H-matrix accelerated second moment analysis for potentials with rough correlation

We consider the efficient solution of partial differential equationsfor strongly elliptic operators with constant coefficients and stochastic Dirichlet data by the boundary integral equation method. The computation of the solution's two-point correlation is well understood if the two-point correlation of the Dirichlet data is known and sufficiently smooth.Unfortunately, the problem becomes much more involved in case of rough data. We will show that the concept of the H-matrix arithmetic provides a powerful tool to cope with this problem. By employing a parametric surface representation, we end up with an H-matrix arithmetic based on balanced cluster trees. This considerably simplifies the implementation and improves the performance of the H-matrix arithmetic. Numerical experiments are provided to validate and quantify the presented methods and algorithms

A case-control study of the effect of infant feeding on celiac disease

Aims: The aim of this study was to investigate the association between the duration of breast-feeding and the age at the first gluten introduction into the infant diet and the incidence and age at onset of celiac disease. Methods: In a case-control study, 143 children with celiac disease and 137 randomly recruited gender- and age-matched control children were administered a standardized questionnaire. Multivariate-adjusted odds ratios (OR) as estimates of the relative risk and corresponding 95% confidence intervals (95% CI) were calculated. Results: The risk of developing celiac disease decreased significantly by 63% for children breast-fed for more than 2 months (OR 0.37, 95% Cl 0.21-0.64) as compared with children breast-fed for 2 months or less. The age at first gluten introduction had no significant influence on the incidence of celiac disease (OR 0.72, 95% Cl 0.29-1.79 comparing first gluten introduction into infant diet >3 months vs. less than or equal to3 months). Conclusions: A significant protective effect on the incidence of celiac disease was suggested by the duration of breast-feeding (partial breastfeeding as well as exclusive breast-feeding). The data did not support an influence of the age at first dietary gluten exposure on the incidence of celiac disease. However, the age at first gluten exposure appeared to affect the age at onset of symptoms. Copyright (C) 2001 S. Karger AG, Basel

The second order perturbation approach for elliptic partial differential equations on random domains

The present article is dedicated to the solution of elliptic boundary value problems on random domains. We apply a high-precision second order shape Taylor expansion to quantify the impact of the random perturbation on the solution. Thus, we obtain a representation of the solution with third order accuracy in the size of the perturbation's amplitude. The major advantage of this approach is that we end up with purely deterministic equations for the solution's moments. In particular, we derive representations for the first four moments, i.e., expectation, variance, skewness and kurtosis. These moments are efficiently computable by means of boundary integral equations. Numerical results are presented to validate the presented approach

On the quasi-Monte Carlo quadrature with Halton points for elliptic PDEs with log-normal diffusion

This article is dedicated to the computation of the moments of the solution to elliptic partial differential equations with random, log-normally distributed diffusion coefficients by the quasi-Monte Carlo method. Our main result is that the convergence rate of the quasi-Monte Carlo method based on the Halton sequence for the moment computation depends only linearly on the dimensionality of the stochastic input parameters. Especially, we attain this rather mild dependence on the stochastic dimensionality without any randomization of the quasi-Monte Carlo method under consideration. For the proof of the main result, we require related regularity estimates for the solution and its powers. These estimates are also provided here. Numerical experiments are given to validate the theoretical findings. This article is dedicated to the computation of the moments of the solution to elliptic partial differential equations with random, log-normally distributed diffusion coefficients by the quasi-Monte Carlo method. Our main result is that the convergence rate of the quasi-Monte Carlo method based on the Halton sequence for the moment computation depends only linearly on the dimensionality of the stochastic input parameters. Especially, we attain this rather mild dependence on the stochastic dimensionality without any randomization of the quasi-Monte Carlo method under consideration. For the proof of the main result, we require related regularity estimates for the solution and its powers. These estimates are also provided here. Numerical experiments are given to validate the theoretical findings

An interpolation-based fast multipole method for higher order boundary elements on parametric surfaces

In this article, a black-box higher order fast multipole method for solving boundary integral equations on parametric surfaces in three spatial dimensions is proposed. Such piecewise smooth surfaces are the topic of recent studies in isogeometric analysis. Due to the exact surface representation, the rate of convergence of higher order methods is not limited by approximation errors of the surface. An element-wise clustering strategy yields a balanced cluster tree and an efficient numerical integration scheme for the underlying Galerkin method. By performing the interpolation for the fast multipole method directly on the reference domain, the cost complexity in the polynomial degree is reduced by one order. This gain is independent of the application of either (mathcal{H})- or (mathcal{H}^2)-matrices. In fact, several simplifications in the construction of (mathcal{H}^2)-matrices are pointed out, which are a by-product of the surface representation. Extensive numerical examples are provided in order to quantify and qualify the proposed method. In this article, a black-box higher order fast multipole method for solving boundary integral equations on parametric surfaces in three spatial dimensions is proposed. Such piecewise smooth surfaces are the topic of recent studies in isogeometric analysis. Due to the exact surface representation, the rate of convergence of higher order methods is not limited by approximation errors of the surface. An element-wise clustering strategy yields a balanced cluster tree and an efficient numerical integration scheme for the underlying Galerkin method. By performing the interpolation for the fast multipole method directly on the reference domain, the cost complexity in the polynomial degree is reducedby one order. This gain is independent of the application of either H - or H 2- matrices. In fact, several simplificationsin the construction of  H 2 -matrices are pointed out, which are a by-product of the surface representation. Extensive numerical examples are provided in order to quantify and qualify the proposed method

