Lefschetz Center for Dynamical Systems Seminar
Abstract: We present a bifurcation theory for dispersion managed solitons. After applying the lens transform to the dispersion-managed nonlinear Schroedinger equation (DMNLS) we derive a Schroedinger-type equation with additional quadratic potential. By analyzing the so-called nonlinear TM-equations and averaging of the lens transformed DMNLS we arrive at qualitatively different equations depending on the sign of the residual dispersion and the value of the local dispersion. In the case of positive residual dispersion and moderate values of the local dispersion the quadratic potential is of trapping type and we are able to apply the global bifurcation theorem of Rabinowitz to the averaged equation.
Center for Fluid Mechanics Seminar
Department Models for Aerodynamics and Energetics, France | |
Abstract: This numerical study is related to the resolution of the Navier-Stokes equations in order to predict the onset of the aerodynamic buffeting phenomenon over supercritical airfoils; the influence of several turbulence models on such a prediction has been considered. This flow instability results from Shock Induced Oscillations (SIO) caused by boundary layer separation and shock wave interaction. Two complementary strategies have been applied: the first one uses a Wall Law approach involving a course grid and a Global Time Stepping (referred to as "WL approach") and the second one considers standard approach coupled with a Dual Time Stepping method involving a thin grid (referred to as "DTS approach"). Comparisons have been performed with the RA16SC1 airfoil database (tested in the ONERA S3MA wind tunnel - M = 0.732 - Re_{c} = 4.2 10^{6}). Physical phenomena evidenced on this supercritical airfoil during aerodynamic buffeting are described in detail. Then, an argued scenario on the origin of the two-dimensional aerodynamic buffeting is proposed.
Division of Engineering and Center for Fluid Mechanics Joint Seminar
Abstract Among various control methodologies for turbulence, active feedback control scheme attracts much attention because of its large control effect with small power input. However, the physical mechanism how the control input interacts with the physical mechanism of turbulence remains to be resolved. In my talk, our recent attempts to explore the dynamics of turbulence in wave space are discussed. It is well known that fine grids or screens can reduce free stream turbulence. But, the conventional notion of local energy transfer can not explain this phenomena, because the scale of the disturbances is much smaller than that of the free stream. In the present study, direct numerical simulations of isotropic turbulence were made in order to investigate additional small-scale turbulence. Detailed characteristics of the energy spectral transfer were examined. It is found that nonlocal energy cascade through isosceles triads toward the wave-number range of the energy input is markedly enhanced, thus the turbulent intensity is decreased faster than a flow field with no energy input. The optimum wave-number for turbulence reduction depends only on the characteristic wave-number of the energy-containing range, once the magnitude of the energy input is given. For wall turbulence, direct numerical simulation of turbulent channel flow at Ret = 110~650 was made. Karhunen-Loeve (KL) decomposition of turbulent fluctuations was carried out in order to examine energy transfer in wave space. It is found that the KL modes at 15 < y+ < 30, which correspond to longitudinal vortices and near-wall streaky structures, play a dominant role in the production of turbulence and wall shear stress at Ret = 110. At Ret = 300, however, the KL modes at 30 < y+ < 75 also make appreciable contribution to the wall shear stress generation. The regeneration mechanism of the near-wall vortices is related to the nonlinear interaction between the KL modes at 15 < y+ < 30 and those at 30 < y+ < 75.
Brown Applied Mathematics Pattern Theory and Vision Seminar
Abstract: We give a development of the theory of lossy data compression from the point of view of statistics. This is partly motivated by the enormous success of the statistical approach in lossless compression. A precise characterization of the fundamental limits of compression performance is given, for arbitrary data sources and with respect to general distortion measures. The emphasis is on non- asymptotic results and results that hold with high probability (and not just on the average).
The starting point for this development is the observation that there is a precise correspondence between compression algorithms and probability distributions (in analogy with the Kraft inequality in lossless compression). This leads us to formulate a version of the celebrated Minimum Description Length (or MDL) principle for lossy data compression. We discuss the consequences of the lossy MDL principle, and we explain how it leads to practical design lessons for vector-quantizer design.
Brown Analysis Seminar
Scientific Computing Seminar
Abstract: Toward our goal of modeling strong earthquakes in seismic regions, we are interested in determining mechanical properties of sedimentary basins (such as the greater Los Angeles Basin) from seismograms of past earthquakes.
As an intermediate step, we consider the inverse wave propagation problem of determining the material properties of a heterogeneous acoustic medium, given a source and observations at receiver locations on its boundary. The inverse problem is formulated as a PDE-constrained optimization problem, in which the objective function is an L2 norm misfit between model and observations, the constraint is the acoustic wave equation with appropriate initial and boundary conditions, and the inversion variable is the density field. Both Tikhonov (H1) and total variation regularization are used to render the inverse problem well-posed.
The Euler-Legrange equations representing first-order optimality yield a coupled three-field system composed of the (forward-in-time) acoustic wave equation, the (backward-in-time) adjoint acoustic wave equation, and the integro-differential inversion equation. Newton solution of an appropriate spatio-temporal discretization of this system for state variables, adjoint variables, and the material field yields a linear system with a large indefinite coefficient matrix. Due to the size (proportional to the product of the number of grid points and time steps) and lack of structure of this system, we invoke a block elimination that produces a Schur complement system in the reduced space of the material field.
We solve the Schur complement system using conjugate gradients; these inner iterations are embedded within an inexact Newton outer iteration. The Schur complement is too large and expensive to be formed explicitly; instead, each CG matrix-vector product requires the solution of a forward and an adjoint wave propagation problem. CG converges rapidly for the Tikhonov- regularized Schur complement, and no preconditioning is needed. On the other hand, TV-regularized Schur complement systems are somewhat slower to converge, and we are currently investigating various strategies for their preconditioning.
To address the difficulties posed by numerous local minima of the objective function, we employ a multiscale algorithm that sweeps through the source frequencies, determining successively higher spectral components of the material field, on successively finer grids. Successive material fields remain in the (increasingly narrowing) basin of attraction of the global minimum. Numerical experiments on synthetic 3D inverse wave propagation problems with up to a million variables on up to 128 Cray T3E processors demonstrate high parallel and algorithmic scalability, and the ability to capture global minima and recover accurate inversion fields.
The work is part of the Quake project (www.cs.cmu.edu/~quake), which is a collaboration with J. Bielak and D. O'Hallaron at CMU, S. Day and H. Magistrale at San Diego State, and J. Shewchuk at UC-Berkeley.
PDE Seminar
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