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 28, 2018
Mathieu Desbrun
▦ Dimensionality Reduction via Geometry Processing ▦
     A common and oft-observed assumption for high-dimensional datasets is that they sample (possibly with added noise) a low-dimensional manifold embedded in a high-dimensional space. In this situation, Non-linear Dimensional Reduction (NLDR) offers to find a low-dimensional embedding of the data (i.e., unfold/unroll the original dataset in R^d for a small value of d) that creates little to no distortion so as to reduce the computational complexity of subsequent data processing.
     Given the clear geometric premises of this task, we revisit a series of well-known NLDR tools (such as ISOMAP, Locally-Linear Embeddings, etc) from a geometric processing standpoint, and introduce two novel geometric manifold learning methods, Parallel Transport Unfolding (PTU) and Spectral Affine Kernel Embedding (SAKE). These contributions leverage basic geometry processing notions such as parallel transport or generalized barycentric coordinates. We show that tapping into discrete differential geometry significantly improves the robustness of NLDR compared to existing manifold learning algorithms. Somehow surprisingly, we also demonstrate that we can trivially derive from NLDR the first eigenbased 3D mesh and point set editing technique, hinting at the generality of our approach.
     Work done in collaboration with Max Budninskiy (Caltech) and Yiying Tong (Michigan State U).

January 23, 2019
Lior Pachter
▦ Computational and algorithmic challenges of biological sequence alignment ▦

     I will briefly review the history of biological sequence alignment, and introduce some of the fundamental challenges that emerge from biological, technical and computational considerations. I will then discuss current state-of-the-art methods, which are based on a variety of computational ideas, with a view towards forthcoming opportunities.

February 20, 2019
Oscar Bruno
▦ Fast spectral time-domain PDE solvers for complex structures: the Fourier-Continuation method ▦
     We present fast spectral solvers for time-domain Partial Differential Equations. Based on a novel Fourier-Continuation (FC) method for the resolution of the Gibbs phenomenon, these methodologies give rise to time-domain solvers for PDEs for general engineering problems and structures. The methods enjoy a number of desirable properties, including spectral time evolution essentially free of pollution or dispersion errors for general PDEs in the time domain, with conditional/unconditional stability for explicit/alternating-direction methods and high order of temporal accuracy. A variety of applications to linear and nonlinear PDE problems will be presented, including the diffusion and wave equations, the Navier-Stokes equations and the elastic wave equation, demonstrating the significant improvements the new algorithms can provide over the accuracy and speed resulting from other approaches.

February 27, 2019
Tapio Schneider
▦ Clouds, Climate Predictions, and Possible Surprises in Warm Climates ▦
     While climate change is certain, precisely how climate will change is less clear. Uncertainties arise from the representation of small-scale processes such as clouds and turbulence. Moreover, these uncertainties are poorly quantified; the ensemble of existing climate models may not span the range of possible climate outcomes. I will illustrate this with an example from the dynamics of stratocumulus clouds, which are crucial for Earth’s energy balance. Large-eddy simulations of stratocumulus clouds show that they can exhibit an instability that leads to dramatic global warming under high greenhouse gas concentrations—an instability that does not seem to be captured by current climate models and whose probability of occurrence cannot be assessed with current models.
     Breakthroughs in the accuracy of climate projections and in the quantification of their uncertainties are now within reach, thanks to advances in the computational and data sciences and in the availability of Earth observations from space and from the ground. To achieve a step change in accuracy of climate projections, we are developing a new Earth system modeling platform. Developed by a university consortium dubbed the Climate Modeling Alliance (CliMA), it will fuse an Earth system model (ESM) with global observations and targeted local high-resolution simulations of clouds and other elements of the Earth system. CliMA capitalizes on advances in data assimilation and machine learning to develop an ESM that automatically learns from diverse data sources, be they observations from space or data generated computationally in high-resolution simulations.

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.

May 7, 2019
James Saunderson
▦ Certifying polynomial nonnegativity via hyperbolic optimization ▦
      Certifying nonnegativity of multivariate polynomials is fundamental to solving optimization problems modeled with polynomials. One well-known way to certify nonnegativity is to express a polynomial as a sum of squares. Furthermore, the search for such a certificate can be carried out via semidefinite optimization. An interesting generalization of semidefinite optimization, that retains many of its good algorithmic properties, is hyperbolic optimization. Are there natural certificates of nonnegativity that we can search for via hyperbolic optimization, and that are not obviously captured by sums of squares? If so, these could have the potential to generate hyperbolic optimization-based relaxations of optimization problems with that may be stronger, in some sense, than semidefinite optimization-based relaxations. In this talk, I will describe one candidate for such "hyperbolic certificates of nonnegativity", and discuss what is known about their relationship with sums of squares.

