| Date |
Speaker |
Title and abstract |
| 10/2/2009 Friday 2pm |
William Rundell (Texas A&M) |
Inverse obstacle problems;
uniqueness and non-continuous dependence on the problem itself Abstract: Radar and sonar, satellite observations CAT scans, indeed most of medical imaging, are ubiquitous examples of the recovery of a hidden or remote object by making measurements from a distance. They are wonderful case studies of the application of mathematics. In all such problems there is a partial differential equation lurking behind the scenes. If we knew the shape, location and material properties of the object then mathematics developed in the latter half of the nineteen century and first half of the twentieth would let us predict, in theory, exactly the kind of measurements we would obtain. The much harder, but interesting part, is the converse; given the measurements, where, and what, was the obstacle? This talk will explore this inverse problem, specifically the issues of uniqueness and determination. What is the least amount of measurements one can make in order to obtain a unique recovery? Are there algorithms that would let us reconstruct the object and what are their limitations? But there will be an additional feature. It is well know that such inverse problems are ill-posed -the solution does not depend continuously on the measured data, but we will point out that seemingly minor changes in the partial differential equation or the boundary conditions can result in completely changed answers for the inverse problem. |
| 10/8/2009 Thursday 3:30pm |
Sergej Rjasanow (Saarland University) |
Mathematical model of the
nonlocal electrostatics abstract |
| 10/9/2009 Friday 2pm, Change of room, this talk will be at ACES 4.304 |
Jianliang Qian (Michigan State) |
Eulerian Gaussian beams for
semi-classical solutions of Schrodinger equations Abstract: We propose Gaussian-beam based Eulerian methods to compute semi-classical solutions of the Schrodinger equation. Traditional Gaussian beam type methods for the Schrodinger equation are based on the Lagrangian ray tracing. Based on the first Eulerian Gaussian beam framework proposed in Leung et al. [S. Leung, J. Qian, R. Burridge, Eulerian Gaussian beams for high frequency wave propagation, Geophysics 72 (2007) SM61-SM76], we develop an Eulerian Gaussian beam method which uses global Cartesian coordinates, level-set based implicit representation and Liouville equations. The resulting method gives uniformly distributed phases and amplitudes in phase space simultaneously. To obtain semi-classical solutions to the Schrodinger equation with different initial wave functions, we only need to slightly modify the summation formula. This yields a very efficient method for computing semi-classical solutions to the Schrodinger equation. We also highlight the importance of initializing Gaussian beam propagation. Numerical experiments indicate that this Eulerian Gaussian beam approach yields accurate semi-classical solutions even at caustics. This is a joint work with S. Leung and R. Burridge. |
| 10/13/2009 Tuesday 3:30pm |
Irina Potapenko (Keldish Institute of Applied Math and Russia Academy of Sciences) |
The time-depended solutions of collisional electron kinetic equation with the heating term allowing the solutions in self-similar variables are considered. Our investigation is concentrated on the evolution and formation of the distribution function tails for long times and establishment of the asymptotic self-similar solutions. A broader class of the heating terms resulting in enhancement of the tail of the distribution function in comparison with Maxwellian is analyzed both analytically and numerically. Also formation of non-stationary non-equilibrium electron and phonon distribution functions in metals under action of a strong pulse electric field is briefly considered. |
| 10/20/2009 Tuesday 3:30pm |
Olof Runborg (KTH) |
A Multiscale Method for the Wave
Equation in Heterogeneous Medium Abstract: We consider the wave equation in a medium with a rapidly varying speed of propagation. We construct a multiscale scheme based on the heterogeneous multiscale method, which can compute the correct coarse behavior of wave pulses traveling in the medium, at a computational cost essentially independent of the size of the small scale variations. This is verified by theoretical results and numerical examples. |
| 11/04/2009 Wednesday 4pm |
Olivier Pinaud | On the moment problem for
Quantum Hydrodynamics This work is motivated by a recent series of papers by Degond, Méhats and Ringhofer about the formal derivation of Quantum Hydrodynamics models from the entropy principle. Starting from the Quantum Liouville equation, they obtain a non-closed system for the first moments of the density operator. Then, as in the Levermore moment method for kinetic equations, the system is closed by introducing the solution to the quantum moment problem which consists in minimizing the quantum free energy under some constraints of density, current or energy. The first step towards the rigorous derivation of such Quantum Hydrodynamics models is the mathematical analysis of the quantum moment problem. We will present in the talk a result of existence for the quantum moment problem under a density constraint and give some elements of the proof. The solution will also be characterized in a simplified setting. This is joint work with Florian Méhats (Université de Rennes, France). |
| 11/11/2009 Wednesday 4pm |
Junping Wang (NSF) |
A maximum value principle for
finite element approximations Abstract: The purpose of this talk is to discuss how the classical maximum value principles in PDEs be extended to their numerical approximations arising from finite element methods. In particular, the discussion will be primarily focused on the second order elliptic problems, and the finite element methods shall include standard Galerkin, P1 non-conforming, and mixed elements. |
| 11/13/2009 Friday 2pm |
Hongyu Liu (University of
Washington) |
Acoustic and Electromagnetic
Cloaking Abstract: In this talk, we shall discuss our recent progress on acoustic and electromagnetic cloaking with transformation media. In wave scattering, a cloaking device is an artificially designed device which makes the target object invisible to wave detections. We shall present our study on both perfect cloaking with singular transformation media and near-invisibility cloaking from a regularization viewpoint. |
| 11/17/2009 Tuesday (SPECIAL DATE) |
Ken Golden (Univ of Utah) |
Climate Change and the
