Lunch Seminars Other Seminars Meetings and Workshops |
Past Events2017: Lunch Seminars
Wednesday, April 19, 2017
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Wednesday, May 17, 2017
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Wednesday, May 24, 2017
Speakers:
October 4, 2017
Venkat Chandrasekaran ▦ Learning regularizers from data ▦
October 18, 2017
Franca Hoffmann ▦ Equilibria in energy landscapes with nonlinear diffusion and nonlocal interaction ▦ We study interacting particles behaving according to a reaction-diffusion equation with nonlinear diffusion and nonlocal attractive interaction. This class of partial differential equations has a very nice gradient flow structure that allows us to make links to homogeneous functionals and variations of well-known functional inequalities. However, the convexity properties of these functionals are not known, and we make use of optimal transport techniques to draw connections between the minimizers of the energy and the equilibria of the equation. Depending on the nonlinearity of the diffusion, the choice of interaction potential and the space dimensionality, we obtain different regimes. Our goal is to understand better the asymptotic behavior of solutions in each of these regimes, starting with the fair-competition regime where attractive and repulsive forces are in balance. No prior knowledge of PDE theory is required as I will give a quick introduction to the notions involved. This is joint work with José A. Carrillo and Vincent Calvez.
October 25, 2017
Houman Owhadi ▦ The game theoretic approach to numerical analysis and algorithm design ▦
November 29, 2017
Joey Teran ▦ Elastoplasticity simulation with the material point method ▦ Hyperelastic constitutive models describe a wide range of materials. Examples include biomechanical soft tissues like muscle, tendon, skin etc. Elastoplastic materials consisting of a hyperelastic constitutive model combined with a notion of stress constraint (or feasible stress region) describe an even wider range of materials. In these models, the elastic potential energy only increases with the elastic part of the deformation decomposition. The evolution of the plastic part is designed to satisfy the stress constraint. A very interesting class of these models arise from frictional contact considerations. I will discuss some recent results and examples in computer graphics and virtual surgery applications. I will also talk about practical simulation of these materials with recent novel Material Point Methods (MPM). 2017: Other Seminars
Thursday, August 31, 2017
Annenberg 213 2:00pm Adam Oberman ▦ A PDE approach to regularization in deep learning ▦
Friday, November 17, 2017
Annenberg 213 12:00pm Leonid Berlyand ▦ Hierarchy of PDE models of cell motility ▦ We consider mathematical PDE models of motility of eukaryotic cells on a substrate and discuss them in a broader context of active materials. Our goal is to capture mathematically the key biological phenomena such as steady motion with no external stimuli and spontaneous breaking of symmetry. We first describe the hierarchy of PDE models of cell motility and then focus on two specific models: the phase-field model and the free boundary problem model. The phase-field model consists of the Allen-Cahn equation for the scalar phase field function coupled with a vectorial parabolic equation for the orientation of the actin filament network. The key mathematical properties of this system are (i) the presence of gradients in the coupling terms and (ii) the mass (volume) preservation constraints. These properties lead to mathematical challenges that are specific to active (out of equilibrium) systems, e.g., the fact that variational principles do not apply. Therefore, standard techniques based on maximum principle and Gamma-convergence cannot be used, and one has to develop alternative asymptotic techniques. The free boundary problem model consists of an elliptic equation describing the flow of the cytoskeleton gel coupled with a convection-diffusion PDE for the density of myosin motors. This PDE system is of Keller-Segel type but in a free boundary setting with nonlocal condition that involves boundary curvature. Analysis of this system allows for a reduction to a Liouville type equation which arises in various applications ranging from geometry to chemotaxis. This equation contains an additional term that presents an additional challenge in analysis. In the analysis of the above models our focus is on establishing the traveling wave solutions that are the signature of the cell motility. We also study breaking of symmetry by proving existence of non-radial steady states. Bifurcation of traveling waves from steady states is established via the Schauder's fixed point theorem for the phase field model and the Leray-Schauder degree theory for the free boundary problem model. These results are obtained in collaboration with J. Fuhrmann, M. Potomkin, and V. Rybalko.
