Alternating least squares as moving subspace correction

In this note we take a new look at the local convergence of alternating optimization methods for low-rank matrices and tensors. Our abstract interpretation as sequential optimization on moving subspaces yields insightful reformulations of some known convergence conditions that focus on the interplay between the contractivity of classical multiplicative Schwarz methods with overlapping subspaces and the curvature of low-rank matrix and tensor manifolds. While the verification of the abstract conditions in concrete scenarios remains open in most cases, we are able to provide an alternative and conceptually simple derivation of the asymptotic convergence rate of the two-sided block power method of numerical algebra for computing the dominant singular subspaces of a rectangular matrix. This method is equivalent to an alternating least squares method applied to a distance function. The theoretical results are illustrated and validated by numerical experiments.Comment: 20 pages, 4 figure

Evolution of speckle during spinodal decomposition

Time-dependent properties of the speckled intensity patterns created by scattering coherent radiation from materials undergoing spinodal decomposition are investigated by numerical integration of the Cahn-Hilliard-Cook equation. For binary systems which obey a local conservation law, the characteristic domain size is known to grow in time $\tau$ as $R = [B \tau]^n$ with n=1/3, where B is a constant. The intensities of individual speckles are found to be nonstationary, persistent time series. The two-time intensity covariance at wave vector ${\bf k}$ can be collapsed onto a scaling function $Cov(\delta t,\bar{t})$, where $\delta t = k^{1/n} B |\tau_2-\tau_1|$ and $\bar{t} = k^{1/n} B (\tau_1+\tau_2)/2$. Both analytically and numerically, the covariance is found to depend on $\delta t$ only through $\delta t/\bar{t}$ in the small-$\bar{t}$ limit and $\delta t/\bar{t} ^{1-n}$ in the large-$\bar{t}$ limit, consistent with a simple theory of moving interfaces that applies to any universality class described by a scalar order parameter. The speckle-intensity covariance is numerically demonstrated to be equal to the square of the two-time structure factor of the scattering material, for which an analytic scaling function is obtained for large $\bar{t}.$ In addition, the two-time, two-point order-parameter correlation function is found to scale as $C(r/(B^n\sqrt{\tau_1^{2n}+\tau_2^{2n}}),\tau_1/\tau_2)$, even for quite large distances $r$. The asymptotic power-law exponent for the autocorrelation function is found to be $\lambda \approx 4.47$, violating an upper bound conjectured by Fisher and Huse.Comment: RevTex: 11 pages + 12 figures, submitted to PR

Enhancing SPH using moving least-squares and radial basis functions

In this paper we consider two sources of enhancement for the meshfree Lagrangian particle method smoothed particle hydrodynamics (SPH) by improving the accuracy of the particle approximation. Namely, we will consider shape functions constructed using: moving least-squares approximation (MLS); radial basis functions (RBF). Using MLS approximation is appealing because polynomial consistency of the particle approximation can be enforced. RBFs further appeal as they allow one to dispense with the smoothing-length -- the parameter in the SPH method which governs the number of particles within the support of the shape function. Currently, only ad hoc methods for choosing the smoothing-length exist. We ensure that any enhancement retains the conservative and meshfree nature of SPH. In doing so, we derive a new set of variationally-consistent hydrodynamic equations. Finally, we demonstrate the performance of the new equations on the Sod shock tube problem.Comment: 10 pages, 3 figures, In Proc. A4A5, Chester UK, Jul. 18-22 200

Underdetermined-order recursive least-squares adaptive filtering: The concept and algorithms

Published versio

Analysis of moving least squares approximation revisited

In this article the error estimation of the moving least squares approximation is provided for functions in fractional order Sobolev spaces. The analysis presented in this paper extends the previous estimations and explains some unnoticed mathematical details. An application to Galerkin method for partial differential equations is also supplied.Comment: Journal of Computational and Applied Mathematics, 2015 Journal of Computational and Applied Mathematic