Wednesdays at 4:15 PM in 380-X.
Least squares methods are of common use when one needs to approximate a function based on its noiseless or noisy observation at n scattered points by a simpler function chosen in an m dimensional space with m less than n. Depending on the context, these points may be randomly drawned according to some distribution, or deterministically selected by the user. In this talk, we shall focus on the analysis of the stability and approximation properties of least squares method. This analysis involves the relative size of m with respect to n as well as the spatial distribution of the samples. Applications with be given in the context of high dimensional sparse polynomial approximation to parametric-stochastic PDE's and approximation of acoustic fields by plane waves.
Singularly perturbed elliptic equations often give rise to solutions that are almost constant except for one or more localized large amplitude excursions, so-called spike layers. Here we consider the corresponding parabolic equations and show the existence of moving spike layer solutions – existing and retain that shape for all negative as well as positive time. This result is made possible by first proving and abstract result on the existence of a normally hyperbolic invariant manifold for a semiflow in Banach space when one only has an approximation of such.
Array imaging in a strongly scattering medium is limited because coherent signals recorded at the array and coming from a reflector to be imaged are weak and dominated by incoherent signals coming from multiple scattering by the medium. We will show how the use of cross correlation techniques and auxiliary, passive receiver arrays can help imaging reflectors buried in the medium. We will clarify the relation between time reversal, cross correlation techniques, coherent interferometry, and virtual source imaging.
Rayleigh-B\'enard convection is the buoyancy-driven flow of a fluid heated from below and cooled from above. Heat transport by convection an important physical process for applications in engineering, atmosphere and ocean science, and astrophysics, and it serves as a fundamental paradigm of modern nonlinear dynamics, pattern formation, chaos, and turbulence theory. Determining the transport properties of high Rayleigh number convection turbulent convection remains a challenge for experiment, simulation, theory, and analysis. In this talk, after a general survey of the theory and applications of thermal convection we describe recent results for rigorous upper limits on the vertical heat transport in two dimensions between stress-free isothermal boundaries derived from the Boussinesq approximation of the Navier-Stokes equations (Rayleigh's original model). These bounds on the heat transport scaling challenge one popular theoretical argument for the asymptotic high Rayleigh number convection. This is joint work with Jared Whitehead.
This talk will concern the analysis of elastic bodies which exhibit residual stress at free equilibria. Examples of such structures arise due to a range of causes: inhomogeneous growth, plastic deformation, swelling or shrinkage driven by solvent absorption. The mathematical description departs from the model of 3d non-Euclidean elastic energy, which measures the deficit of the metric induced by a deformation from a given Riemannian metric with non-zero curvature. I will discuss different adaptations of the classical variational nonlinear elasticity theory results to the present setting. These will include: scaling laws for thin films, dimension reduction, and thin shells with midsurface of arbitrary geometry.
In this talk we will present a study on the kernel functions of the Dirichlet-Neumann maps for dissipative systems in a half space. We start from the consideration of the Green’s function for an initial-boundary value problems for linear dissipative systems. With the fundamental solutions of the dissipative systems, one can reduce the initial-boundary value problems into boundary value problems so that the well-posedness of the system gives linear algebraic systems over the polynomials in the Fourier and Laplace variables for the Dirichlet-Neumann datum at the boundary, where Fourier variables are in the directions of boundary, and the Laplace is for the time variable. In order to invert the Dirichlet-Neumann map from the transformation variables to the space-time variables we introduce a path, which contains the spectral information of the systems, in the complex plan for the time Laplace variable. On this path, the Laplace-Fourier variables can be recombined, through the Cauchy’s complex contour integral, into a form resemble to that for a whole space problem. Thus, the classical results for the whole space problem can be used to obtain the pointwise spae-time structure for long wave components of the kernel function of the Dirichlet-Neumann map for points within a finite Mach region. We also apply direct energy estimates to yield the pointwise structure of the kernel functions in any high Mach number region. Finally, we have obtained exponentially sharp estimates for the kernel function in the space-time variables. For example, the kernel functions for both D’Alermbert wave equation with dissipation and a linearized compressible Navier-Stokes equation can be expressed explicitly in space-time variables with errors which decay exponentially in both space-time variables. This gives a globally quantitative and qualitative wave propagations at boundary.
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