| January
15 |
Speaker:
Anna Mazzucato, IMA and Yale |
| |
Title:
Function Spaces and Non-linear PDE's |
| |
Abstract:
I will discuss some applications of the theory of function
spaces to non-linear PDEs, specifically the Navier-Stokes
equation. Besov, Zygmund classes, along with BMO, appear
rather naturally in connection with pseudo-differential
calculus for non-regular symbols and the paraproduct of
J. M. Bony. Morrey spaces model interesting physical examples,
such as vortex rings. |
| January
22 |
Speaker:
Jianliang Qian, IMA |
| |
Title:
Eulerian Methods for Viscosity and Non-Viscosity
Solutions of Hamilton-Jacobi Equations Arising from Wave
Propagations |
| |
Abstract:
The geometric optics approximation to wave propagation reduces
the anisotropic wave equation to a static Hamilton-Jacobi
equation, the anisotropic eikonal equation. This equation
has as solutions three different coupled wave modes, i.e.,
quasi-longitudinal (qP) and two transverse (qSV and qSH)
waves. Viewed in the slowness space, this eikonal equation
describes a sextic surface consisting of three sheets for
the three waves. Observing that the qP sheet is always convex,
we propose a paraxial formulation for computing the viscosity
solution to quasi-P wave propagation. This formulation use
partial information about characteristic directions to guarantee
a stable evolution in any preferred spatial direction; furthermore,
this incorporated in down-n-out (DNO) and post-sweeping
(PS) update yields an Eulerian method with O(N) complexity,
where N is the number of points in the computational domain.
Since the qSV slowness sheet is nonconvex, the qSV waves
have cusps, which form a class of multi-valued, non-viscosity
solutions. We introduce a level set based Eulerian approach
which is able to accurately capture the qSV waves with cusps.
The work on qP waves is joint with W. W. Symes, and the
work on qS waves is joint with L.-T. Cheng and S. Osher. |
| January
29 |
Speaker:
Lev Truskinovsky, Aerospace Engineering and Mechanics
-- UMN |
| |
Title:
Dynamics of Phase Boundaries in Discrete Lattices |
| |
Abstract:By
using a prototypical discrete model we study the motion
of a generic crystalline
defect and explicitly compute the functional relation
between the macroscopic driving force and the velocity
of the defect. Although the adopted model is purely conservative
and contains information only about elasticity of the
constitutive elements, the resulting kinetic relation
provides a quantitative description of a dissipative process.
The apparent dissipation is due to the micro-instabilities
and induced radiation of the high frequency waves, which
are invisible at the macro level. The mechanism is believed
to be generic and accounting for a considerable fraction
of inelastic irreversibility in deforming solids. Joint
work with Olga Martynenko. |
| February
5 |
Speaker:Miao-Jung
Ou, IMA |
| |
Title:
A Uniqueness Theorem of the 3-Dimensional Acoustic
Scattering Problem in a Shallow Ocean with a Fluid-like
Seabed |
| |
Abstract:We
show that under the assumption of out-going radiation conditions
at infinity, the time-harmonic acoustic scattered field
off a sound-soft solid in a shallow ocean with a fluid-like
seabed is unique in $C^2(M_1)\cap C^2(M_2)\cap C(R_h^3 \setminus
\Omega)$. Here $M_1$ is the water part, $M_2$ the seabed,
$R_h^3$ the waveguide and $\Omega$ is the solid object.
The associated modal problem is studied and a representation
formula for the solution in terms of the Green's function
is derived. |
| February19 |
Speaker:Daniel
Kern, IMA |
| |
Title:
Compartmental Model for Cancer Evolution |
| |
Abstract:
A model is presented that examines the role of drug resistance
in the evolution of cancer subject to treatment with a single
cytotoxic (chemotherapeutic) agent. The model starts from
a single cell and generates the evolution of the cancer.
The roles of natural occurring, and acquired, resistance
from chemotherapy can be seen throughout the development
of the cancer or treatment history. Numerical examples illustrate
the effects of resistance on chemotherapy treatment scheduling. |
| February26 |
Speaker:Prof.
Peter Linch, Met Eireann |
| |
Title:
Resonant Triads and Swinging Springs |
| |
Abstract:
1: An Interesting Analogy. A Powerful Equivalence.The oscillations
of the atmosphere fall into two categories, thelow frequency
Rossby waves and the high frequency gravity waves.
The Swinging Spring is a simple mechanical system also having
low frequencyand high frequency oscillations. There
are several illuminating analogies between the spring's
behavior and that of the atmosphere. The equivalencebetween
the systems allows us to deduce properties of atmospheric
motion from the behavior of the spring. The talk will include
a demonstration with a real-life swinging spring and a Java
applet illustrating some characteristics of its motion.
