which it is assumed that each f_u belongs to a normed space F_u, and that bounds B_u on \|f_u\|_{F_u} are known. This contrasts with the assumption in many papers that weights \gamma_u, appearing in the norm of the infinite-dimensional function space, are somehow known. Often such weights \gamma_u were determined by minimizing an error bound depending on the B_u, the \gamma_u *and* the chosen algorithm, resulting in weights that depend on the algorithm. In contrast, in this paper only the bounds B_u are assumed known. We give two examples in which we specialize the MDM: in the first case F_u is the |u|-fold tensor product of an anchored reproducing kernel Hilbert space, and in the second case it is a particular non-Hilbert space for integration over an unbounded domain.status: publishe

The applicability of genetically modified microorganisms in bioremediation of contaminated environments

Inżynieria genetyczna, będąca nowoczesną technologią, pozwala na projektowanie mikroorganizmów zdolnych do rozkładu określonego typu zanieczyszczeń. Konstruowanie GMMs na potrzeby bioremediacji jest możliwe dzięki poznaniu mechanizmów degradacji związków toksycznych, szlaków metabolicznych, enzymów katabolicznych oraz odpowiednich genów. Do detekcji i wizualizacji GMMs w środowisku służą różne metody molekularne: FISH, in situ PCR, DGGE, TGGE, T-RFLP, ARDRA oraz markery selekcyjne (lux, gfp, lacZ, xylE). W celu zminimalizowania ryzyka wynikającego z uwolnienia GMMs do środowiska stosowane są pewne bariery genetyczne. Mają one na celu ograniczenie przeżywalności rekombinantów oraz transferu genów do mikroorganizmów autochtonicznych. W artykule omówiono zasady projektowania GMMs oraz przedstawiono przykłady ich praktycznego wykorzystania w bioremediacji zanieczyszczonych środowisk.Genetic engineering is a modern technology, which Allowi to design microorganisms capable of degrading specific contaminants. The construction of GMMs for bioremediation purposes is possible because many degradative pathways, enzyme and their respective genes are known and biochemical reactions are well understood. For selection and identification of GMMs in the environment many molecular techniques were developed. They include FISH, in situ PCR, DGGE, TGGE, T-RFLP, ARDRA and marker genes (lux, gfp, lacZ, xylE). In order to reduce potential risk of the use of GMMs in the environment some genetic barriers were created. They limit survival of the recombinants and gene transfer into autochthonous microorganisms. In this review the construction and practical applications of GMMs in bioremediation studies are discussed