Professor Dany Leviatan, School of Mathematics, Tel-Aviv University
Multivariate polynomial approximation in Convex sets, and applications to image compression
We shall begin with a discussion of two fundamental estimates in the approximation
by multivariate polynomials in convex domains in Rd, the Bramble-Hilbert lemma and
the Whitney inequality.
The Bramble-Hilbert lemma is frequently applied in the analysis of Finite Element
Methods (FEM) used for numerical solutions of PDEs. However, this classical estimate
depends on the geometry of the domain and may ‘blow-up’ for simple examples such
as a sequence of triangles of equivalent diameter that become thinner and thinner.
Thus, in FEM applications one usually requires that the mesh has ‘quasi-uniform’
geometry. This assumption is too restrictive when one tries to obtain estimates of
nonlinear approximation methods that use piecewise polynomials. We show that it is
possible to obtain estimates where the constant is independent of the geometry of the
domain. We will apply it first to obtain the Whitney inequality for convex domains
(with constants that are independent of the geometry of the domain).
Our results allow us to describe a nonlinear algorithm for image compression making
use of a new family of what we call geometric wavelets. We will describe the
algorithm give some estimates on its rate of approximation to the image, and compare
the compression by diadic wavelets and by geometric wavelets of some familiar images.
If time allows we will apply the above estimates to characterization of nonlinear
multivariate approximation by piecewise polynomials on families of nested triangulations
of Rd into simplices. Again it is essential here that we do not have to pay attention
to how slim the simplices might become.
3:30 PM, Math 3206
Joint Math/CSCAMM Seminar
TUESDAY, February 1st
at 3:30PM Math Colloquium Room #3206
Professor Donald Estep, Department of Mathematics, Colorado State University
Fast and Reliable Methods for Determining the Evolution of Uncertain
Parameters in Differential Equations
A very common problem in science and engineering is the
determination of the effects of uncertainty or variation in
parameters and data on the output of a nonlinear operator. For
example, such variations may describe the effect of experimental
error or may arise as part of a sensitivity analysis of the model.
The Monte-Carlo Method is a widely used tool for understanding such
effects that employs random sampling of the input space in order to
produce a pointwise representation of the output. It is a robust and
easily implemented tool. Unfortunately, it generally requires
sampling the operator very many times at a significant cost.
Moreover, it provides no robust measure of the error of information
computed from a particular representation. In this paper, we present
an alternative approach for ascertaining the effects of variations
and uncertainty in parameters in a differential equation that is
based on techniques borrowed from a posteriori error analysis
for finite element methods. The generalized Green's function is used
to describe how variation propagates into the solution around
localized points in the parameter space. This information can be
used either to create a higher order method or produce an error
estimate for information computed from a given representation. In
the latter case, this provides the basis for adaptive sampling. Both
the higher order method and the adaptive sampling methods are
generally orders of magnitude faster than Monte-Carlo methods in a
variety of situations.
TUESDAY, February 8th
at 3:30PM Math Colloquium Room #3206
Professor Alex Mahalov, Arizona State University
Global Regularity of the 3D Navier-Stokes Equations
with Uniformly Large Initial Vorticity
We prove existence on infinite time intervals of regular solutions
to the classical incompressible 3D Navier-Stokes Equations for fully
three-dimensional periodic and almost periodic initial data characterized
by uniformly large vorticity in R^3 and in bounded cylindrical
domains; smoothness assumptions for initial data are the same as
in local existence theorems. There are no conditional
assumptions on the properties of solutions at later times,
nor are the global solutions close to any 2D manifold.
The global existence is proven using techniques of fast
singular oscillating limits and the Littlewood-Paley
dyadic decomposition. The approach is based on the
computation of singular limits of rapidly oscillating
operators and cancellation of oscillations in the nonlinear
interactions for the vorticity field. With nonlinear
averaging methods in the context of almost periodic
functions, we obtain fully 3D limit resonant Navier-Stokes
equations. We establish the global regularity of the latter
without any restriction on the size of 3D initial data.
With strong convergence theorems, we bootstrap this
into the global regularity of the 3D Navier-Stokes
Equations with uniformly large initial vorticity.
We review applications of our mathematical techniques
to numerical analysis of highly oscillatory PDE's arising
in geophysical fluid dynamics. Global regularity
of the 3D Navier-Stokes Equations of Geophysics is proven
for all domain aspect ratios and all small Froude and
Joint Meteorology/CSCAMM Seminar
THURSDAY, February 10th
at 3:30PM Auditorium (Rm. 2400),
2nd floor of the New Wing of the CSS Bldg
Professor Alex Mahalov, Arizona State University
Characterization and High Resolution Numerical Simulations
of Stratospheric Clear Air and Optical Turbulence
We present high resolution (1024 vertical levels)
numerical simulations on massively parallel architectures
of stratospheric optical and clear air turbulence (CAT).
The stratospheric CAT for altitudes from 10 to 30 km is
characterized by patchy high frequency fluctuations in the
stratospheric wind fields and long-lived energetic
vortex structures with several hundred meters scale.