2: Resonant
Motions and Stepwise Precession of the Spring. The three-dimensional
motion of the swinging spring is investigated using a
perturbation approach. If the Lagrangian is approximated
by keeping terms up to cubic order, the system has three
independent constants of motion; it is therefore completely
integrable. When the ratio of the vertical and horizontal
oscillations is approximately two-to-one, an interesting
resonance phenomenon occurs, in which energy is transferred
periodically between predominantly vertical and predominantly
horizontal oscillations. The motion has two distinct
characteristic times, that of the oscillations and that
of the resonance envelope, and a multiple time-scale analysis
is found to be productive. The modulation equations
are the well-known Three-Wave Equations that also apply
to many other physical systems. As the oscillations change
from horizontal to vertical and back again, it is observed
that each horizontal excursion is in a different direction.
Expressions for the precession of the swing-plane are
derived. The approximate solutions are compared to numerical
integrations of the exact equations, and are found to
give a realistic description of the motion.
3: Rossby Wave
Triads and the Swinging Spring The relationship between
the spring dynamics and large-scale Rossby waves in the
atmosphere will be described. Rossby waves are of
fundamental importance for atmospheric dynamics. The nonlinear
interactions between these waves determine the primary
characteristics of the energy spectrum. These interactions
take place between triplets of waves known as `resonant
triads' and, for small amplitude, they are described by
the three-wave equations. The characteristic stepwise
precession of the swing-plane, so obvious from observation
of the physical spring pendulum, is also found for the
Rossby triads. This phenomenon has not been previously
noted and is an example of the insight coming from the
mathematical equivalence of the two systems. The implications
of the precession for predictability of atmospheric motions
are considered. The pattern of breakdown of unstable Rossby
waves is very sensitive to unobservable details of the
perturbations, making accurate prediction very difficult. |
| March
12 |
Speaker:Jamylle
Carter, IMA |
| |
Title:
A Dual Optimizer for Total Variation-Based Image
Restoration |
| |
Abstract:This
talk will describe a computational technique for the inverse
problem of edge-preserving image restoration. We solve an
equivalent dual form of a variational partial differential
equation. Images restored using this dual approach will
have crisp edges (discontinuities), whereas images recovered
under earlier primal methods may contain blurred edges.
Joint work with Tony Chan, Pep Mulet, and Lieven Vandenberghe. |
| March
26 |
Speaker:Prof.
O'Malley, University of Washington and IMA |
| |
Title:
Shock motion for certain advection-reaction-diffusion
equations. |
| |
Abstract:
The talk will report on joint work with Karl
Knaub, generalizing earlier work with Jacques Laforgue and
Michael Ward. It considers the long time asymptotic solutions
for some singularly perturbed parabolic equations in one
bounded spatial dimension. We show, in particular, how the
tail behavior of travelling wave solutions to the stretched
problem predicts shock motion featuring either exponential
or algebraic asymptotics. |
| April
2 |
Speaker:Vittorio
Cristini, IMA and UMN Dept. of Chem. Eng. |
| |
Title:
A mathematical and computer model of cancer growth |
| |
Abstract:I
will present and discuss the current status of a mathematical
and computer model of cancer growth under development in
my research group. The motivation of this work is to identify
and analyze diagnostic and treatment strategies through
direct in silico simulations. Once completed, this sophisticated
computer model will provide a physician with a tool that
simulates the development of a tumor corresponding to a
specific patient's clinical history. In particular, the
efficacy of different treatment strategies will be assessed
by direct simulation. The
development of a realistic computer model is now possible
due to the recent advances by our group in adaptive numerical
modeling and simulation techniques for complex evolving
microstructures. These new techniques are capable of describing,
for example,
the
complex shape of a solid carcinoma characterized by invasive
fingering and metastasization.
The
model will include all phases of growth that have been
identified in the biological and biomedical literature:
1. the
initial, diffusion-driven growth to a dormant multicell
spheroid state, regulated by the transport of nutrient
chemical species;
2. necrosis,
and the formation of a core of dead tumor cells, due to
the transport of growth inhibitor factors;
3. angiogenesis,
or blood vessel formation in the tumor, triggered by tumor
angiogenetic factors, and endothelial cell migration and
proliferation in response to the growth of the multicell
spheroid;
4. vascular
growth of malignant carcinoma with invasive fingering
and metastasization.
I will
present the current status of the model, capable of simulating
phases 1, 2, and 4, but not the process of angiogenesis.
I will then demonstrate how the assumptions and predictions
of the model have been validated and refined, for the
case of avascular growth (1 and 2), by direct comparison
with experimental observations of in vitro and in vivo
tumor growth. I will also discuss how angiogenesis will
be included in our model. |
| April
9 |
Speaker:
Prof. John Lowengrub, UMN School of Mathematics |
| |
Title:
Mathematical Modeling and Numerical Simulation of Microstructured
Materials |
| |
Abstract:
Microstructured materials, such as emulsions and polymer
blends, crystals and metallic alloys, blood and biological
tissues, are fundamental to many industrial and biomedical
applications. These diverse materials share the
common feature that the microscale and macroscale are linked.
