CMX is a research group aimed at the development and analysis of novel algorithmic ideas underlying emerging applications in the physical, biological, social and information sciences.  We are distinguished by a shared value system built on the development of foundational mathematical understanding, and the deployment of this understanding to impact on emerging key scientific and technological challenges.


Faculty

Venkat Chandrasekaran
Mathieu Desbrun
Thomas Hou
Houman Owhadi
Peter Schröder
Andrew Stuart
Joel Tropp

Von Karman
Instructors

Franca Hoffmann
Ka Chun Lam

Postdoctoral
Researchers

Alfredo Garbuno-Inigo
Bamdad Hosseini
Pengfei Liu
Krithika Manohar
Melike Sirlanci

Grad Students

Max Budninskiy
Utkan Candogan
JiaJie Chen
De Huang
Nikola Kovachki
Matt Levine
Riley Murray
Florian Schaefer
Yong Shen Soh
Yousuf Soliman
Armeen Taeb
Gene R. Yoo
Shumao Zhang

Fall Quarter 2018



Lunch Seminars

(Will be held at 12 noon in Annenberg 213, unless otherwise specified.)

October 17, 2018
Dimitris_Giannakis
Data-driven 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 measure-preserving, 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 data-driven 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, skew-adjoint operators acting on a suitable RKHS of observables. These "compactified" generators have well-defined, 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, data-driven 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
Operator-adapted wavelets for finite-element differential forms

     We introduce an operator-adapted multiresolution analysis for finite-element differential forms. From a given continuous, linear, bijective, and self-adjoint positive-definite operator L, a hierarchy of basis functions and associated wavelets for discrete differential forms is constructed in a fine-to-coarse fashion and in quasilinear time. The resulting wavelets are L-orthogonal across all scales, and can be used to derive a Galerkin discretization of the operator such that the resulting stiffness matrix becomes block-diagonal, with uniformly well-conditioned and sparse blocks. Because our approach applies to arbitrary differential p-forms, we can derive both scalar-valued and vector-valued wavelets block-diagonalizing 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 divergence-freeness. We demonstrate the benefits of the operator-adapted multiresolution decomposition for coarse-graining and model reduction of linear and nonlinear partial differential equations.

November 7, 2018
Matt Thomson
topic tba

November 28, 2018
Mathieu Desbrun
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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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Other Events

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 bounded-confidence models (which were introduced originally to attempt to model how extremism can take root in a population).





Past Events

Lunch Seminars Other Seminars Meetings & Workshops