The main mechanism of formation of stratospheric
anisotropic CAT is wave-induced windshears in synergy
with saturated inertio-gravity wave fields; lateral
directional shear induced by gravity waves is a key
instability mechanism in layers as thin as a few
hundred meters. From the fundamental fluid dynamics
perspective, this is related to 3D instabilities
and turbulent dynamics of helical velocity profiles
(U(z),V(z),0) embedded in a vertically variable backgroud
stratification N(z); the conventional Ri_g=0.25 criterion
does not hold for such flows for which lateral shear
is the key instability mechanism. The structute of the
turbulent velocity, temperature and vorticity fields
are analyzed, and are compared with existing observational
and numerical studies of stably stratified shear flows
in the atmosphere.
Scientific Computing in Chemistry and Materials Science:
Algorithms for Direct Dynamics Simulations and the Non-Equilibrium Dynamics of Sliding Surfaces
Scientific computing is an important approach
for studying the atomistic dynamics of chemical
reactions and of a broad range of problems in
materials science. Atomic-level simulations
require a potential energy surface for the
system of interest, and recently it has become
possible to obtain this surface and its gradient
directly from an electronic structure theory
calculation. Such simulations, referred to as
direct dynamics, require substantial
computational resources and there is a need to
enhance numerical integration algorithms used in
the simulations to make the application of this
computational approach more practical. The
details of these simulations, the enhancements
which are needed, and an example chemical
reaction application will be described. An
important problem for nano-materials is the
friction at the interface of sliding surfaces.
An atomic-level simulation of the friction at
the interface of sliding hydroxylated alumina
surfaces will be discussed. Significant
non-Boltzmann characteristics are found in the
heat flow from the sliding interface.
Professor Eric Vanden-Eijnden, Courant Institute of Mathematical Sciences
Metastability in complex systems
The evolution of many complex system can be represented as a navigation
over some energy landscape in the presence of small noise. The system
stays confined for a long time within a metastable basin corresponding
to a region of rather low energy, then suddenly hops over an energy
barrier to another basin, etc. This is the mechanism by which e.g.
conformation changes in molecules, chemical reactions, or phase
transitions arise. Direct numerical simulations fail in these
situations because of the huge separation between the time-scale that
needs to be resolved in the simulations, and the time-scale over which
the transitions occur. For systems with relatively smooth energy
landscapes, the corresponding effective dynamics can be described
within the framework of large deviation theory which provides the most
probable transition path between the metastable basins and the rates of
the transitions. When the energy landscape is non-smooth and entropic
effects are important, large deviation theory becomes inadequate. I
will describe generalization of the theory to these situations. I will
also describe numerical techniques that can be developed based on such
framework to explicitly obtain the transition pathways, the free
energy, and the rates. These techniques will be illustrated on examples
arising from materials science, molecular dynamics, and biology.
A new transform for improved lossy compression of color images(*)
We introduce the eidochromatic
transform as a tool for improved lossy coding of color images. Many
current image-coding formats (such as JPEG 2000) utilize both a
color-component transform and a wavelet or other spatial
transform (relating values of a single image component at proximate,
but different image locations).
The eidochromatic transform further reduces redundancy by relating image values simultaneously across color components and in the two spatial dimensions.
Our approach is to introduce an additional transform step following the
color-component and spatial transforms. In tests, this step reduced
the overall static entropy of the chrominance components of quantized
transformed images by up to 40% or more. Combined with JPEG 2000's
modeling and coding method, the eidochromatic transform was found to
reduce the size of lossily coded color images by up to 27% overall.
Professor Giovanni Russo,
Department of Mathematics, University of Catania
Computation of Strained Epitaxial Growth in Three Dimensions by
Kinetic Monte Carlo
A numerical method for computation of heteroepitaxial
growth in the presence of strain is presented.
The model used is based on a solid-on-solid model with a cubic
lattice. Elastic effects are incorporated using a ball and spring
type model. The growing film is evolved using Kinetic Monte Carlo (KMC)
and it is assumed that the film is in mechanical equilibrium.
The strain field in the substrate is computed by an exact
solution which is efficiently evaluated using the fast Fourier transform.
The strain field in the growing film is computed directly. The resulting
coupled system is solved iteratively using the conjugate gradient method.
Finally we introduce various approximations in the implementation of KMC
to improve the computation speed. Numerical results show that layer-by-layer
growth is unstable if the misfit is large enough resulting in the formation
of three dimensional islands.
Further development using multigrid approach will be addressed.
Dr. Vladimir Krasnopolsky, ESSIC, University of Maryland and NCEP/NOAA
in collaboration with
Michael Fox-Rabinovitz and Dmitry Chalikov,
ESSIC, University of Maryland
Application of Neural Network Techniques for Approximating Complex Multidimensional Mappings: NN Emulations of Time Consuming Components in Numerical Models
Neural Network (NN) techniques provide an effective tool for fast and accurate approximations of complex multidimensional mappings. This approach has been applied to develop accurate and fast NN emulations of time consuming components in numerical climate and weather prediction models. Calculation of model physics is the “bottleneck” of any climate and weather prediction model. Model physics calculations take from 70% to 90+% of the total calculation time. Model physics parameterizations can be considered as continuous or almost continuous mappings and accurately emulated by NNs, which are 102 - 105 times faster than the original parameterizations of model physics.
