April 17, 2019
Thomas Anderson
▦ "Fast hybrid" frequency/time techniques for efficient and parallelizable high-order transient wave scattering simulation ▦
     We propose a frequency/time hybrid integral-equation (though other frequency-domain methods are readily usable) method for the time-dependent wave equation in two and three-dimensional spatial domains. Relying on Fourier transformation in time, the method utilizes a fixed (time-independent) number of frequency-domain integral-equation solutions to evaluate time domain solutions for arbitrarily long times. The approach relies on two main elements, namely, 1) A smooth time-windowing methodology that enables accurate band-limited representations for arbitrary long time signals, and 2) A novel Fourier transform approach which, in a time-parallel manner and without causing spurious periodicity effects, delivers numerically-dispersionless spectrally-accurate solutions. The algorithm can tackle complex physical structures, it enables parallelization in time in a straightforward manner, and it allows for time leaping--that is, solution sampling at any given time T at O(1)-bounded sampling cost. The proposed frequency/time hybridization strategy, which generalizes to any linear partial differential equation in the time domain for which frequency-domain solutions can be obtained (including e.g. the time-domain Maxwell equations or time domain problems posed with dispersive media), provides significant advantages over other available alternatives such as volumetric discretization and convolution-quadrature approaches.

April 24, 2019
Heather Zinn-Brooks
▦ Influence of media on opinion dynamics in social networks ▦
     Many people rely on online social networks as sources for news and information, and the spread of media content with ideologies across the political spectrum both influences online discussions and impacts actions offline. To examine such phenomena, we generalize bounded-confidence models of opinion dynamics on a social network by incorporating media accounts as influencers. We quantify partisanship of content as a continuous parameter on an interval, and we present higher-dimensional generalizations to incorporate content quality and increasingly nuanced political positions. We simulate our model with one and two ideological dimensions, and we use the results of our simulations to quantify the ``entrainment'' of non-media account content to the ideologies of media accounts in networks. We maximize media impact over a social network by tuning the number of media accounts that promote the content and the number of followers of the accounts. Through our numerical computations, we find that the entrainment of the non-media content's ideology to the media ideology depends on structural features of networks such as size, mean number of followers, and the receptiveness of nodes to different opinions. We then introduce content quality --- a key novel contribution of our work --- into our model. We incorporate multiple media sources with ideological biases and qualities that we draw from real media sources and demonstrate that our model can produce distinct communities (``echo chambers") that are polarized in both ideology and quality. Our model provides a step toward understanding content quality in spreading dynamics, with important ramifications for how to mitigate the spread of undesired content and promote the spread of desired content.

April 11, 2019
• CMX Special Seminar •

Annenberg 213
4:00pm

Richard Nickl
▦ Statistical Guarantees for the Bayesian Approach to Inverse Problems ▦

     Bayes methods for inverse problems have become very popular in applied mathematics in the last decade. They provide reconstruction algorithms as well as in-built `uncertainty quantification’ via Bayesian credible sets, and particularly for Gaussian priors can be efficiently implemented by MCMC methodology. For linear inverse problems, they are closely related to classical penalised least squares methods and thus not fundamentally new, but for non-linear and non-convex problems, they give genuinely distinct and computable algorithmic alternatives that cannot be studied by variational analysis or convex optimisation techniques. In this talk we will discuss recent progress in Bayesian Non-Parametric statistics that allows to give rigorous statistical guarantees for posterior mean reconstructions in non-linear non-convex inverse problems arising in some elliptic PDE models and in non-Abelian (`neutronspin’) X-ray tomography.

Southern California Applied Mathematics Symposium (SOCAMS 2019)

• Meeting Poster