Mathematics of Transport in Sea Ice Sea ice is both an indicator and agent of climate change. It also hosts extensive microbial communities which sustain life in the polar oceans. Fluid flow through porous sea ice mediates a broad range of processes such as the growth and decay of seasonal ice, the evolution of melt ponds and sea ice reflectance, and biomass build-up. We'll discuss recent mathematical advances using percolation theory, hierarchical models, and diffusion processes in understanding the fluid permeability of sea ice and the thermal evolution of its microstructure. Our work will help in predicting how global warming may affect Earth's sea ice packs and how polar ecosystems may respond. Related results on electromagnetic properties will help in monitoring ice thickness and in remote sensing of the polar marine environment. Video from a 2007 Antarctic expedition where we measured fluid and electrical transport in sea ice will be shown. |
| 11/20/2009 Friday 2pm |
Wolfgang Bangerth (Texas A&M) |
Numerics for Inverse Problems in
Biomedical Imaging In many of the modern biomedical imaging modalities, the measurable signal can be described as the solution of a partial differential equation that depends nonlinearly on the tissue properties (the "parameters") one would like to image. Consequently, there are typically no explicit solution formulas for these so-called "inverse problems" that can recover the parameters from the measurements, and the only way to generate body images from measurements is through numerical approximation. The resulting parameter estimation schemes have the underlying partial differential equations as side-constraints, and the solution of these optimization problems often requires solving the partial differential equation thousands or hundred of thousands of times. The development of efficient schemes is therefore of great interest for the practical use of such imaging modalities in clinical settings. In this talk, the formulation and efficient solution strategies for such inverse problems will be discussed, and we will demonstrate its efficacy using examples from our work on Optical Tomography, a novel way of imaging tumors in humans and animals. The talk will conclude with an outlook to even more complex problems that attempt to automatically optimize experimental setups to obtain better images. |
| 11/20/2009 Friday 3pm |
Harald van Brummelen (Eindhoven University of Technology) | Goal-oriented adaptivity for a
1D prototype of the Boltzmann equation Abstract: With the perpetual trend towards smaller and smaller scales in science and engineering, fluid-flow problems in the transitional molecular/continuum regime have rapidly gained prominence over the past years. Accurate numerical simulation of flow problems in the transitional regime poses a fundamental challenge, on account of the large difference between the molecular free path and a typical length scale of observation. To bridge the gap between the molecular length scale and the continuum length scale in numerical simulations, many heuristic approaches have recently appeared in the literature to couple molecular-dynamics models to continuum models, such as the Navier-Stokes equations. The appropriateness of such a direct connection between a molecular model and a continuum model is arguable, however, because the range of validity of the models is highly disparate. A suitable model for transitional molecular/continuum flows is provided by the Boltzmann equation. In the Boltzmann equation, the flow is characterized by a one-particle probability-density function, which measures the probability that a molecule resides in a certain subset of the position/momentum space. The Boltzmann equation itself is an integro-differential equation that governs the evolution of the one-particle probability-density. The Boltzmann equation is in principle valid downto the molecular scale, while on the other hand it encapsulates all conventional continuum models, such as the compressible and incompressible Navier-Stokes equations, in the sense that with appropriate scalings of the macroscopic length and time scales, limit solutions of the Boltzmann equation correspond to solutions of these continuum equations. Essentially, the Boltzmann equation is connected to the continuum equations by the fact that solutions of the Boltzmann equation converge to a particular class of solutions, the so-called Maxwell-Boltzmann equilibrium distributions. Direct numerical simulation of the Boltzmann equation is prohibited by its high-dimensional setting: for a problem in d spatial dimensions, the corresponding position/momentum domain is 2d dimensional. However, it is anticipated that in many cases the computational complexity can be significantly reduced by means of adaptive low (d) dimensional approximations based on Boltzmann moment closures. In many applications, interest is in fact restricted to one particular goal functional. For instance, in micro-scale heat-transfer problems, it is ultimately only the heat flux across a certain part of the boundary that is of interest. This class of problems provides fertile ground for goal-oriented adaptive-refinement strategies. An essential impediment in the development of goal-oriented adaptive-refinement techniques for the Boltzmann equation, is the fact that the convergence-to-equilibrium property which forms the basis of the hierarchical modeling process only occurs for d=2,3. Hence, to test our ideas, we would have to consider problems in 4 or 6 dimensions. To bypass this complication, we have developed a 1D prototype of the Boltzmann equation, which exhibits all the characteristic features of the Boltzmann equation, including the weak-convergence-to-equilibrium property. The underlying molecular model is based on random collisions, which conserve energy but not momentum. In the presentation, I will give an overview of transitional molecular/continuum flows, from the perspective of hierarchical modeling and model adaptivity. I will then elaborate the 1D prototype of the Boltzmann equation that we have developed, and derive its characteristic properties, such as an entropy inequality. Finally, I will present numerical result obtained by a discontinuous Galerkin finite-element discretization of the prototype, including recent results for goal-oriented error estimation. |
| 12/2/2009 Wednesday 4pm |
John Schotland (UPenn) | Optical Tomography There is considerable interest in the development of optical methods for biomedical imaging. The physical problem consists of recovering the optical properties of a medium in which light propagates by multiple scattering. This talk will review recent work on related inverse scattering problems for the radiative transport equation and fast image reconstruction algorithms for large data sets. Numerical simulations and experimental data from model systems are used to illustrate the results. |