Analysis of Energy-Based Blended Quasicontinuum Approximations

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) 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 $\ell^2$ strain error for the non-blended QCE energy (QCE), which has low order $\text{O}(\epsilon^{1/2})$ where $\epsilon$ is the atomistic length scale, 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. 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.Comment: 26 pages, 1 figur

Optimal Calibration for Multiple Testing against Local Inhomogeneity in Higher Dimension

Based on two independent samples X_1,...,X_m and X_{m+1},...,X_n drawn from multivariate distributions with unknown Lebesgue densities p and q respectively, we propose an exact multiple test in order to identify simultaneously regions of significant deviations between p and q. The construction is built from randomized nearest-neighbor statistics. It does not require any preliminary information about the multivariate densities such as compact support, strict positivity or smoothness and shape properties. The properly adjusted multiple testing procedure is shown to be sharp-optimal for typical arrangements of the observation values which appear with probability close to one. The proof relies on a new coupling Bernstein type exponential inequality, reflecting the non-subgaussian tail behavior of a combinatorial process. For power investigation of the proposed method a reparametrized minimax set-up is introduced, reducing the composite hypothesis "p=q" to a simple one with the multivariate mixed density (m/n)p+(1-m/n)q as infinite dimensional nuisance parameter. Within this framework, the test is shown to be spatially and sharply asymptotically adaptive with respect to uniform loss on isotropic H\"older classes. The exact minimax risk asymptotics are obtained in terms of solutions of the optimal recovery

Improving Sparse Representation-Based Classification Using Local Principal Component Analysis

Sparse representation-based classification (SRC), proposed by Wright et al., seeks the sparsest decomposition of a test sample over the dictionary of training samples, with classification to the most-contributing class. Because it assumes test samples can be written as linear combinations of their same-class training samples, the success of SRC depends on the size and representativeness of the training set. Our proposed classification algorithm enlarges the training set by using local principal component analysis to approximate the basis vectors of the tangent hyperplane of the class manifold at each training sample. The dictionary in SRC is replaced by a local dictionary that adapts to the test sample and includes training samples and their corresponding tangent basis vectors. We use a synthetic data set and three face databases to demonstrate that this method can achieve higher classification accuracy than SRC in cases of sparse sampling, nonlinear class manifolds, and stringent dimension reduction.Comment: Published in "Computational Intelligence for Pattern Recognition," editors Shyi-Ming Chen and Witold Pedrycz. The original publication is available at http://www.springerlink.co

Analysis of the quasi-nonlocal approximation of linear and circular chains in the plane

We give an analysis of the stability and displacement error for linear and circular atomistic chains in the plane when the atomistic energy is approximated by the Cauchy-Born continuum energy and by the quasi-nonlocal atomistic-to-continuum coupling energy. We consider atomistic energies that include Lennard-Jones type nearest neighbor and next nearest neighbor pair-potential interactions. Previous analyses for linear chains have shown that the Cauchy-Born and quasi-nonlocal approximations reproduce (up to the order of the lattice spacing) the atomistic lattice stability for perturbations that are constrained to the line of the chain. However, we show that the Cauchy-Born and quasi-nonlocal approximations give a finite increase for the lattice stability of a linear or circular chain under compression when general perturbations in the plane are allowed. We also analyze the increase of the lattice stability under compression when pair-potential energies are augmented by bond-angle energies. Our estimates of the largest strain for lattice stability (the critical strain) are sharp (exact up to the order of the lattice scale). We then use these stability estimates and modeling error estimates for the linearized Cauchy-Born and quasi-nonlocal energies to give an optimal order (in the lattice scale) {\em a priori} error analysis for the approximation of the atomistic strain in $\ell^2_\epsilon$ due to an external force.Comment: 27 pages, 0 figure