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Mon Oct 15

Applied and Computational Math Colloquium

3:35pm - Vincent Hall 207
Modeling gas, fluid, and particle transport in the lung airway system: the curse of scales
Marcel Filoche, Ecole Polytechnique, Paris

The pulmonary airway system is a highly hierarchical tree-like 3D network in charge of transferring oxygen from the upper airways down to the alveolar region where the gas exchange occurs between air and blood. It is also the place of numerous particle and fluid transport processes. Given the complexity of the anatomy and of the physics involved, numerical simulation is a remarkable tool for investigating the properties of this organ seen as a "transport system", understanding its behavior and possible failures, and predicting the outcome of therapies. However, the range of scales represents here a huge challenge: from the meter scale of the entire organ size down to the micron scale of the mucus layer, about 5 to 6 orders of magnitude of length scale are crossed.

In this talk, we will present several examples of numerical models able to capture the various aspects of the lung airway system, from the simplest linear approach of gas transport to more advanced computational fluid dynamics simulation. We will show, in particular, how the range of scales involved and the difficulty to access the actual parameters in the patient imposes to use a hierarchy of models and a diversity of numerical techniques. We will explain how these models can be used either to reach a general understanding of the system or to design patient specific diagnosis and therapy.

Mon Oct 29

Applied and Computational Math Colloquium

3:35pm - Vincent Hall 207
Preconditioning systems of PDEs: from robust estimates to fast iterations, with applications to simulations of the brain
Ragnar Winther, University of Oslo

We will present a set up for simulations of the so-called glymphatic system, which is believed to
be an important mechanism for clearance of waste from the brain. Accumulation of waste can be coupled to
brain shrinkage and conditions like Alzheimers disease. An approach to design mathematical models for the
glymphatic system is to consider the brain as a poroelastic material, with complex geometry, bathed in water, and where various parameters of the model will vary over extreme ranges. A challenge in discretizing such systems in to design methods which behave well with respect variations of the parameters. In particular, we will explain how parameter robust stability estimates for the PDE systems can be used as a key tool to design effective algorithms for the corresponding discrete systems.

Mon Nov 19

Applied and Computational Math Colloquium

3:35pm - Vincent Hall 207
Divide and conquer algorithms and software for large Hermitian eigenvalue problems
Yousef Saad, Department of Computer Science and Engineering, University of Minnesota

The solution of large symmetric real (or Hermitian complex) eigenvalue
problems is central to applications ranging from electronic structure
calculations to the study of vibrations in mechanical systems. A few
of these applications require the computation of a large number of
eigenvalues and associated eigenvectors. For example, when dealing
with excited states in quantum mechanics, it is not uncommon to seek a
few tens of thousands of eigenvalues of matrices of sizes in the tens
of millions. In such situations it is imperative to resort to
`spectrum slicing' strategies, i.e., strategies that extract slices of
the spectrum independently. The presentation will discuss a few of
these techniques that are based on a combination of filtering
(polynomial, rational) and standard projection methods (Lanczos,
subspace iteration). The talk will also discuss our recently released
code named EVSL (for eigenvalues slicing library) that implements
these ideas.

Mon Dec 03

Applied and Computational Math Colloquium

3:35pm - Vincent Hall 207
Interfaces with singularities: understanding phase transitions in nematic liquid crystals
Dmitry Golovaty, Department of Mathematics, University of Akron

A nematic liquid crystal is essentially a liquid that exhibits some orientational order. The liquid crystalline state is typically observed in materials composed of highly asymmetric molecules for temperatures in the intermediate range when the material is neither a classical liquid nor a solid. In this talk we will be interested in describing a phase transition between the nematic and the isotopic state when the nematic order melts and the liquid crystal turns into a regular isotopic fluid. The experimental observations of this process indicate that the transition proceeds via evolution of interfaces separating different phases where the interfaces are generally not smooth. Our goal in this talk is to explain presence of these phase boundary singularities from a mathematical point of view .

The nematic-to-isotopic phase transition can be described within the so called Landau-de Genes variational theory for a tensor-valued order parameter. Because this theory is rather complex, in order to develop an initial understanding of transitions between the ordered and disordered states, we propose a simpler toy model based on the modified Ginzburg-Landau-type energy defined over vector fields on the plane. The corresponding variational model consists of anisotropic gradient terms and a potential that vanishes on two disconnected sets. While this model may not quantitatively describe the nematic state, the topology of the simplified target set retains the crucial features of the full problem. The principal observation from the study of the simplified model is that the phase boundary singularities can be explained by large disparity between the elastic constants that specify the gradient contribution to the energy. In the talk we will present a combination of rigorous analysis and numerics that leads to this conclusion.

Mon Mar 11

Applied and Computational Math Colloquium

3:35pm - Vincent Hall 313
Applied and Computational Math Colloquium
Mon Apr 01

Applied and Computational Math Colloquium

3:35pm - Vincent Hall 313
Quadratic Wasserstein Metrics for Nonlinear Inverse Problems
Kui Ren, Columbia University

In the absence of analytical reconstruction methods, numerical solutions of nonlinear inverse problems have been mostly based on least-square formulations where solutions are sought by minimizing the $L^2$ difference between model predictions and measured data. We present here recent computational studies of some nonlinear inverse problems where quadratic Wasserstein distances are used to measure the discrepancy between model predictions and measured data. Related numerical and theoretical issues will be discussed.

Mon Sep 09

Applied and Computational Math Colloquium

3:35pm - Vincent Hall 207
Emergent behavior in collective dynamics
Eitan Tadmor, University of Maryland

Collective dynamics is driven by alignment that tend to self-organize the crowd and by different external forces that keep the crowd together. Different emerging equilibria are self-organized into clusters, flocks, tissues, parties, etc.

I will overview recent results on the hydrodynamics of large-time, large-crowd collective behavior, driven by different “rules of engagement”. In particular, I address the question how short-range interactions lead, over time, to the emergence of long-range patterns, comparing geometric vs. topological interactions.