Thu, 02 Jun 2022

14:00 - 15:00
Virtual

Balanced truncation for Bayesian inference

Elizabeth Qian
(Caltech)
Abstract

We consider the Bayesian inverse problem of inferring the initial condition of a linear dynamical system from noisy output measurements taken after the initial time. In practical applications, the large dimension of the dynamical system state poses a computational obstacle to computing the exact posterior distribution. Balanced truncation is a system-theoretic method for model reduction which obtains an efficient reduced-dimension dynamical system by projecting the system operators onto state directions which simultaneously maximize energies defined by reachability and observability Gramians. We show that in our inference setting, the prior covariance and Fisher information matrices can be naturally interpreted as reachability and observability Gramians, respectively. We use these connections to propose a balancing approach to model reduction for the inference setting. The resulting reduced model then inherits stability properties and error bounds from system theory, and yields an optimal posterior covariance approximation. 

Thu, 27 Jan 2022

16:00 - 17:00
Virtual

Learning Homogenized PDEs in Continuum Mechanics

Andrew Stuart
(Caltech)
Further Information
Abstract

Neural networks have shown great success at learning function approximators between spaces X and Y, in the setting where X is a finite dimensional Euclidean space and where Y is either a finite dimensional Euclidean space (regression) or a set of finite cardinality (classification); the neural networks learn the approximator from N data pairs {x_n, y_n}. In many problems arising in the physical and engineering sciences it is desirable to generalize this setting to learn operators between spaces of functions X and Y. The talk will overview recent work in this context.

Then the talk will focus on work aimed at addressing the problem of learning operators which define the constitutive model characterizing the macroscopic behaviour of multiscale materials arising in material modeling. Mathematically this corresponds to using machine learning to determine appropriate homogenized equations, using data generated at the microscopic scale. Applications to visco-elasticity and crystal-plasticity are given.

Wed, 02 Feb 2022

16:00 - 17:00
N3.12

Higher Teichmüller spaces

Nathaniel Sagman
(Caltech)
Abstract

The Teichmüller space for a closed surface of genus g is the space of marked complex/hyperbolic structures on the surface. Teichmüller space also identifies with the space of Fuchsian representations of the fundamental group into PSL(2,R) (mod conjugation). Higher Teichmüller theory concerns special representations of surface (or hyperbolic) groups into higher rank Lie groups of non-compact type.

Fri, 06 Nov 2020

15:00 - 16:00
Virtual

Level-set methods for TDA on spatial data

Michelle Feng
(Caltech)
Abstract

In this talk, I will give a brief introduction to level-set methods for image analysis. I will then describe an application of level-sets to the construction of filtrations for persistent homology computations. I will present several case studies with various spatial data sets using this construction, including applications to voting, analyzing urban street patterns, and spiderwebs. I will conclude by discussing the types of data which I might imagine such methods to be suitable for analyzing and suggesting a few potential future applications of level-set based computations.

 

Mon, 19 Nov 2018

14:15 - 15:15
L4

Zed-hat

Sergei Gukov
(Caltech)
Abstract

The goal of the talk will be to introduce a class of functions that answer a question in topology, can be computed via analytic methods more common in the theory of dynamical systems, and in the end turn out to enjoy beautiful modular properties of the type first observed by Ramanujan. If time permits, we will discuss connections with vertex algebras and physics of BPS states which play an important role, but will be hidden "under the hood" in much of the talk.

 

Thu, 14 Jun 2018

14:00 - 15:00
L4

Applied Random Matrix Theory

Prof. Joel Tropp
(Caltech)
Abstract

Random matrices now play a role in many areas of theoretical, applied, and computational mathematics. Therefore, it is desirable to have tools for studying random matrices that are flexible, easy to use, and powerful. Over the last fifteen years, researchers have developed a remarkable family of results, called matrix concentration inequalities, that balance these criteria. This talk offers an invitation to the field of matrix concentration inequalities and their applications.

Thu, 06 Nov 2014

12:00 - 13:00
L4

Towards an effective theory for nematic elastomers in a membrane limit

Paul Plucinsky
(Caltech)
Abstract
 

For nematic elastomers in a membrane limit, one expects in the elastic theory an interplay of material and structural non-linearities. For instance, nematic elastomer material has an associated anisotropy which allows for the formation of microstructure via nematic reorientation under deformation. Furthermore, polymeric membrane type structures (of which nematic elastomer membranes are a type) often wrinkle under applied deformations or tractions to avoid compressive stresses. An interesting question which motivates this study is whether the formation of microstructure can suppress wrinkling in nematic elastomer membranes for certain classes of deformation. This idea has captured the interest of NASA as they seek lightweight and easily deployable space structures, and since the use of lightweight deployable membranes is often limited by wrinkling.

 

In order to understand the interplay of these non-linearities, we derive an elastic theory for nematic elastomers of small thickness. Our starting point is three-dimensional elasticity, and for this we incorporate the widely used model Bladon, Terentjev and Warner for the energy density of a nematic elastomer along with a Frank elastic penalty on nematic reorientation. We derive membrane and bending limits taking the thickness to zero by exploiting the mathematical framework of Gamma-convergence. This follows closely the seminal works of LeDret and Raoult on the membrane theory and Friesecke, James and Mueller on the bending theory.

 

Thu, 14 Jun 2007

14:00 - 15:00
Comlab

Dynamic depletion of vortex stretching and nonlinear stability of 3D incompressible flows

Prof Tom Hou
(Caltech)
Abstract
Whether the 3D incompressible Euler or Navier-Stokes equations

can develop a finite time singularity from smooth initial data has been

an outstanding open problem. Here we review some existing computational

and theoretical work on possible finite blow-up of the 3D Euler equations.

We show that the geometric regularity of vortex filaments, even in an

extremely localized region, can lead to dynamic depletion of vortex

stretching, thus avoid finite time blowup of the 3D Euler equations.

Further, we perform large scale computations of the 3D Euler equations

to re-examine the two slightly perturbed anti-parallel vortex tubes which

is considered as one of the most attractive candidates for a finite time

blowup of the 3D Euler equations. We found that there is tremendous dynamic

depletion of vortex stretching and the maximum vorticity does not grow faster

than double exponential in time. Finally, we present a new class of solutions

for the 3D Euler and Navier-Stokes equations, which exhibit very interesting

dynamic growth property. By exploiting the special nonlinear structure of the

equations, we prove nonlinear stability and the global regularity of this class of solutions.

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