Efficient NN emulations are developed and tested/validated under the condition of preserving the quality and accuracy of the original parameterizations. In addition to fast and accurate emulation of the original parameterizations, NN also provides the entire Jacobian for a very little computational cost.
The NN emulations for the National Center for Atmospheric Research (NCAR) Community Atmospheric Model (CAM) radiation parameterizations are presented and discussed as examples of the developed approach. High accuracy and greatly improved, as compared with the original parameterization, computational efficiency of the NN emulations are demonstrated (they are about 80 times faster). The results of climate simulations obtained for parallel runs of NCAR CAM with the original radiation parameterizations and with their NN emulations are discussed. Both simulations show very close results. The major properties of simulated climate are well preserved.
Developing NN emulations for other more complex model physics components like the nonlinear wave-wave interaction in the ocean wind-wave model is discussed. The developed NN emulation is 105 times faster than the original parameterization in this case.
These successful experiments show the potential of using adaptive machine learning techniques for emulating complex and time consuming components of numerical models.
Dr. Vladislav Panferov, Department of Mathematics and Statistics, McMaster University
Regular small data solutions of the Boltzmann equation
in one space dimension
In this talk I will present a recent work about propagation
of L∞ bounds for spatially one-dimensional
(plane-wave) solutions of the nonlinear Boltzmann
equation. This problem is relevant for the study of
regularity of general weak solutions of the Boltzmann
equation, as provided by the DiPerna-Lions theory. The
result is based on a remarkable regulatization property
of the averaged ``gain'' term. Essentially, we find that,
under certain truncations, propagation of Lp, p>1 bounds
of the averages, implies the propagation of the L∞
bounds of the solutions. It is curious that the
logarithmic-type condition given by the bound on the
Boltzmann entropy functional appears as a sort of a critical
case in this estimate. As a consequence of this approach we
are able to show the global in time existence (and uniqueness)
of small data regular solutions in a bounded interval for
the Boltzmann equation with cut-off hard potentials, subject
to a large velocity truncation in the collision term. The
important case of large data prompts further investigation.
The Multiscale Structure of Non-Differentiable Image Appearance Manifolds
The images generated by varying the underlying articulation parameters
of an object (pose, attitude, light source position, and so on) can be
viewed as points on a low-dimensional "image appearance manifold"
(IAM) in a high-dimensional ambient space. In this talk, we will
expand on the observation that typical IAMs are not differentiable, in
particular if the images contain sharp edges. However, all is not
lost, since IAMs have an intrinsic multiscale geometric structure. In
fact, each IAM has a family of approximate tangent spaces, each one
good at a certain resolution. We will focus on the particular inverse
problem of estimating, from a given image on or near an IAM, the
underlying parameters that produced it. Putting the multiscale
structural aspect to work, we develop a new algorithm for
high-accuracy parameter estimation based on a coarse-to-fine Newton
iteration through the family of approximate tangent spaces. This
algorithm is reminiscent of recently proposed algorithms for
multiscale image registration and super-resolution. This is joint work with Michael Wakin, Hyeokho Choi, David Donoho, and
Professor Patrick Wolfe, Division of Engineering and Applied Sciences, Harvard University
Stochastic Computation and Applications to Statistical Signal Processing
Many problems arising in science and engineering are effectively ones of statistical inference,
and in all but the simplest cases the associated models may not admit analytical solutions.
In this talk I will describe simulation-based Monte Carlo methods for inference, in particular two
important classes of algorithms for stochastic computation: a batch methodology known as Markov chain
Monte Carlo and an on-line one termed sequential Monte Carlo. Many interpretations are possible,
but I shall frame my discussion in terms of the Bayesian paradigm, whereby all inference stems
from a description of the (posterior) probability distribution associated with a given model after
having observed the data in question. I will illustrate these simulation methodologies with examples
of my own research into statistical audio signal processing, in which case they are used to obtain
point estimates of salient parameters. This application area is not only interesting and important
in its own right, but also provides a convenient test bed for more generally applicable techniques of
time series modeling.
Nematic polymers are high aspect ratio macromolecules, either rods or platelets. They are utilized in high performance materials for a diversity of properties including mechanical, thermal, barrier, and electrical. Such macromolecule ensembles in solutions, and similar geometric colloidal suspensions, exhibit remarkable response to shear-dominated flow. Bulk phases undergo an isotropic-nematic first order phase transition. Weak flows such as shear drive this transition to create a myriad of responses, including steady and unsteady bulk modes. The result for these nano-composite films is a combination of anisotropy and heterogeneity---in the performance features of materials, which is somehow dictated by the morphology of the high aspect ratio macromolecular ensemble. This "map" between composition, flow conditions, and ultimate material properties is the basis of the lecture. Progress to date, and the challenges ahead to theory and computation, will be highlighted.