Agenda

Abstracts

• Individualized prediction through the optimal transport barycenter problem: Esteban Tabak, Courant Institute, NYU

  Abstract

  This talk will discuss an optimal transport-based methodology to generate samples from a conditional probability distribution rho(x | z) only known through a finite number of samples {x^i, z^i}. For example, the variables x may represent the effect of a medical treatment and the factors z may characterize the patient and treatment under consideration. The factors can either be known explicitly, such a patient’s age or a drug’s dosage, be up to discovery, such as a subset of individuals who react to treatment in a particular way, or only be known by association, such as those characteristics shared by a particular social group or even the idiosyncratic response to treatment of each individual patient. The procedure is based on the push-forward of the z-indexed family of distributions rho(x|z) to their barycenter mu(y) through particle flows. In addition to its use as a conditional generator, the methodology serves to eliminate batch effects when consolidating data bases, to uncover hidden factors (broadly extending classical methodologies such as k-means clustering and principal component analysis) and to predict and optimize trajectories under treatment. 

• Around random matrices using free probability tools: Marwa Banna, Science Division, NYUAD

  Abstract

  In this talk, I give an introduction to free probability and random matrix theory and highlight the connection between the two domains. I will particularly show how free probability tools can be used to obtain different results on random matrices such as: regularity properties of limiting spectral distributions and obtaining quantitative bounds for such limit theorems. 

• Mathematical models of phase separation and their analysis: Maurizio Grasselli, Math Department, Polytechnic University of Milan

  Abstract

  Phase separation, namely, the creation of two (or more) distinct phases from a single homogeneous mixture (think of oil and vinegar), is a phenomenon which characterizes many important processes. In particular, it has recently become a paradigm in Cell Biology. Phase separation arises from the competition between the mixing entropy of the mixture and the demixing effects due to the reciprocal attraction of the components of the same species. This requires a temperature below some critical threshold that depends on the mixture itself. A suitable free energy functional, whose variables are the concentration of the mixture components, accounts for the competing terms. The evolution is then governed by the corresponding conserved gradient flow which is the well-known and widely studied Cahn-Hilliard equation. In its original form, the Cahn-Hilliard equation is a fourth-order nonlinear partial differential equation which was proposed in 1958 to model phase separation in binary alloys. Since then, it has been used to describe, theoretically and numerically, many different processes characterized by phase separation. An alternative to the Cahn-Hilliard equation has  been proposed by J. Rubinstein and P. Sternberg in 1992. This is a second-order reaction-diffusion equation of Allen-Cahn type with a nonlocal constraint which preserves the total mass. This the reason why it is usually called conserved Allen-Cahn equation. I would like to give a brief introduction to the modeling and an overview of some recent results obtained for the Cahn-Hilliard equation and for the conserved Allen-Cahn equation.

• Optimal Mass Transport meets Stochastic Thermodynamics: a geometric approach to power harvesting from anisotropic fluctuations: Tryphon Georgiou, Deptartment of Mechanical Aerospace Engineering, University California Irvine

  Abstract

  In contrast to the classical concept of a Carnot engine that alternates contact between heat baths of different temperatures, naturally occurring processes usually harvest energy from anisotropy, being exposed simultaneously to chemical and thermal fluctuations of different intensities. In these cases, the enabling mechanism responsible for transduction of energy is the presence of a non-equilibrium steady state (NESS). A suitable stochastic model for such a phenomenon is the Brownian gyrator – a two-degree of freedom stochastically driven system that exchanges energy and heat with the environment. In this context, we exploit the relationship between optimal mass transport and stochastic thermodynamics to present a geometric view of the energy harvesting mechanism, that entails a forced periodic trajectory of the system state on the thermodynamic manifold. Dissipation and work output are expressed accordingly as path integrals of a controlled process, and fundamental limitations on power and efficiency are expressed in geometric terms via a relationship to an isoperimetric problem. The analysis presented provides guiding principles for building autonomous engines that extract work from anistropic fluctuations.

