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This work proposes and analyzes a Smolyak-type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients and forcing terms (input data of the model). To compute solution statistics, the sparse grid stochastic collocation method uses approximate solutions, produced here by finite elements, corresponding to a deterministic set of points in the random input space. This naturally requires solving uncoupled deterministic problems as in the Monte Carlo method. If the number of random variables needed to describe the input data is moderately large, full tensor product spaces are computationally expensive to use due to the curse of dimensionality. In this case the sparse grid approach is still expected to be competitive with the classical Monte Carlo method. Therefore, it is of major practical relevance to understand in which situations the sparse grid stochastic collocation method is more efficient than Monte Carlo. This work provides error estimates for the fully discrete solution using L q norms and analyzes the computational efficiency of the proposed method. In particular, it demonstrates algebraic convergence with respect to the total number of collocation points and quantifies the effect of the dimension of the problem (number of input random variables) in the final estimates. The derived estimates are then used to compare the method with Monte Carlo, indicating for which problems the former is more efficient than the latter. Computational evidence complements the present theory and shows the effectiveness of the sparse grid stochastic collocation method compared to full tensor and Monte Carlo approaches.

In this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms (input data of the model). The input data are assumed to depend on a finite number of random variables. The method consists in a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It can be seen as a generalization of the stochastic Galerkin method proposed in [I. Babuška, R. Tempone, and G. E. Zouraris, SIAM J. Numer. Anal., 42 (2004), pp. 800-825] and allows one to treat easily a wider range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the \"probability error\" with respect to the number of Gauss points in each direction in the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method.

Triple collocation analysis (TCA) enables estimation of error variances for three or more products that retrieve or estimate the same geophysical variable using mutually independent methods. Several statistical assumptions regarding the statistical nature of errors (e.g., mutual independence and orthogonality with respect to the truth) are required for TCA estimates to be unbiased. Even though soil moisture studies commonly acknowledge that these assumptions are required for an unbiased TCA, no study has specifically investigated the degree to which errors in existing soil moisture data sets conform to these assumptions. Here these assumptions are evaluated both analytically and numerically over four extensively instrumented watershed sites using soil moisture products derived from active microwave remote sensing, passive microwave remote sensing, and a land surface model. Results demonstrate that non-orthogonal and error cross-covariance terms represent a significant fraction of the total variance of these products. However, the overall impact of error cross correlation on TCA is found to be significantly larger than the impact of non-orthogonal errors. Because of the impact of cross-correlated errors, TCA error estimates generally underestimate the true random error of soil moisture products.

This work proposes and analyzes an anisotropic sparse grid stochastic collocation method for solving partial differential equations with random coefficients and forcing terms (input data of the model). The method consists of a Galerkin approximation in the space variables and a collocation, in probability space, on sparse tensor product grids utilizing either Clenshaw—Curtis or Gaussian knots. Even in the presence of nonlinearities, the collocation approach leads to the solution of uncoupled deterministic problems, just as in the Monte Carlo method. This work includes a priori and a posteriori procedures to adapt the anisotropy of the sparse grids to each given problem. These procedures seem to be very effective for the problems under study. The proposed method combines the advantages of isotropic sparse collocation with those of anisotropic full tensor product collocation: the first approach is effective for problems depending on random variables which weigh approximately equally in the solution, while the benefits of the latter approach become apparent when solving highly anisotropic problems depending on a relatively small number of random variables, as in the case where input random variables are Karhunen—Loève truncations of \"smooth\" random fields. This work also provides a rigorous convergence analysis of the fully discrete problem and demonstrates (sub)exponential convergence in the asymptotic regime and algebraic convergence in the preasymptotic regime, with respect to the total number of collocation points. It also shows that the anisotropic approximation breaks the curse of dimensionality for a wide set of problems. Numerical examples illustrate the theoretical results and are used to compare this approach with several others, including the standard Monte Carlo. In particular, for moderately large-dimensional problems, the sparse grid approach with a properly chosen anisotropy seems to be very efficient and superior to all examined methods.

This paper summarizes the Collection-6 refinements in the Moderate Resolution Imaging Spectroradiometer (MODIS) operational cloud-top properties algorithm. The focus is on calibration improvements and on cloud macrophysical properties including cloud-top pressure–temperature–height and cloud thermodynamic phase. The cloud phase is based solely on infrared band measurements. In addition, new parameters will be provided in Collection 6, including cloud-top height and a flag for clouds near the tropopause. The cloud parameters are improved primarily through 1) improved knowledge of the spectral response functions for the MODIS 15-μm carbon dioxide bands gleaned from comparison of coincident MODIS and Atmospheric Infrared Sounder (AIRS) radiance measurements and 2) continual comparison of global MODIS andCloud–Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO)instantaneous cloud products throughout the course of algorithm refinement.Whereas the cloud-top macrophysical parameters were provided through Collection 5 solely at 5-km spatial resolution, these parameters will be available additionally at 1-km spatial resolution in Collection 6.

This work proposes and analyzes a stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms. These input data are assumed to depend on a finite number of random variables. The method consists of a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space, and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It treats easily a wide range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the \"probability error\" with respect to the number of Gauss points in each direction of the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method. Finally, we include a section with developments posterior to the original publication of this work. There we review sparse grid stochastic collocation methods, which are effective collocation strategies for problems that depend on a moderately large number of random variables.

We introduce a multistep Legendre–Gauss spectral collocation method for the non-linear Volterra integral equations of the second kind. This method is easy to implement and possesses high order accuracy. In addition, it is very suitable for long time calculations. We also derive the optimal convergence of the hp-version of the multistep collocation method under the L2-norm. Numerical experiments confirm the theoretical expectations.

This article discusses the extent to which methods normally associated with corpus linguistics can be effectively used by critical discourse analysts. Our research is based on the analysis of a 140-million-word corpus of British news articles about refugees, asylum seekers, immigrants and migrants (collectively RASIM). We discuss how processes such as collocation and concordance analysis were able to identify common categories of representation of RASIM as well as directing analysts to representative texts in order to carry out qualitative analysis. The article suggests a framework for adopting corpus approaches in critical discourse analysis.

A study was conducted to identify the scope and timing of maturational constraints in three linguistic domains within the same individuals, as well as the potential mediating roles of amount of second language (L2) exposure and language aptitude at different ages in different domains. Participants were 65 Chinese learners of Spanish and 12 native speaker controls. Results for three learner groups defined by age of onset - 3-6, 7-15, and 16-29 years - confirmed previous findings of windows of opportunity closing first for L2 phonology, then for lexis and collocation and, finally, in the mid-teens, for morphosyntax. All three age functions exhibited the discontinuities in the rate of decline with increasing AO associated with sensitive periods. Significant correlations were found between language aptitude, measured using the LLAMA test (Meara, 2005), and pronunciation scores, and between language aptitude and lexis and collocation scores, in the 16-29 group.

Research on agglomeration finds that either a higher survival rate of incumbent firms or a higher founding rate of new entrants, or both, can sustain an industry cluster. The conditioning effects of time on the two distinct mechanisms of survival and founding are, however, rarely examined. We argue that the forces driving geographic concentration vary across the industry life cycle. Data from Ontario's winery industry from 1865 to 1974 demonstrates a dynamic model of geographic concentration: agglomeration attracts more new entry in the growth stage only, whereas it contributes to firm survival in the mature stage only. The results not only establish the importance of understanding the temporal dynamics underlying agglomeration externalities, but also provide a possible explanation for the mixed empirical results found in previous studies.