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).

November 15, 2018
• Special CMX Seminar •

Annenberg 314
4:00pm

Sam Stechmann
▦ Multi-model Communication and Data Assimilation for Mitigating Model Error and Improving Forecasts ▦

     Models for weather and climate prediction are complex, and each model typically has at least a small number of phenomena that are poorly represented, such as perhaps the Madden–Julian Oscillation (MJO) or El Niño–Southern Oscillation (ENSO) or sea ice. Furthermore, it is often a very challenging task to modify and improve a complex model without creating new deficiencies. On the other hand, it is sometimes possible to design a low-dimensional model for a particular phenomenon, such as the MJO or ENSO, with significant skill, although the model may not represent the dynamics of the full weather– climate system. Here a strategy is proposed to mitigate these model errors by taking advantage of each model’s strengths. The strategy involves inter-model data assimilation, during a forecast simulation, whereby models can exchange information in order to obtain more faithful representations of the full weather–climate system. As an initial investigation, the method is examined using a simplified scenario of linear models, involving a system of stochastic partial differential equations (SPDEs) as an imperfect tropical climate model and stochastic differential equations (SDEs) as a low-dimensional model for the MJO. It is shown that the MJO prediction skill of the imperfect climate model can be enhanced to equal the predictive skill of the low-dimensional model. Such an approach could provide a route to improving global model forecasts in a minimally invasive way, with modifications to the prediction system but without modifying the complex global physical model itself. The methods may also be applicable to other settings where multiple models can be used together to improve predictions.

November 29, 2018
• CMX Special Seminar •

Annenberg 213
12:00pm

Hau-Tieng Wu
▦ Diffusion to fuse sensors — with sleep dynamics analysis as an example ▦

     Quantifying dynamics of a nonlinear and complex system, like our physiological system, from multivariate time series recorded from multimodal sensors is challenging. We develop a new diffusion geometry based sensor fusion algorithm, called alternating diffusion map, to capture the “common” nonlinear information. In addition to its theoretical/statistical properties under the Riemannian manifold model and its relationship with the traditional canonical correlation analysis, its application to sleep dynamics study will be discussed. We show that this unsupervised approach not only leads to a compatible automatic annotation results with the state-of-the-art approach based on deep neural network, but also provides interpretable sleep dynamics features for the ongoing sleep boosting study. If time permits, how to extract spectral features by a nonlinear-type time-frequency analysis will be discussed.

December 13, 2018
• CMX Special Seminar •

Annenberg 213
4:30pm

Lisa Maria Kreusser
▦ An Anisotropic Interaction Model for Simulating Fingerprints ▦

     Motivated by the formation of fingerprint patterns we consider a class of interaction models with short-range repulsive, long-range attractive forces whose orientations depend on an underlying stress field. This stress field introduces an anisotropy leading to complex patterns which do not occur in the associated isotropic models. The transition between the isotropic and the anisotropic model can be characterized by one of the model parameters and we study the variation of this parameter both analytically and numerically. We analyze the steady states and their stability by considering the particle model and the associated mean-field equations. Besides, we propose a bio-inspired model to simulate fingerprint patterns as stationary solutions by choosing the underlying tensor field appropriately.

January 28, 2019
• CMX Special Seminar •

Annenberg 213
12:00pm

Camille Carvalho
▦ Accurate evaluation of near-fields in plasmonic structures ▦

     Plasmonic structures are commonly made of dielectrics and metals. At optical frequencies metals exhibit unusual electromagnetic properties like a dielectric permittivity with a negative real part whereas dielectrics have a positive one. This configuration allows the propagation of electromagnetic surface waves strongly oscillating at the metal-dielectric interface, and hyper-oscillating if the interface presents corners. Standard numerical methods to study surface plasmons excitation do not always take into account the multiple scales inherent in electromagnetic problems which may lead to inaccurate predictions. In this presentation we present some techniques (using Finite Element method, or Boundary Integral Equation methods) to accurately compute and efficiently take into account the multiple scales of 2D light scattering problems in plasmonic structures.