• On the identication of cavities in a nonlinear model arising from cardiac electrophysiology: Elena Beretta, Science Division, NYUAD

  Abstract

  Detecting ischemic regions from noninvasive (or minimally invasive) measurements is of primary importance to prevent lethal ventricular ischemic tachycardia. This is usually performed by recording the electrical activity of the heart, by means of either body surface or intracardiac measurements. Mathematical and numerical models of the cardiac electrophysiology can be used to shed light on the potentialities of electrical measurements in detecting ischemias. More specifically, the goal is to combine boundary measurements of (body-surface or intracavitary) potentials and a mathematical description of the electrical activity of the heart in order to identify the position, the shape and the size of heart ischemias and/or infarctions. The ischemic region is a non-excitable tissue that can be modeled as an electrical insulator and the cardiac electrical activity can be comprehensively described in terms of the monodomain model, consisting of a boundary value problem for a semilinear reaction-diffusion equation. I

  n my talk, I will analyze the case of an insulated heart neglecting the coupling with the torso and considering the steady-state version of the monodomain model. This results in the challenging inverse problem of detecting cavities for a semilinear equation with a single measurement of the transmembrane potential. With minimal regularity assumptions on the cavities, I will first show well-posedness of the direct problem, prove uniqueness of the inverse problem and propose a novel reconstruction algorithm by means of a phas-eld approach justified via Gamma-convergence. Finally, I will describe some recent results that we have obtained for the inverse problem in the case of the time dependent monodomain model system.

• Normalizing Flows, Optimal Transport, and Bayesian Inference: Giulio Trigila, Baruch College, CUNY

  Abstract

  Computational Bayesian inference aims to estimate the (posterior) conditional distribution of the parameters characterizing a statistical model given observations from the model. This inverse problem is often challenging to solve due to the high dimensionality of the parameter space and to the fact that the numerical simulation of the model at hand is computationally expensive. A class of approaches that gained popularity during the last few years in the computational Bayesian inference community is based on finding a map from the prior distribution of the parameters to the posterior. We will find this map by means of a new class of Normalizing Flows that, leveraging on the theory of Optimal Transport, alleviates some of the issues characterizing current state-of-the-art methods.

• Stochastic flows and advection: Yves Le Jan, Science Division, NYUAD

  Abstract

  We review some aspects of the theory of stochastic differential equations related to advection models.

   

• A diffuse interface model for tumor growth: long-time dynamics and optimal control: Cecilia Cavaterra, Science Division, NYUAD, Math Deptartment, University of Milan

  Abstract

  Modeling tumor growth dynamics has recently become an important issue in applied  mathematics. Here we investigate the long-time dynamics and an optimal control problem for a diffuse interface model describing the growth of a tumor in presence of a nutrient and surrounded by host tissues. The state system consists of a Cahn-Hilliard type equation
  for the tumor cell fraction coupled with a reaction-diffusion equation for the nutrient. The possible medication that serves to eliminate tumor cells is introduced into the system through the nutrient. In this setting, the control variable acts as an external source in the nutrient equation.

  First, we analyze the problem of “long-time treatment” under a suitable given mass source and prove the convergence of any global solution to a single equilibrium. Second, we consider the “finite-time treatment” that corresponds to an optimal control problem. Here the objective cost functional depends on a free variable, representing the unknown treatment time to be optimized. We prove the existence of an optimal control and obtain first order necessary optimality conditions for both the drug concentration and the treatment time.

• Stability and instability in fluid motions: Weiren Zhao, Science Division, NYUAD

  Abstract

  In this talk, I will present some recent progress in the asymptotic stability and instability results of shear flow in both inviscid and viscous fluids. The inviscid damping and enhanced dissipation phenomenon are the main stability mechanisms for stabilizing the system. I will also discuss a new dynamical approach for the instability problem to show the threshold at which enhanced dissipation cannot be maintained.