Professor Uri Keich, Computer Science
Department, Cornell University
Estimating the
Significance of Sequence Motifs
Efficient and accurate statistical
significance evaluation is an essential
requirement of motif-finding tools. One such
widely used significance criterion is the
E-value of the motif's information content or
entropy score. Current computation schemes used
in popular motif-finding programs can
unwittingly provide poor approximations. We
present an approach to a fast and reliable
estimation of this E-value that can be applied
more generally.
Unfortunately, this improvement did not
completely solve the motif significance
estimation problem. In particular, we more
recently found that relying on these E-values
when searching for relatively weak motifs can
lead to undesirable results. This motivated our
design of a novel, parametric approach for
analyzing the significance of sequence motifs.
Dr. Siddhartha Mishra, Centre of
Mathematics for Applications, University of
Oslo
Finite Volume Methods
for Wave Propagation in Stratified
Magneto-atmospheres
We consider equations of
Magneto-Hydro dynamics (MHD) along with
gravitational source terms which serve as a
model for the propagation of waves in the solar
atmosphere. Numerical methods for this model
involve the design of suitable finite volume
schemes. Numerical issues addressed include
handling the divergence constraint in the
equations by a suitable upwind discretization of
the Godunov-Powell source term.
Approximate Riemann solvers of the HLL-type
based on a splitting between the fluid and
magnetic field parts of the equation has been
developed and will be compared with other
existing solvers. Non-reflecting characteristic
type boundary conditions have been designed to
treat wave propagation through the artificial
top boundary. All the above issues will be
described and a series of numerical examples
will be presented during the lecture.
Free Boundary Problems:
Viscous Fluids Interacting with Elastic Shells
The moving boundary problem in
continuum mechanics is one of the most beautiful
problems in nature. It appears when considering
the motion of two types of materials in liquid
crystals, elastic solids, porous media, and etc.
This seminar talk is concerned with a moving
boundary value problem consisting of a viscous
incompressible fluid moving and interacting with
a elastic shell. Two types of elastic shells, a
fluid-shell type of shell which is usually used
to model bio-membrane, and a solid-shell type of
shell which is widely used in solid mechanics,
will be considered here. We will learn how to
model these problems in terms of the PDEs, and
the fundamental difficulties in both theoretic
and numerical points of view. Finally, a brief
presentation of the idea of solving the problem
and a well-posedness result will be discussed.
Professor Luminita Vese, Department of
Mathematics, UCLA
Minimization Models and
Algorithms for Image Segmentation, Image
Decomposition, and Texture Modeling
This talk is devoted to some
computational methods for image segmentation,
cartoon-texture separation and texture modeling,
in a variational and partial differential
equations approach. A multilayer curve evolution
model will be presented for image segmentation
that is well adapted for volumetric MRI brain
data and is more efficient than previous models.
Also, computational methods for modeling
oscillations and image decomposition into
cartoon and texture will be presented. Images
and in particular oscillatory patterns are
represented by more refined texture norms, such
as dual norms or generalized functions. The
difficulty arising in the minimization of such
models will be addressed, and theoretical and
experimental results will be presented.
Professor Richard James, Aerospace
Engineering and Mechanics, University of
Minnesota
Lessons on Structure
from the Structure of Viruses
As the most primitive organisms,
occupying the gray area between the living and
nonliving, viruses are the least complex
biological system. One can begin to think about
them in a quantitative way, while still being at
some level faithful to biochemical processes. We
make some observations about their structure,
formalizing in mathematical terms some
rules-of-construction discovered by Watson and
Crick and Caspar and Klug. We call the resulting
structures objective structures. It is then seen
that objective structures include many of the
most important structures studied in science
today: carbon
nanotubes, the capsids, necks, tails and other
parts of many viruses, the cilia of some
bacteria, DNA octahedra, buckyballs, actin and
collagen and many other common proteins, and
certain severely bent and twisted beams. The
rules defining them relate to the basic
invariance group of quantum mechanics. We give
simple formulas for all such
structures. Some of the nonperiodic structures
revealed by the formulas exhibit beautifully
subtle relations of symmetry. This common
mathematical structure paves the way toward many
interesting calculations for such structures:
simplified schemes for exact molecular dynamics
of such structures (objective MD), phase
transformations between them (as in
bacteriophage T4), new x-ray methods for direct
determination of structure not relying on
crystallization, and a theory of their growth by
self-assembly.
Dr. Amit Singer, Department of
Mathematics, Yale University
Geometric Inversion
Problems with Case Studies in Structural Biology and
Sensor Networks
In many applications, the main goal
is to obtain a global low dimensional representation
of the data, given some local noisy geometric
constraints. In this talk we will show how all
(seemingly unrelated) problems listed below can be
solved by constructing suitable operators on their
data. Those operators are different from the graph
Laplacian, and can be regarded as its extension. The
solutions involve only the computation of a few
eigenvectors of sparse matrices corresponding to the
data operators.
