Abstract. We present a comprehensive error analysis of two prototypical atomistic-tocontinuum coupling methods of blending type: the energy-based and the force-based quasicontinuum methods.Our results are valid in two and three dimensions, for finite range many-body interactions (e.g., EAM type), and in the presence of lattice defects (we consider point defects and dislocations). The two key ingredients in the analysis are (i) new force and energy consistency error estimates; and (ii) a new technique for proving energy norm stability of a/c couplings that requires only the assumption that the exact atomistic solution is a stable equilibrium.

Copies of full items can be used for personal research or study, educational, or not-forprofit purposes without prior permission or charge. Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way.Publisher's statement: NOTICE: this is the author's version of a work that was accepted for publication in Computer Methods in Applied Mechanics and Engineering. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published Computer Methods in Applied Mechanics and Engineering, Volume 253(1). pp.160-168. http://dx.doi.org/10.1016/j.cma.2012.09.007 A note on versions:The version presented here may differ from the published version or, version of record, if you wish to cite this item you are advised to consult the publisher's version. Please see the 'permanent WRAP url' above for details on accessing the published version and note that access may require a subscription. Abstract. We formulate an energy-based atomistic-to-continuum coupling method based on blending the quasicontinuum method [14] for the simulation of crystal defects. We present optimal choices of approximation parameters (blending function and finite element grid) and confirm our analytical predictions in numerical tests.

Umbrella sampling efficiently yields equilibrium averages that depend on exploring rare states of a model by biasing simulations to windows of coordinate values and then combining the resulting data with physical weighting. Here, we introduce a mathematical framework that casts the step of combining the data as an eigenproblem. The advantage to this approach is that it facilitates error analysis. We discuss how the error scales with the number of windows. Then, we derive a central limit theorem for averages that are obtained from umbrella sampling. The central limit theorem suggests an estimator of the error contributions from individual windows, and we develop a simple and computationally inexpensive procedure for implementing it. We demonstrate this estimator for simulations of the alanine dipeptide and show that it emphasizes low free energy pathways between stable states in comparison to existing approaches for assessing error contributions. Our work suggests the possibility of using the estimator and, more generally, the eigenvector method for umbrella sampling to guide adaptation of the simulation parameters to accelerate convergence. Published by AIP Publishing. [http://dx

Background: The underlying goal of microarray experiments is to identify gene expression patterns across different experimental conditions. Genes that are contained in a particular pathway or that respond similarly to experimental conditions could be co-expressed and show similar patterns of expression on a microarray. Using any of a variety of clustering methods or gene network analyses we can partition genes of interest into groups, clusters, or modules based on measures of similarity. Typically, Pearson correlation is used to measure distance (or similarity) before implementing a clustering algorithm. Pearson correlation is quite susceptible to outliers, however, an unfortunate characteristic when dealing with microarray data (well known to be typically quite noisy.)

Abstract. The development of patch test consistent quasicontinuum energies for multi-dimensional crystalline solids modeled by many-body potentials remains a challenge. The original quasicontinuum energy (QCE) [28] has been implemented for many-body potentials in two and three space dimensions, but it is not patch test consistent. We propose that by blending the atomistic and corresponding Cauchy-Born continuum models of QCE in an interfacial region with thickness of a small number k of blended atoms, a general quasicontinuum energy (BQCE) can be developed with the potential to significantly improve the accuracy of QCE near lattice instabilities such as dislocation formation and motion.In this paper, we give an error analysis of the blended quasicontinuum energy (BQCE) for a periodic one-dimensional chain of atoms with next-nearest neighbor interactions. Our analysis includes the optimization of the blending function for an improved convergence rate. We show that the ℓ 2 strain error for the non-blended QCE energy (QCE), which has low order O(ε 1/2 ) where ε is the atomistic length scale [13,29], can be reduced by a factor of k 3/2 for an optimized blending function where k is the number of atoms in the blending region. The QCE energy has been further shown to suffer from a O(1) error in the critical strain at which the lattice loses stability [16]. We prove that the error in the critical strain of BQCE can be reduced by a factor of k 2 for an optimized blending function, thus demonstrating that the BQCE energy for an optimized blending function has the potential to give an accurate approximation of the deformation near lattice instabilities such as crack growth.