Deformation Measurements at the Sub-Micron Size Scale: II. Refinements in the Algorithm for Digital Image Correction

Improvements are proposed in the application of the Digital Image Correlation method, a technique that compares digital images of a specimen surface before and after deformation to deduce its sureface (2-D) displacement field and strains. These refinements, tested on translations and rigid body rotations were significant with regard to the computer efficiency and covergence properties of the method. In addition, the formulation of the algorithm was extended so as to compute the three-dimensional surface displacement field from Scanning Tunneling Microscope tomographies of a deforming specimen. The reolsution of this new displacement measuring method at the namometer scale was assessed on translation and uniaxial tensile tests and was found to be 4.8 nm for in-plane displacement components and 1.5 nm for the out-of-plane one spanning a 10 x 10 μm area

Comparison of different nonlinear solvers for 2D time-implicit stellar hydrodynamics

Time-implicit schemes are attractive since they allow numerical time steps that are much larger than those permitted by the Courant-Friedrich-Lewy criterion characterizing time-explicit methods. This advantage comes, however, with a cost: the solution of a system of nonlinear equations is required at each time step. In this work, the nonlinear system results from the discretization of the hydrodynamical equations with the Crank-Nicholson scheme. We compare the cost of different methods, based on Newton-Raphson iterations, to solve this nonlinear system, and benchmark their performances against time-explicit schemes. Since our general scientific objective is to model stellar interiors, we use as test cases two realistic models for the convective envelope of a red giant and a young Sun. Focusing on 2D simulations, we show that the best performances are obtained with the quasi-Newton method proposed by Broyden. Another important concern is the accuracy of implicit calculations. Based on the study of an idealized problem, namely the advection of a single vortex by a uniform flow, we show that there are two aspects: i) the nonlinear solver has to be accurate enough to resolve the truncation error of the numerical discretization, and ii) the time step has be small enough to resolve the advection of eddies. We show that with these two conditions fulfilled, our implicit methods exhibit similar accuracy to time-explicit schemes, which have lower values for the time step and higher computational costs. Finally, we discuss in the conclusion the applicability of these methods to fully implicit 3D calculations.Comment: Accepted for publication in A&

Rigorous approximation of diffusion coefficients for expanding maps

We use Ulam's method to provide rigorous approximation of diffusion coefficients for uniformly expanding maps. An algorithm is provided and its implementation is illustrated using Lanford's map.Comment: In this version Lanford's map has been used to illustrate the computer implementation of the algorithm. To appear in Journal of Statistical Physic

Sample-path solutions for simulation optimization problems and stochastic variational inequalities

inequality;simulation;optimization

Limits of spiked random matrices II

The top eigenvalues of rank $r$ spiked real Wishart matrices and additively perturbed Gaussian orthogonal ensembles are known to exhibit a phase transition in the large size limit. We show that they have limiting distributions for near-critical perturbations, fully resolving the conjecture of Baik, Ben Arous and P\'{e}ch\'{e} [Duke Math. J. (2006) 133 205-235]. The starting point is a new $(2r+1)$-diagonal form that is algebraically natural to the problem; for both models it converges to a certain random Schr\"{o}dinger operator on the half-line with $r\times r$ matrix-valued potential. The perturbation determines the boundary condition and the low-lying eigenvalues describe the limit, jointly as the perturbation varies in a fixed subspace. We treat the real, complex and quaternion ($\beta=1,2,4$) cases simultaneously. We further characterize the limit laws in terms of a diffusion related to Dyson's Brownian motion, or alternatively a linear parabolic PDE; here $\beta$ appears simply as a parameter. At $\beta=2$, the PDE appears to reconcile with known Painlev\'{e} formulas for these $r$-parameter deformations of the GUE Tracy-Widom law.Comment: Published at http://dx.doi.org/10.1214/15-AOP1033 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org