Many unconditionally energy stable schemes for the physical models will lead to a highly nonlinear elliptic PDE systems which arise from time discretization of parabolic equations. I will discuss two efficient and practical preconditioned solvers- Preconditioned Steepest Descent (PSD) solver and Preconditioned Nonlinear Conjugate Gradient (PNCG) solver - for the nonlinear elliptic PDE systems. The main idea of the preconditioned solvers is to use a linearized version of the nonlinear operator as a pre-conditioner, or in other words, as a metric for choosing the search direction. Based on certain reasonable assumptions of the linear pre-conditioner, a geometric convergence rate is shown for the nonlinear PSD iteration. Numerical simulations for some important physical application problems - including Cahn-Hillilard equation, epitaxial thin film equation with slope selection, square phase field crystal equation and Functionalized Cahn-Hilliard equation- are carried out to verify the efficiency of the solvers.
In this talk, we will explore two applications: The first considers simplicial cohomology as a tool to investigate and eliminate inequity in kidney paired donation (KPD). The KPD pool is modeled as a graph wherein cocycles represent fair organ exchanges. Helmholtz decomposition is used to split donation utilities into gradient and harmonic portions. The gradient portion yields a preference score for cocycle allocation. The harmonic portion is isomorphic to the 1-cohomology and is used to guide a new algorithmic search for exchange cocycles. We examine correlation between a patient’s chance to obtain a kidney and their score under various allocation methods and conclude by showing that traditional methods are biased, while our new algorithm is not.
The second considers the persistent homology of a smoothed noisy dynamic. The machinery of persistent homology yields topological structure for discrete data within a metric space. Homology in dynamical systems can capture important features such as periodicity, multistability, and chaos. We consider a hidden Markov dynamic and compare particle filter to optimal smoothing a posteriori. We conclude with a stability theorem for the convergence of the persistent homology of the particle filtered path to that of the optimal smoothed path.
Adaptive acquisition of correct features from massive datasets
is at the core of modern data analysis. One particular interest in
medicine is the extraction of hidden dynamics from an observed time series
composed of multiple oscillatory signals. The mathematical and statistical
problems are made challenging by the structure of the signal which
consists of non-sinusoidal oscillations with time varying amplitude and
time varying frequency, and by the heteroscedastic nature of the noise. In
this talk, I will discuss recent progress in solving this kind of problem.
Based on the cepstrum-based nonlinear time-frequency analysis and manifold
learning technique, a particular solution will be given along with its
theoretical properties. I will also discuss the application of this method
to two medical problems – (1) the extraction of a fetal ECG signal from a
single lead maternal abdominal ECG signal; (2) the simultaneous extraction
of the instantaneous heart rate and instantaneous respiratory rate from a
PPG signal during exercise. If time permits, an extension to multiple-time
series will be discussed.
Ideas that challenge the status quo either evaporate and are forgotten, or eventually
become the new status quo. Mathematically, an ODE model was developed by Strogatz et al. for
the propagation of one idea moving through one group of a large number of interacting individuals
(a 'city'). Recently, the Strogatz model was extended to include interacting multiple cities at SUMMER@ICERM 2016
at Brown University. The one and two city models are analyzed to determine the circumstances under
which there can be consensus. The case of three or more cities is analyzed to determine when, and under
what conditions, clustering occurs. Preliminary results will be presented.
The study of interfacial dynamics between two different components has become the key role to understand the behavior
of many interesting systems. Indeed, two-phase flows composed of fluids exhibiting different microscopic structures
are an important class of engineering materials. The dynamics of these flows are determined by the coupling among
three different length scales: microscopic inside each component, mesoscopic interfacial morphology and macroscopic
hydrodynamics. Moreover, in the case of complex fluids composed by the mixture between isotropic (newtonian fluid)
and nematic (liquid crystal) flows, its interfaces exhibit novel dynamics due anchoring effects of the liquid crystal
molecules on the interface.
In this talk I will introduce a PDE system to model mixtures composed by isotropic fluids and nematic liquid
crystals, taking into account viscous, mixing, nematic, stretching and anchoring effects and reformulating the corre-
sponding stress tensors in order to derive a dissipative energy law. Then, I will present new linear unconditionally
energy-stable splitting schemes that allows us to split the computation of the three pairs of unknowns (velocity- pres-
sure, phase field-chemical potential and director vector-equilibrium) in three different steps. The fact of being able
to decouple the computations in different linear sub-steps maintaining the discrete energy law is crucial to carry out
relevant numerical experiments under a feasible computational cost and assuring the accuracy of the computed results.
Finally, I will present several numerical simulations in order to show the efficiency of the proposed numerical
schemes, the influence of the shape of the nematic molecules (stretching effects) in the dynamics and the importance
of the interfacial interactions (anchoring effects) in the equilibrium configurations achieved by the system.
This contribution is based on joint work with Francisco Guill´ en-Gonzal´ ez (Universidad de Sevilla, Spain) and Mar´ıa
Angeles Rodr´ıguez-Bellido (Universidad de Sevilla, Spain)
Department of Mathematics
Michigan State University
619 Red Cedar Road
C212 Wells Hall
East Lansing, MI 48824
Phone: (517) 353-0844
Fax: (517) 432-1562
College of Natural Science