Graduate Student Pizza Seminar
Abstract: Much of uncertainty quantification to date has focused on determining the effect of variables modeled probabilistically, and with a known distribution, on some physical or engineering system. We develop methods to obtain information on the system when the distributions of some variables are known exactly, others are known only approximately, and perhaps others are not modeled as random variables at all. The main tool used is the duality between risk-sensitive integrals and relative entropy, and we obtain explicit bounds on standard performance measures (variances, exceedance probabilities) over families of distributions whose distance from a nominal distribution is measured by relative entropy. The evaluation of the risk-sensitive expectations is based on polynomial chaos expansions, which help keep the computational aspects tractable.
Center for Fluid Mechanics, Division of Applied Mathematics Fluids, Thermal and Chemical Processes Group, School of Engineering Joint Seminar Series
Abstract: The primary cilium is found for all non-dividing mammalian cells. Since its discovery a century ago, only recently has more understanding of the biological role of primary cilia been gained. In this work slender-body formulation is utilized to describe the dynamics of the primary cilium, modeled as an elastic filament attached to a solid wall or membrane. Comparison with the experimental data will be provided. Coupling between the filament/membrane system and the mechanosensitive channel (MscL) show how the primary cilium functions as a probe of the extracellular flow.
Abstract: We first give a brief overview of a series of results on the construction of Markov semigroup for the 3D Navier-Stokes equation. Since uniqueness is an open problem, the construction is not obvious. We show that under appropriate conditions on the non degeneracy of the noise, it is possible to do this in a constructive way and obtain a transition semigroup which is (and even strong) Feller. This is the result of joint works with G. Da Prato and C. Odasso. Exponential convergence to equilibrium follows from an argument due to C. Odasso. Then we consider physically more realistic noises which do not satisfy the above non degeneracy condition and prove that the projections of the distribution onto finite dimensional spaces of any (weak) solution have a density with respect to the Lebesgue measure. Since Malliavin calculus does not seem to be applicable, we use and extend a recent idea due to N. Fournier and J. Printems. This is a joint work with M. Romito.
Boston/Brown PDE Seminar
Abstract: I will look at two- and three-dimensional pattern formation in a reverse microemulsion consisting of nanometer diameter droplets of water containing the reactants of the Belousov-Zhabotinsky oscillating chemical reaction dispersed in oil. This system allows the physical structure (size, spacing) of the droplets and their chemical composition to be controlled independently, enabling one to generate a remarkable variety of stationary and moving patterns, including Turing structures, ordinary and antispirals, packet waves and spatiotemporal chaos. I will show examples of patterns and discuss two alternative mechanisms by which they may arise, one involving diffusion of different species at very different rates, the other via cross diffusion, whereby gradients in the concentration of one species influence the rate of diffusion of another species. I will also discuss pattern formation in one and two-dimensional arrays of micrometer diameter aqueous BZ droplets in oil prepared using microfluidic techniques.
Boston/Brown PDE Seminar
Abstract: A posteriori error estimation and adaptive error control based on the formulation and solution of an adjoint problem are well established for problems involving a single type of physics. For systems that exhibit sufficiently complicated physics or a range of scales so that they severely challenge standard solution techniques, operator decomposition provides an attractive way to decompose the problem into components with relatively simple physics or into behaviors that occur over a modest range of scales. Operator decomposition creates additional challenges for adaptive error control since errors in one component may limit the accuracy in another component, yet a global adjoint solution is typically not available. In this talk, I will describe an a posteriori analysis of operator decomposition methods for coupled elliptic systems. This analysis takes into account the accuracy with which individual components are solved as well as the global effects of operator decomposition. The estimates obtained provide the means for adaptive error control. Extensions of these ideas that are currently under investigation include coupled elliptic-parabolic and coupled parabolic systems arising in studies of cardiac physiology. Time-dependent problems present additional challenges and I will describe a one approach we have developed which we call "block adaptivity".
Scientific Computing Seminar
Abstract: Stability and convergence analysis for fully discrete pseudo spectral numerical schemes to nonlinear PDEs are presented in this talk, such as viscous Burgers' equation and incompressible Navier-Stokes equations. Related applications to incompressible Euler equation and quasi-geostrophic equation will also be addressed, in both 2-D and 3-D, for smooth and vortex sheet initial data. In addition, high order time stepping schemes, including Adams Bashforth-Adams Moulton multi-step schemes up to fourth order accuracy and high order explicit SSP schemes, will be explored in detail. Unconditional stability is established for the implicit time stepping algorithms.
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