Cryo Electron Microscopy for protein structuring:
reconstructing the three-dimensional structure of a
molecule from projection images taken at random
unknown orientations (unlike classical tomography,
where orientations are known).
NMR spectroscopy for protein structuring: finding
the global positioning of all hydrogen atoms in a
molecule from their local distances. Distances
between neighboring hydrogen atoms are estimated
from the spectral lines corresponding to the short
ranged spin-spin interaction.
Sensor networks: finding the global positioning from
local distances.
Detecting the slow manifold in stochastic chemical
reactions:
finding the slow coordinates in large multi-scaled
dynamical systems
from experimental or simulation data.
Non linear independent component analysis: de-mixing
statistically independent processes that were mixed
by an unknown smooth non-linear functions.
Joint work with Ronald Coifman, Yoel Shkolnisky
and Fred Sigworth.
October 12
3.00PM
3206 Math Bldg
note special
place & time
Joint Seminar with Mathematics Department.
Professor Nira Dyn, School of Mathematical
Sciences, Tel-Aviv University
Subdivision Schemes for the
Refinement of Geometric Objects
Subdivision schemes are efficient
computational methods for the design,
representation and approximation of surfaces of
arbitrary topology in 3D.
Subdivision schemes generate curves/surfaces from
discrete data by repeated
refinements. This talk is mainly concerned with
”classical” schemes refining
control points, and their applications in geometric
modeling. The relation of
subdivision schemes to the construction of wavelets
is also discussed. The last part of the talk reviews
subdivision schemes refining other objects, such as
compact sets and nets of curves. Examples of various
schemes are presented
Dr. Istvan Szunyogh, Institute for
Physical Science and Technology, University of
Maryland
Development of a Data
Assimilation System to Estimate the State of
Large Spatio-temporally Chaotic Systems
This talk is an overview of the
research that led to the current algorithmic and
code design of the Local Ensemble Transform
Kalman Filter (LETKF) data assimilation system.
The LETKF estimates the time evolving state of
the system based on observations and a numerical
model of the system. To the best of our
knowledge, the LETKF is the computationally most
efficient ensemble-based data assimilation
system for large observational data sets. The
LETKF has been successfully tested on models of
the atmosphere, the oceans and a laboratory
system to simulate coupled convective cells. In
this talk, we demonstrate the potentials of the
LETKF with an implementation on a
state-of-the-art numerical model of the
atmospheric global circulation. We pay special
attention to the role the process of data
assimilation plays in the predictability of the
system.
Professor David Jacobs, Computer Science,
University of Maryland
Modeling Lighting to
Capture the Variability of Images
Variations in lighting can have a
significant impact on the appearance of an object.
Because the set of possible lighting conditions is
high-dimensional, it is a challenge to represent
this variability in a computationally tractable way.
We will discuss three ways of doing this for the
problem of determining the lighting that best fits a
known, 3D model to a new image. First, we show that
we can model the reflection of light by diffuse
(non-shiny) objects as a low-pass filter. This
implies that the set of images produced by such
objects is well represented by a low-dimensional,
linear subspace. This is not the case for specular
(shiny objects); high frequency components of the
lighting significantly affect their appearance. In
this case, it is important to also enforce the
constraint that lighting is everywhere non-negative,
meaning that we must model the set of images of an
object as a convex subspace of a higher-dimensional
space. Using an extension of Szego’s eigenvalue
distribution theorem to spherical harmonics, we show
that we can enforce this non-negative lighting
constraint using semi-definite programming. Finally,
we consider the set of images produced in scenes
with cast shadows. We show that these images can be
captured when we represent lighting as a combination
of a low-dimensional subspace and a sparse subset of
the set of all images produced by point sources of
light. This provides a compact and computationally
tractable representation. We will discuss
applications of these results to face recognition,
3D reconstruction, and lighting recovery for
computer graphics.
Collaborators: Ronen Basri, The Weizmann Institute
Xue Mei, the University of Maryland
Margarita Osadchy, NEC Labs
Ravi Ramamoorthi, Columbia University
Sameer Shirdhonkar, the University of Maryland
Professor Bala Balachandran, Department of Mechanical Engineering, University of Maryland
Microsystems: Mechanics and Nonlinear Phenomena
Analytical, numerical, and experimental efforts carried out to understand nonlinear phenomena exhibited by micro-resonator and micro-resonator arrays will be presented in this talk. The nonlinear phenomena considered include jumps, oscillations about non-flat positions caused by buckling, and intrinsic localized modes. The distributed-parameter systems used to model the oscillators and the weakly nonlinear analysis undertaken to explain buckling influenced oscillations will be discussed. The phenomenon of intrinsic localized modes in coupled oscillator arrays will also be explored, and it is discussed as to how one can take advantage of them for different applications.
