Applied and Computational Mathematics Colloquium

Back to Current Schedule Fall 2016 - Spring 2017

Past Talks

• 10/28/16

  Todd Arbogast (UT Austin)

  • Mixed Methods for Two-Phase Darcy-Stokes Mixtures of Partially Melted Materials with Regions of Zero Porosity

  The Earth's mantle (or, e.g., a glacier) involves a deformable solid matrix phase within which a second phase, a fluid, may form due to melting processes. The system is modeled as a dual-continuum mixture, with at each point of space the solid matrix being governed by a Stokes flow and the fluid melt, if it exists, being governed by a Darcy law. This system is mathematically degenerate when the porosity (volume fraction of fluid) vanishes. Assuming the porosity is given, we develop a mixed variational framework for the mechanics of the system by carefully scaling the Darcy variables by powers of the porosity. We prove that the variational problem is well-posed, even when there are regions of one and two phases (i.e., even when there are regions of positive measure where the porosity vanishes). We then develop an accurate mixed finite element method for solving this Darcy-Stokes system and prove a convergence result. Numerical results are presented that illustrate and verify the convergence of the method. (joint work w/ Marc A. Hesse and Abraham L. Taicher) (host: B. Cockburn)

• 10/21/16

  Jie Shen (Purdue)

  • Efficient spectral methods for solving a class of fractional PDEs

  We consider spectral approximations of fractional PDEs in bounded and unbounded domains. For fractional PDEs based on Riemann-Liouville or Caputo derivatives in bounded domains, two main difficulties are: (i) they are non-local operators; and (ii) they lead to singular solutions at domain boundaries. We introduce a family of generalized Jacobi functions (GJFs) such that their fractional derivatives can be easily computed, and present spectral methods which are as efficient as the spectral methods for regular PDEs. We also present error estimates which show that spectral convergence rate in properly weighted Sobolev spaces can be achieved despite the fact that the solutions have singularities at the endpoints. For fractional PDEs based on fractional Laplacians in unbounded domains, the solutions decay very slowly at infinity and no transparent boundary condition is available for domain truncation, we develop efficient spectral-collocation and spectral-Galerkin methods using Hermite functions to solve thes problems in unbounded domains directly. (host: B. Cockburn)

• 10/14/16

  Selim Esodoglu (University of Michigan)

  • Threshold dynamics for multiphase, anisotropic mean curvature motion

  Motion by mean curvature of networks of surfaces arises in many applications, including computer vision (in the context of the Mumford-Shah segmentation model) and materials science (in the context of Mullins’ model for grain networks in polycrystalline materials). It describes gradient flow for a cost function that measures the surface area of interfaces that partition a domain into disjoint regions. Junction conditions that stipulate behavior along curves where three of more surfaces may meet, along with inevitable topological changes to the network, make computing these evolutions particularly challenging. Anisotropic versions of the flow, where the corresponding cost function is a normal dependent version of surface area, are of great interest in the materials science context. In 1989, Merriman, Bence, and Osher proposed an elegant algorithm known as threshold dynamics that approximates this evolution in the simplest two-phase, isotropic setting. It generates the approximation simply by alternating two efficient operations: Convolution, and thresholding. Since then, much has been written on whether such a simple algorithm can be extended, while maintaining its simplicity and efficiency, to (1) arbitrary networks, and (2) anisotropic surface energies. Even the two-phase, anisotropic situation remained unsolved. Drawing on a recent variational formulation of threshold dynamics (joint work with Felix Otto), we will give a fairly complete answer to the fully anisotropic, multiphase question. Based on joint works with Matt Elsey, Matt Jacobs, and Pengbo Zhang.(host: F. Santosa)

• 10/07/16

  Philip Maini (Oxford University) NOTE: SPECIAL LOCATION - LIND 305

  • Ordway Lecture: Collective Cell Motion in Biology

  The phenomenon of collective cell motion is widespread in biology. Cells move as individuals, as loosely signalling groups, as units in a sheet, etc. In this talk I will review a number of applications in normal development and disease. The modelling approaches will range from partial differential equations to hybrid discrete cell-based and particular applications will cover cranial neural crest cell migration, the role of heterogeneity in cancer cell migration and epithelial sheet dynamics. (host:H. Othmer)

