FacultyVenkat ChandrasekaranMathieu Desbrun Thomas Hou Houman Owhadi Peter Schröder Andrew Stuart Joel Tropp Von Karman
Franca Hoffmann 
Lunch Seminars(Will be held at 12 noon in Annenberg 213, unless otherwise specified.)
October 17, 2018
Dimitris_Giannakis ▦ Datadriven approaches for spectral analysis of ergodic dynamical systems ▦ It is a remarkable fact, realized in the work of Koopman in the 1930s, that a general deterministic dynamical system can be characterized through its action on spaces of observables (functions of the state) through intrinsically linear evolution operators, acting by composition with the flow map. In the setting of measurepreserving, ergodic dynamics, these operators form a unitary group, whose spectral properties are useful for coherent pattern extraction and prediction of observables, among many applications. In this talk, we will discuss methods for datadriven approximation of the spectra of such unitary groups, focusing, in particular, on systems with mixing (chaotic) dynamics and continuous spectra. These methods utilize techniques from reproducing kernel Hilbert space (RKHS) theory to approximate the generator of the unitary Koopman group (an unbounded operator with complicated spectral behavior), through compact, skewadjoint operators acting on a suitable RKHS of observables. These "compactified" generators have welldefined, purely atomic spectral measures, which are shown to converge to the spectrum of the generator in a limit of vanishing regularization parameter. In addition, the spectral measures of the compactified generators identify coherent observables under the dynamics through corresponding eigenfunctions, and have an associated functional calculus, allowing one to approximate functions of the generator. In particular, exponentiating the generator leads to an approximation of the unitary Koopman operator, which can be used to perform prediction of observables. The RKHS structure also allows stable, datadriven formulations of this framework that converge under fairly general assumptions on the system and observation modality. We illustrate this approach with applications to toy dynamical systems and examples drawn from climate dynamics.
October 24, 2018
Max Budninskiy ▦ Operatoradapted wavelets for finiteelement differential forms ▦ We introduce an operatoradapted multiresolution analysis for finiteelement differential forms. From a given continuous, linear, bijective, and selfadjoint positivedefinite operator L, a hierarchy of basis functions and associated wavelets for discrete differential forms is constructed in a finetocoarse fashion and in quasilinear time. The resulting wavelets are Lorthogonal across all scales, and can be used to derive a Galerkin discretization of the operator such that the resulting stiffness matrix becomes blockdiagonal, with uniformly wellconditioned and sparse blocks. Because our approach applies to arbitrary differential pforms, we can derive both scalarvalued and vectorvalued wavelets blockdiagonalizing a prescribed operator. We also discuss the generality of the construction by pointing out that it applies to various types of computational grids, offers arbitrary smoothness orders of basis functions and wavelets, and can accommodate linear differential constraints such as divergencefreeness. We demonstrate the benefits of the operatoradapted multiresolution decomposition for coarsegraining and model reduction of linear and nonlinear partial differential equations.
November 7, 2018
Matt Thomson ▦ topic tba ▦
November 28, 2018
Mathieu Desbrun ▦ topic tba ▦
January 16, 2109
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January 23, 2019
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January 30, 2019
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February 20, 2019
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February 27, 2019
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October 19, 2018
• Special CMX Seminar • Annenberg 105 2:00pm Mason Porter ▦ Social Contagions and Opinions on Networks ▦ Diseases, rumors, memes, "alternative facts," and many other things spread on networks, whose structure has a significant effect on spreading processes. In this talk, I will give an introduction to spreading processes on networks. I will then discuss several types of "threshold" contagion models, in which spreading occurs when some kind of peer pressure matches or exceeds some kind of internal resistance of nodes. I will also briefly discuss other types of opinion models, such as boundedconfidence models (which were introduced originally to attempt to model how extremism can take root in a population). 