Professor Michelle Girvan, Department of
Physics and IPST, University of Maryland
Finding, Evaluating,
and Generating Community Structure in Complex
Networks
I will discuss a set of algorithms
for discovering community structure in networks—
natural divisions of network nodes into densely
connected subgroups. These algorithms all share
two definitive features: first, they involve
iterative removal of edges from the network to
split it into communities, the edges removed
being identified using one of a number of
possible “betweenness” measures, and second,
these measures are, crucially, recalculated
after each removal. I will also discuss a
measure for the strength of the community
structure found by the algorithms, which gives
an objective metric for choosing the number of
communities into which a network should be
divided.
I'll demonstrate that the algorithms are highly
effective at discovering community structure in
both computer-generated and real-world network
data, and show how they can be used to shed
light on the sometimes dauntingly complex
structure of networked systems. Finally, I will
discuss a simple model for the formation of
community structure in social networks.
Professor Kenneth Karlsen, Centre of
Mathematics for Applications, University of Oslo
Convergent Numerical
Schemes for the Camassa-Holm Equation
The Korteweg-de Vries (KdV) equation models weakly
nonlinear unidirectional long waves, and arises in
various physical contexts. For example, it models
surface waves of small amplitude and long wavelength
on shallow water. The Camassa-Holm equation entered
the arena in the early 1990s. In one interpretation,
it models the propagation of unidirectional shallow
water waves on a flat bottom. The Camassa-Holm
equation is a water wave equation of quadratic order
in an asymptotic expansion for unidirectional
shallow water waves described by the incompressible
Euler equations, while the KdV equation appears as first
order in this expansion. In another interpretation
it describes finite length, small amplitude radial
deformation waves in cylindrical compressible
hyperelastic rods, and in this context the equation
is often referred to as the hyperelastic-rod wave
equation. The Camassa-Holm equation possesses many
interesting properties, like its bi-Hamiltonian
structure and complete integrability. Moreover, it
has an infinite number of non-smooth solitary wave
solutions called peakons. Related equations are the
Hunter-Saxton, variational wave, and
Degasperis-Procesi equations.
From a mathematical analysis point of view, the
Camassa-Holm equation is rather well understood.
However, much less is known when it comes to the
design and analysis of numerical schemes. In
particular, it has been rather difficult to
construct numerical schemes for which one can prove
the convergence to a (non-smooth) solution of the
Camassa-Holm equation. This statement is
particularly accurate in the case of general H^1
initial data and peakon-antipeakon interactions. In
this talk we present finite dfference schemes for
the Camassa-Holm equation that can handle general
H^1 initial data. The form of the difference
schemes are judiciously chosen to ensure that they
satisfy a total energy inequality. We prove that the
difference schemes converge strongly in H^1
towards a dissipative weak solution of the Camassa-Holm
equation. Similar results hold for the
Hunter-Saxton, variational wave, and
Degasperis-Procesi equations.
This is joint work with Giuseppe Maria Coclite
(Bari) and Nils Henrik Risebro (Oslo).
Professor Alexander Vladimirsky,
Department of Mathematics, Cornell University
Causality and
Efficiency: Non-Iterative Numerical Methods
Our knowledge of the direction of
information flow is fundamental for many efficient
numerical methods (e.g., time-marching for
evolutionary PDEs). However, for many problems
(including first-order static nonlinear PDEs) the
direction of information flow might be a priori
unknown even if it is otherwise well-defined. This
leads to a common use of iterative methods, which
can be unnecessarily inefficient.
For certain systems of nonlinear equations, the
"causality" present in the problem can be used to
uncover the direction of information flow at
runtime. Exploiting causality to effectively
de-couple nonlinear systems is the fundamental idea
behind Dijkstra's classical method for finding
shortest paths on graphs.
We will use a continuous analogue of this principle
to build efficient methods for a wide class of
causal problems. We will consider examples in
continuous and hybrid optimal control (e.g., optimal
traveling on foot and using the buses), in
anisotropic front propagation (e.g., first-arrivals
and multiple-arrivals in seismic imaging), in
optimal control under uncertainty (e.g., optimal
traveling when the map is not quite known), in
Markov decision processes (e.g., stochastic shortest
paths on graphs), and in dynamical systems (e.g.,
approximation of "geometrically stiff" invariant
manifolds).
