Brown University
Joint Materials/Solid Mechanics Seminar Series
Ecole Normale Superieure | |
Abstract: This talk emphasizes on the interplay between mechanical stresses and growth in pattern generation in biology. Two questions are investigated. How does mechanical stresses drive growth? How does growth induced stresses generate patterns? In the first part, I consider single cells having walls (such as plant cells or bacteria). They are modeled as thin elastic shells inflated by the osmotic pressure. Growth is accounted for by analogy with plastic deformations. Two scaling laws are derived and are in agreement with compiled data from the biology litterature. Possible cell shapes are computed accordingly. In the second part, I show results concerning thin elastic plates having a larger natural length at the edge (joint work with Basile Audoly, Laboratoire de Modelisation en Mecanique, Paris VI). This problem is motivated by the tearing of plastic sheets and by the shape of plant leaf edges, where a cascade of wrinkles with smaller and smaller wavelengths can be observed. We have reproduced this cascade numerically and analysed its basic features.
Brown Applied Mathematics Pattern Theory and Vision Seminar
Abstract: The problem of computing marginal statistics of probability distributions defined over graphs with cycles, occurs in many fields: error-correcting coding, machine learning, communication theory, computer vision, and statistical physics. Because exact computations are often difficult, approximate algorithms based on local message passing have been developed. One such algorithm is the sum-product (loopy belief propagation) algorithm. In this talk we present a new framework for analyzing the sum-product algorithm. Our analysis relies on the computation tree which represents an unwrapping of the original graph with respect to the sum-product algorithm. We show that the convergence and quality of sum-product algorithm is related to the question of uniqueness of the Gibbs measure defined on the infinite computation tree. The stationary points of the Bethe variational problem correspond to different Markov chains defined on this tree. We present easily testable conditions that insure convergence and we compute rates of convergence. This framework gives new insights into the mechanics of the sum-product algorithm.
Brown Analysis Seminar
Scientific Computing Seminar
Abstract: I will present and discuss a framework for computer- aided multiscale analysis, which enables models at a "fine" (microscopic/stochastic) level of description to perform modeling tasks at a "coarse" (macroscopic, systems) level.
These macroscopic modeling tasks, yielding information over long time and large space scales, are accomplished through appropriately initialized calls to the microscopic simulator for only short times and small spatial domains: "patches" in macroscopic space-time.
Traditional modeling approaches first involve the derivation of macroscopic evolution equations (balances closed through constitutive relations). An arsenal of analytical and numerical techniques for the efficient solution of such evolution equations (usually, Partial Differential Equations, PDEs) is then brought to bear on the problem. Our equation-free (EF) approach, introduced in PNAS (2000) when successful, can bypass the derivation of the macroscopic evolution equations {it when these equations conceptually exist but are not available in closed form).
We discuss how the mathematics-assisted development of a computational superstructure may enable alternative descriptions of the problem physics (e.g. Lattice Boltzmann (LB), kinetic Monte Carlo (KMC) or Molecular Dynamics (MD) microscopic simulators, executed over relatively short time and space scales) to perform systems level tasks (integration over relatively large time and space scales, "coarse" bifurcation analysis, but also optimization and control tasks) directly.
In effect, the procedure constitutes a systems identification based, "closure on demand" computational toolkit, bridging microscopic/stochastic simulation with traditional continuum scientific computation and numerical analysis. We illustrate these "numerical enabling technology" ideas through examples from chemical kinetics (LB, KMC), rheology (Brownian Dynamics), homogenization and the computation of "coarsely self-similar" solutions, and discuss various features, limitations and potential extensions of the approach.
PDE Seminar
Woods Hole Oceanographic Institution, Woods Hole MA, 02543, jpedlosky@whoi.edu | |
Abstract: The baroclinic instability of a zonal current on the beta plane is studied in the context of the two-layer model when the shear of the basic current is a periodic function of time. The basic shear is contained in a zonal channel and is independent of the meridional direction. The instability properties are studied in the neighborhood of the classical, steady shear, threshold for marginal stability. It is shown that the linear problem shares common features with the behavior of the well-known Mathieu equation. That is, the oscillatory nature of the shear tends to stabilize an otherwise unstable current while on the contrary, the oscillation is able to destabilize a current whose time-averaged shear is stable. Indeed, this parametric instability can destabilize a flow that at every instant possesses a shear that is sub-critical with respect to the standard stability threshold. This is a new source of growing disturbances. Similar behavior for the linear problem is found at parameter settings well below the stability threshold as well. The nonlinear problem is studied in the same near neighborhood of the marginal curve. When the time averaged flow is unstable the presence of the oscillation in the shear produces both periodic finite amplitude motions and aperiodic behavior. Generally speaking, the aperiodic behavior appears when the amplitude of the oscillating shear exceeds a critical value depending on frequency and dissipation. When the time averaged flow is stable, i.e. sub-critical, the presence of finite amplitude aperiodic motion occurs when the amplitude of the oscillating part of the shear is large enough to lift the flow into the unstable domain for at least part of the cycle of oscillation. A particularly interesting phenomenon occurs when the time averaged flow is stable and the oscillating part is too small to ever render the flow unstable according to the standard criteria. Nevertheless, in this regime parametric instability occurs for ranges of frequency that expand as the amplitude of the oscillating shear increases. The amplitude of the resulting unstable wave is a function of frequency and the magnitude of the oscillating shear. For some ranges of shear amplitude and oscillation frequency there exist multiple solutions. It is suggested that the nature of the response of the finite amplitude behavior of the baroclinic waves in the presence of the oscillating mean flow may be indicative of the role of seasonal variability in shaping eddy activity in both the atmosphere and the ocean.
Department of Mathematics Colloquium
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