• 9/30/16

  Amit Singer (Princeton University) NOTE: SPECIAL LOCATION - LIND 305

  • Vector diffusion maps and the graph connection Laplacian

  Vector diffusion maps (VDM) is a mathematical framework for organizing and analyzing high-dimensional datasets that generalizes diffusion maps and other nonlinear dimensionality reduction methods, such as LLE, ISOMAP, and Laplacian eigenmaps. Whereas weighted undirected graphs are commonly used to describe networks and relationships between data objects, in VDM each edge is endowed with an orthogonal transformation encoding the relationship between the data at its vertices. The graph structure and orthogonal transformations are summarized by the graph connection Laplacian. In manifold learning, VDM can infer topological properties from point cloud data such as orientability, and graph connection Laplacians converge to their manifold counterparts (Laplacians for vector fields and higher order forms) in the large sample limit. The graph connection Laplacian satisfies a Cheeger-type inequality that provides a theoretical performance guarantee for the popular spectral algorithm for rotation synchronization, a problem with many applications in robotics and computer vision. The application to 2D class averaging in cryo-electron microscopy will serve as our main motivation. (host: G. Lerman)

• 9/16/16

  Anita Layton (Duke University)

  • Unraveling Kidney Physiology, Pathophysiology and Therapeutics: A Modeling Approach

    The kidney not only filters metabolic wastes and toxins from the body,but it also regulates the body's water balance, electrolyte balance, and acid-base balance, blood pressure, and blood flow. Despite intense research, aspects of kidney functions remain incompletely understood. I will discuss how our group use mathematical modeling techniques to address a host of previously unanswered questions in renal physiology and pathophysiology: Why is the mammalian kidney so susceptible to hypoxia, despite receiving ~25% of the cardiac output? What are the mechanisms underlying the development of acute kidney injury in a patient who has undergone cardiac surgery performed on cardiopulmonary bypass? What is the effect of inhibiting sodium-glucose transport, a novel treatment for reducing renal glucose update in diabetes, on renal NaCl transport and oxygen consumption? (host: J. Foo)

• 9/9/16

  Boaz Nadler (Weizmann Institute of Science)

  • Unsupervised Ensemble Learning

  In various applications, one is given the advice or predictions of several classifiers of unknown reliability, over multiple questions or queries. This scenario is different from standard supervised learning where classifier accuracy can be assessed from available labeled training or validation data, and raises several questions: Given only the predictions of several classifiers of unknown accuracies, over a large set of unlabeled test data, is it possible to a) reliably rank them, and b) construct a meta-classifier more accurate than any individual classifier in the ensemble? In this talk we'll show that under various independence assumptions between classifier errors, this high dimensional data hides simple low dimensional structures. Exploiting these, we will present simple spectral methods to address the above questions, and derive new unsupervised spectral meta-learners. We'll prove these methods are asymptotically consistent when the model assumptions hold, and present their empirical success on a variety of unsupervised learning problems. (host: Gilad Lerman)

• 3/9/16

  Randall J. LeVeque (University of Washington)

  • Wave Propagation Methods and Biomedical Applications

  The study of shock waves propagating in biological tissue and bone has several biomedical applications. In lithotripsy, focused shock waves are used to pulverize kidney stones without surgery, while in Extracorporeal Shock Wave Therapy (ESWT), focused shock waves of smaller amplitude are used to stimulate healing and bone growth. On the negative side, blast-induced traumatic brain injury (TBI) affects countless veterans and civilians who have survived nearby explosions. In this talk I will describe some of these applications and efforts to obtain a better understanding of the affect of wave propagation on biological media. High-resolution wave propagation algorithms can robustly handle interfaces between different materials, while methods recently developed for poroelasticity may be valuable in studying wave propagation in bone.