The phenomena at microscopic scale, such as the morphological
instability of crystalline precipitates and drop deformation,
break-up and coalescence determine the microstructure and
its time evolution; thus affecting the rheology and mechanical
properties of the materials on the macroscale.
In this talk,
I will focus on mathematical and numerical modeling at
the microscale. In particular, I will present a class
of physically-based models of complex (multicomponent)
fluid flows which incorporate buoyancy, viscosity, compressibility
and surface tension at interfaces. The models are capable
of describing systems with both miscible and immiscible
components. In addition, the models allow topological
transitions such as pinchoff and reconnection of interfaces
to occur without relying on ad hoc 'cut and connect' or
smoothing procedures.
Results will
be presented for a variety of physically interesting flows.
To validate the model and numerical algorithms, we examine
the pinchoff of liquid/liquid threads (Rayleigh instability)
and jets and compare the numerical results to theory and
experiments. We then consider the development of a complex,
three dimensional microstructure in which the
flow components fully interpenetrate one another to yield
a sponge-like microstructure. Such co-continuous microstructures
have many important industrial applications.
Finally, I
will demonstrate how these models and numerical techniques
originally developed for fluid flows may be adapted to
also investigate the behavior of complex materials and
biological systems. |
| April
16 |
Speaker:
Toshio Yoshikawa, IMA |
| |
Title:
Transition from Mechanical Equilibrium
to Thermal Equilibrium of a Chain with Non-Monotone Stress-Strain
Relation |
| |
Abstract:
I will discuss a mechanical property
of a chain wich consists of unit masses and identical springs.
The spring is governed by the stress-strain relation: F(r)=r-H(r-1),
where H(x) is the Heaviside function. If
a heavy mass is attached to this chain and the springs
are stretched equally, the chain starts to contract. The
system stays in the state of mechanical equilibrium until
some point of time. After this time, the light masses
start to oscillate with high frequency.
I
will show that after this time the chain is in the state
of thermal equilibrium. By using Takahashi's method for
one-dimensional gas with nearest neighbor interactions,
I will derive the formulae for Gibbs free energy, entropy
and stress-strain relation. These relations lead to the
stress-strain relation in adiabatic change. I will show
that numerical experimental data satisfy the condition
of adiabatic change (constant entropy) and the theoretical
stress-strain relation. |
| May
7 |
Speaker:
Valeriy Shcherbakov |
| |
Title:
Rock- and paleomagnetism:
1. How the rocks record the geomagnetic field.
2. Geomagnetic field behaviour for the last 3 billions years. |
| |
Abstract:The
geomagnetic field is recorded in rocks essentially in the
same way, as the sound or visual images are recorded on
magnetic tapes. However, a rock is not a subject of high
technology, so the task is hard enough to record even a
single vector within required accuracy and to ensure an
enormous time stability of the natural remanence majnetization
(NRM). The signal must survive over millions or even billions
of years. Both major types of rocks, volcanics and sediments,
may carry the NRM. But the mechanisms of their acquisition,
and properties of NRM are quite different. Igneous
rocks acquire NRM through cooling from well above 1000 C
to the Earth surface temperature. Sediments get the remanence
simply by physical orientation of the magnetic grains during
deposition and compaction. The evolution of the virtual
dipole moment (VDM) over the last 3 billions years is analysed
on the basis of Paleointensity Database. The C_2 test showed,
at a 90% significance level, a bimodal VDM distribution
over the 400 millions years. The behaviour of the VDM is
characterised by a succession of high and low field. |
| May
14 |
Speaker:
Santiago Betelu, IMA |
| |
Title:
Singularities at the free surface of viscous fluids |
| |
Abstract:
We study the flow in the neighborhood of singular points
(such as cusps) of the free surface of a viscous fluid.
The objective is to know when singularities may appear starting
from smooth initial conditions. At the free surface, the
tangential viscous stress is zero and the normal viscous
stress is balanced with the Laplace pressure.
