Workshop on Lunar Breccias and Soils and Their Meteoritic analogs

Lunar soils and breccia studies are used in studying the evolution of meteorite parent bodies. These studies are compared to lunar soils and breccias

Parameter estimation by implicit sampling

Implicit sampling is a weighted sampling method that is used in data assimilation, where one sequentially updates estimates of the state of a stochastic model based on a stream of noisy or incomplete data. Here we describe how to use implicit sampling in parameter estimation problems, where the goal is to find parameters of a numerical model, e.g.~a partial differential equation (PDE), such that the output of the numerical model is compatible with (noisy) data. We use the Bayesian approach to parameter estimation, in which a posterior probability density describes the probability of the parameter conditioned on data and compute an empirical estimate of this posterior with implicit sampling. Our approach generates independent samples, so that some of the practical difficulties one encounters with Markov Chain Monte Carlo methods, e.g.~burn-in time or correlations among dependent samples, are avoided. We describe a new implementation of implicit sampling for parameter estimation problems that makes use of multiple grids (coarse to fine) and BFGS optimization coupled to adjoint equations for the required gradient calculations. The implementation is "dimension independent", in the sense that a well-defined finite dimensional subspace is sampled as the mesh used for discretization of the PDE is refined. We illustrate the algorithm with an example where we estimate a diffusion coefficient in an elliptic equation from sparse and noisy pressure measurements. In the example, dimension\slash mesh-independence is achieved via Karhunen-Lo\`{e}ve expansions

Breakdown of self-similarity at the crests of large amplitude standing water waves

We study the limiting behavior of large-amplitude standing waves on deep water using high-resolution numerical simulations in double and quadruple precision. While periodic traveling waves approach Stokes's sharply crested extreme wave in an asymptotically self-similar manner, we find that standing waves behave differently. Instead of sharpening to a corner or cusp as previously conjectured, the crest tip develops a variety of oscillatory structures. This causes the bifurcation curve that parametrizes these waves to fragment into disjoint branches corresponding to the different oscillation patterns that occur. In many cases, a vertical jet of fluid pushes these structures upward, leading to wave profiles commonly seen in wave tank experiments. Thus, we observe a rich array of dynamic behavior at small length scales in a regime previously thought to be self-similar.Comment: 4 pages, 5 figures. Final version accepted for publicatio

Occupation Time for Classical and Quantum Walks

This is a personal tribute to Lance Littlejohn on the occasion of his 70th birthday. It is meant as a present to him for many years of friendship. It is not written in the “Satz-Beweis” style of Edmund Landau or even in the format of a standard mathematics paper. It is rather an invitation to a fairly new, largely unexplored, topic in the hope that Lance will read it some afternoon and enjoy it. If he cares about complete proofs he will have to wait a bit longer; we almost have them but not in time for this volume. We hope that the figures will convince him and other readers that the phenomena displayed here are interesting enough

Lithium motion in the anode material LiC6 as seen via time-domain 7Li NMR

Since the commercialization of rechargeable lithium-ion energy storage systems in the early 1990s, graphite intercalation compounds (GICs) have served as the number one negative electrode material in most of today's batteries. During charging the performance of a battery is closely tied with facile Li insertion into the graphite host structure. So far, only occasionally time-domain nuclear magnetic resonance (NMR) measurements have been reported to study Li self-diffusion parameters in GICs. Here, we used several NMR techniques to enlighten Li hopping motions from an atomic-scale point of view. Li self-diffusion in the stage-1 GIC LiC6 has been studied comparatively by temperature-variable spin-spin relaxation NMR as well as (rotating frame) spin-lattice relaxation NMR. The data collected yield information on both the relevant activation energies and jump rates, which can directly be transformed into Li self-diffusion coefficients. At room temperature the Li self-diffusion coefficient turns out to be 10−15m2s−1, thus, slightly lower than that for layer-structured cathode materials such as Lix≈0.7TiS2. © 2013 American Physical Society

Shape Optimization of Swimming Sheets

The swimming behavior of a flexible sheet which moves by propagating deformation waves along its body was first studied by G. I. Taylor in 1951. In addition to being of theoretical interest, this problem serves as a useful model of the locomotion of gastropods and various micro-organisms. Although the mechanics of swimming via wave propagation has been studied extensively, relatively little work has been done to define or describe optimal swimming by this mechanism.We carry out this objective for a sheet that is separated from a rigid substrate by a thin film of viscous Newtonian fluid. Using a lubrication approximation to model the dynamics, we derive the relevant Euler-Lagrange equations to optimize swimming speed and efficiency. The optimization equations are solved numerically using two different schemes: a limited memory BFGS method that uses cubic splines to represent the wave profile, and a multi-shooting Runge-Kutta approach that uses the Levenberg-Marquardt method to vary the parameters of the equations until the constraints are satisfied. The former approach is less efficient but generalizes nicely to the non-lubrication setting. For each optimization problem we obtain a one parameter family of solutions that becomes singular in a self-similar fashion as the parameter approaches a critical value. We explore the validity of the lubrication approximation near this singular limit by monitoring higher order corrections to the zeroth order theory and by comparing the results with finite element solutions of the full Stokes equations

Shape optimization of a sheet swimming over a thin liquid layer

Motivated by the propulsion mechanisms adopted by gastropods, annelids and other invertebrates, we consider shape optimization of a flexible sheet that moves by propagating deformation waves along its body. The self-propelled sheet is separated from a rigid substrate by a thin layer of viscous Newtonian fluid. We use a lubrication approximation to model the dynamics and derive the relevant Euler-Lagrange equations to simultaneously optimize swimming speed, efficiency and fluid loss. We find that as the parameters controlling these quantities approach critical values, the optimal solutions become singular in a self-similar fashion and sometimes leave the realm of validity of the lubrication model. We explore these singular limits by computing higher order corrections to the zeroth order theory and find that wave profiles that develop cusp-like singularities are appropriately penalized, yielding non-singular optimal solutions. These corrections are themselves validated by comparison with finite element solutions of the full Stokes equations, and, to the extent possible, using recent rigorous a-priori error bounds