Multilevel Accelerated Quadrature for PDEs with Log-Normally Distributed Diffusion Coefficient

This article is dedicated to multilevel quadrature methods for the rapid solution of stochastic partial differential equations with a log-normally distributed diffusion coefficient. The key idea of such approaches is a sparse-grid approximation of the occurring product space between the stochastic and the spatial variable. We develop the mathematical theory and present error estimates for the computation of the solution's moments with focus on the mean and the variance. Especially, the present framework covers the multilevel Monte Carlo method and the multilevel quasi-Monte Carlo method as special cases. The theoretical findings are supplemented by numerical experiments. This article is dedicated to multilevel quadrature methods forthe rapid solution of stochastic partial differential equationswith a log-normally distributed diffusion coefficient. The key ideaof such approaches is a sparse-grid approximation of the occurring product space between the stochastic and the spatial variable. We develop the mathematical theory and present error estimates for the computation of the solution's moments with focus on the mean and the variance. Especially, the present framework covers the multilevel Monte Carlo method and the multilevel quasi-Monte Carlo method as special cases. The theoretical findings are supplemented by numerical experiments

H-matrix based second moment analysis for rough random fields and finite element discretizations

We consider the efficient solution of strongly elliptic partial differential equations with random load based on the finite element method. The solution's two-point correlation can efficiently be approximated by means of an H- matrix, in particular if the correlation length is rather short or the correlation kernel is nonsmooth. Since the inverses of the finite element matrices which correspond to the differential operator under consideration can likewise efficiently be approximated in the H- matrix format, we can solve the correspondent H- matrix equation in essentially linear time by using the H -matrix arithmetic. Numerical experiments for three-dimensional finite element discretizations for several correlation lengths and different smoothness are provided. They validate the presented method and demonstrate that the computation times do not increase for nonsmooth or shortly correlated data

Analysis of the domain mapping method for elliptic diffusion problems on random domains

In this article, we provide a rigorous analysis of the solution to elliptic diffusion problems on random domains. In particular, based on the decay of the Karhunen-Lo`eve expansion of the domain perturbation field, we establish decay rates for the derivatives of the random solution that are independent of the stochastic dimension. For the implementation of a related approximation scheme, like quasi-Monte Carlo quadrature, stochastic collocation, etc., we propose parametric finite elements to compute the solution of the diffusion problem on each individual realization of the domain generated by the perturbation field. This simplifies the implementation and yields a non-intrusive approach. Having this machinery at hand, we can easily transfer it to stochastic interface problems. The theoretical findings are complemented by numerical examples for both, stochastic interface problems and boundary value problems on random domains. In this article, we provide a rigorous analysis of the solution to elliptic diffusion problems on random domains. In particular, based on the decay of the Karhunen-Loeve expansion of the domain perturbation field, we establish decay rates for the derivativesof the random solution that are independent of the stochastic dimension. For the implementation of a related approximation scheme, like quasi-Monte Carlo quadrature, stochastic collocation, etc., we propose parametric finite elements to compute the solution of the diffusion problem on each individual realization of the domain generated by the perturbation field. This simplifies the implementation and yields a non-intrusive approach. Having this machinery at hand, we can easily transfer it to stochastic interface problems. The theoretical findings are complemented by numerical examples for both, stochastic interface problems and boundary value problems on random domains

Estimating dynamic R&D demand : an analysis of costs and long-run benefits

Using firm-level data from the German manufacturing sector, we estimate a dynamic, structural model of the firm’s decision to invest in R&D and quantify the cost and longrun benefit of this investment. The model incorporates and quantifies linkages between the firm’s R&D investment, product and process innovations, and future productivity and profits. The dynamic model provides a natural measure of the long-run payoff to R&D as the difference in expected firm value generated by the R&D investment. For the median productivity firm, investment in R&D raises firm value by 3.0 percent in a group of hightech industries but only 0.2 percent in low-tech industries. Simulations of the model show that cost subsidies for R&D can significantly affect R&D investment rates and productivity changes in the high-tech industries