We compute the asymptotic flow in corners and cusps, and
we construct exact time dependent solutions using complex
variable techniques. We also consider 2D singularities on
thin films of fluid spreading on a plane substrate. We show
physically meaningful solutions describing how a plane film
separated in two halves is gradually joined by a singular
point travelling at finite speed. |
| May
21 |
Speaker:
Prof. Rachel Kuske, UMN School of Mathematics |
| |
Title:
Stochastic dynamics in models
sensitive to noise |
| |
Abstract:
Many systems which are
sensitive to noise exhibit dynamical features from both
the underlying deterministic behavior and the stochastic
elements. Then the stochastic effects are obscured in this
mix of dynamics. Several different methods have recently
been applied to separate the "deterministic" and "stochastic"
dynamics. These approaches lead to simplified approximate
models which can be analyzed or simulated efficiently,
providing useful measures of the noise sensitivity. The
methods combine projection methods and the identification
of important scaling relationships to exploit features common
to these systems, such as the presence of multiple time
scales, limited regions of strong noise-sensitivity, and
resonances. These approaches are valuable for studying a
variety of problems, including stochastic delay-differential
equations, noisy bursters, and meta-stable interfaces. The
approach will be outlined in one or two of these applications,
and the generalization to other areas will be discussed.
The results have interesting connections to classical probabilistic
methods, dynamical systems analysis, and singular perturbation
theory. |
| May
28 |
Speaker:
Shalom Michaeli, Radiology Dept., School of Medicine,
UMN. |
| |
Title:
CANCELLED!!! |
June 4 |
Abstract:
Speaker:
Prof. Mark Bebbington, Massey
University
Title:
Some Stochastic Models for
Seismicity Abstract:
The lecture will first
outline how point processes can be used to model earthquake
occurrence data and the sorts of results that can be achieved
from such models. The stress release process is a stochastic
version of the simple elastic-rebound hypothesis. For
this model it is possible to calculate an upper bound
on its forecasting performance using entropy gains. An
example will be given for north China. The stress release
model can be extended spatially to provide quantitative
estimates of linkages between regions. This will be illustrated
using
data from central Japan. The linked
stress release process gives rise to a number of problems,
only partially resolved, with features such as fitting,
robustness, and stability. As a final example, it will
be shown that the stress release process can provide cycles
of accelerating strain release. |
| June
18 |
Speaker:
Prof. Gerard Schuster, University
of Oregon |
| |
Title:
Similarities between Optical
and Seismic Imaging |
| |
Abstract:
Optical imaging has its
roots in the inventions of the telescope (1608 by Lippershey)
and microscope (1595 by Jansen) nearly 4 centuries ago.
It's first major success was Gallileo's discovery of the
Gallilean moons of Jupiter in 1609, and since then has propelled
major advances in the physical, medical and engineering
sciences. Luckily, the governing equations of electromagnetic
wave propagation, namely the Helmholtz equation, is the
same asthe governing equation of acoustic wave propagation.
This means that the imaging methods used by seismologists
are very similar to those used by optical physicists, and
so both communities can greatly benefit by drawing from
their overlapping pools of knowledge. As examples, I show
the following similarities between seismic and optical imaging.
1).
Principles of seismic and optical lenses. The seismic
method of poststack migration in the 15 degree approximation
is mathematically equivalent to the focusing operation
of an optical lens.
2).
Lens design. The point spread response (PSF) of both seismic
and optical lenses can be determined by use of the Array
theorem, a tool widely used in Fourier optics for lens
design.
3).
Broken lenses. Deblurring the images due to flawed lenses,
such as the early Hubble telescope lens, can be accomplished
by applying a PSF deconvolution filter to the image.
Seismologists use a similar deblurring
filter known as a migration deconvolution filter to deblur
their images.
4).
Multiple lenses. Light received at widely separated lenses
can be combined to yield high resolution images of stellar
objects. Similarly, sound waves received at separated
seismic lenses can be combined to yield reflectivity images
of the earth.
|
| June
25 |
Speaker:
Prof. Steven Jaume, College
of Charleston |
| |
Title:
Earthquake Communication 101:
Elastic Stress Transfer and
Its Role in Future Earthquake Occurrence |
| |
Abstract:
For the past 10 years numerous
case studies have illuminated the large and potentially
dominant role of elastic stress transfer from past earthquakes
on the location and timing of future earthquakes.
First I will review the basic physics of elastic stress
transfer and how it is commonly implimented in seismology.
I will then present the results of some of these case studies
illustrating stress transfer's role in the location of aftershocks
and the location and timing of future large earthquakes.
Finally I will present some problems and limitations of
this model and make (and take) suggestions for future studies. |
| August
20 |
Speaker:
Prof. John Donaldson, University
of Tasmania |
| |
Title:
Analytical and Numerical Solutions
of Differential and Differential
Delay Equations |
| |
Abstract:
The Louie, Wake et al continuous
mathematical model for the combined growth of Rye and Clover
includes distributed delay terms. A search for numerical
solutions raises questions on the relationship between the
continuous and discrete analogues of differential equations
and more generally between the analytic and numerical solutions
of differential equations. In the latter situation,
it is shown how invariants can be used to make the numerical
solutions more faithful to the analytic solution. Observations
on the discrete analogues of the logistic and logistic delay
equation indicate quite different behaviour patterns and
indicate a need for a deeper understanding of the relationships. |