January 31, 2019
• CMX Seminar Series: Optimal Transport •

Location tba
2:30pm

Wuchen Li
▦ Learning via Wasserstein information geometry  ▦

     In this talk, I review several differential structures from optimal transport (Wasserstein metric). Based on it, I will introduce the Wasserstein natural gradient in parametric models. The L2-Wasserstein metric tensor in probability density space is pulled back to the one on parameter space, under which the parameter space forms a Riemannian manifold. The Wasserstein gradient flows and proximal operator in parameter space are derived. We demonstrate that the Wasserstein natural gradient works efficiently in several machine learning examples, including Boltzmann machine, generative adversary models (GANs) and variational Bayesian statistics.

February 28, 2019
• CMX Seminar Series: Optimal Transport •

Location tba
2:30pm

Chenchen Mou
▦ Mean field games on graphs ▦

     Mean field game theory is the study of the limit of Nash equilibria of differential games when the number of players tends to infinity. It was introduced by J.-M. Lasry and P.-L. Lions, and independently by P. Caines, M. Huang and R. Malhame. A fundamental object in the theory is the master equation, which fully characterizes the limit equilibrium. In this talk, we will introduce Mean field game and master equations on graphs. We will construct solutions to both equations and link them to the solution to a Hamilton-Jacobi equation on graphs.

March 7, 2019
• CMX Special Seminar •

Annenberg 213
12:00pm

Gitta Kutyniok
▦ Deep Learning and Modeling: Taking the Best out of Both Worlds ▦

     Inverse problems in imaging such as denoising, recovery of missing data, or the inverse scattering problem appear in numerous applications. However, due to their increasing complexity, model-based methods are often today not sufficient anymore. At the same time, we witness the tremendous success of data-based methodologies, in particular, deep neural networks for such problems. However, at the same time, pure deep learning approaches often neglect known and valuable information from the modeling world. In this talk, we will provide an introduction to this problem complex and then focus on the inverse problem of computed tomography, where one of the key issues is the limited angle problem. For this problem, we will demonstrate the success of hybrid approaches. We will develop a solver for this severely ill-posed inverse problem by combining the model-based method of sparse regularization by shearlets with the data-driven method of deep learning. Our approach is faithful in the sense that we only learn the part which cannot be handled by model-based methods, while applying the theoretically controllable sparse regularization technique to all other parts. We further show that our algorithm significantly outperforms previous methodologies, including methods entirely based on deep learning.

March 7, 2019
• CMX Special Seminar •

Annenberg 314
4:00pm

Zbigniew J. Jurek
▦ Selfdecomposability and S-Selfdecomposability: A view towards simulations? ▦

     Click here for abstract

March 14, 2019
• CMX Seminar Series: Optimal Transport •

Annenberg 314
2:30pm

Flavien Leger
▦ The Schrödinger bridge problem ▦

We present the Schrödinger bridge problem and discuss its connections with optimal transport, mean field games and stochastic optimal control.
     

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.

May 16, 2019
• CMX Special Seminar •

Annenberg 213
4:00pm

C.-C. Jay Kuo
▦ Interpretable Convolutional Neural Networks(CNNs)via Feedforward Design  ▦

      Given a convolutional neural network (CNN) architecture, its network parameters are determined by backpropagation (BP) nowadays. The underlying mechanism remains to be a black-box after a large amount of theoretical investigation. In this talk, I describe a new interpretable and feedforward (FF) design with the LeNet-5 as an example. The FF-trained CNN is a data-centric approach that derives network parameters based on training data statistics layer by layer in one pass. To build the convolutional layers, we develop a new signal transform, called the Saab (Subspace approximation with adjusted bias) transform. The bias in filter weights is chosen to annihilate nonlinearity of the activation function. To build the fully-connected (FC) layers, we adopt a label-guided linear least squared regression (LSR) method. The classification performances of BP- and FF-trained CNNs on the MNIST and the CIFAR-10 datasets are compared. The computational complexity of the FF design is significantly lower than the BP design and, therefore, the FF-trained CNN is ideal for mobile/edge computing. We also comment on the relationship between BP and FF designs by examining the cross-entropy values at nodes of intermediate layers.