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IMA POSTDOC SEMINAR
January - August 2002

Complete List

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.

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