Friday, December 8, 2017
• Special ACM and CMX Seminar • Annenberg 213 2:00pm Gabriel Acosta ▦ Numerical methods for fractional laplacians ▦ The aim of this talk is to review some recent numerical techniques to deal with equations involving fractional laplacian operators. We focus mainly on finite element approaches for two different definitions associated to the fractional laplacian: the so-called integral version, involving a hypersingular kernel and the spectral version of the operator. Some regularity results, needed for the error analysis, are also discussed. |
2018: Lunch Seminars
January 17, 2018
Basile Audoly ▦ The Non-linear mechanics of slender deformable bodies ▦ We discuss some challenges arising in the mechanics of slender (quasi-1D) deformable bodies, such as a thin thread of polymer, curly hair, or a carpenter's tape for example. Slender bodies can exhibit a number of complex and intriguing behaviors that are accessible through simple experiments. The analysis of slender bodies exposes one to many of the fundamental concepts of 3D non-linear mechanics, albeit in a simpler setting where explicit analytical solutions and fast numerical methods can be proposed. Based on examples, we review some problems arising in the analysis of deformable bodies, including the derivation of accurate 1D mechanical models by dimensional reduction, the solution of non-linear 1D models by analytical or numerical methods, and the analysis of material or geometrical instabilities.
January 24 2018
Bamdad Hosseini ▦ Non-Gaussian prior measures in Bayesian inverse problems: from theory to applications ▦
February 28, 2018
Joel Tropp ▦ Applied random matrix theory ▦ 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.
April 18, 2018
Andrew Stuart ▦ The Legacy of Rudolph Kalman ▦ In 1960, Rudolph Kalman published what is arguably the first paper to develop a systematic, principled approach to the use of data to improve the predictive capability of the mathematical models developed to understand the world around us. As our ability to gather data grows at an enormous rate, the importance of this work continues to grow too. The lecture will describe the paper and developments that have stemmed from it, revolutionizing fields such space-craft control, weather prediction, oceanography and oil recovery; the potential applicability in new application domains such as medical imaging and machine learning will also be demonstrated. Some mathematical details will be provided, but limited to simple concepts such as optimization and iteration; the talk is designed to be broadly accessible to anyone with an interest in quantitative science.
April 25, 2018
Peter Schröder ▦Putting Connections to Work▦ When dealing with vector fields and their smoothness we are naturally led to consider a connection (or covariant derivative) and objects such as the connection Dirichlet energy and the connection Laplacian. I will use problems from physical and geometric modeling to illustrate how a connection can encode interesting application data, leading to optimization problems as simple as minimizing the (connection) Dirichlet energy. The subject has rich connections to many areas of physics and mathematics and I will give an introduction which does not require an advanced differential geometry background.
May 16, 2018
Tom Hou ▦Computer-Assisted Analysis of 3D Euler Singularity▦ Whether the 3D incompressible Euler equation can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This question is closely related to the Clay Millennium Problem on 3D Navier-Stokes Equations. In a recent joint work with Dr. Guo Luo, we provided convincing numerical evidence that the 3D Euler equation develops finite time singularities. Inspired by this finding, we have recently developed an integrated analysis and computation strategy to analyze the finite time singularity of a regularized 3D Euler equation. We first transform the regularized 3D Euler equation into an equivalent dynamic rescaling formulation. We then study the stability of an approximate self-similar solution. By designing an appropriate functional space and decomposing the solution into a low frequency part and a high frequency part, we prove nonlinear stability of the dynamic rescaling equation around the approximate self-similar solution, which implies the existence of the finite time blow-up of the regularized 3D Euler equation. This is a joint work with Jiajie Chen, De Huang, and Dr. Pengfei Liu.
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). 2018: Other Seminars
April 26, 2018
• Special ACM and CMX Seminar • Annenberg 213 12:00pm Hau-tieng Wu ▦ Single-Channel Blind Source Separation for Medical Time Series Challenges. ▦ Acquisition of correct features from massive datasets is at the core of data science. A particular interest in medicine is extracting hidden dynamics from a single channel time series composed of multiple oscillatory signals, which could be viewed as a single-channel blind source separation problem. The mathematical and statistical problems are made challenging by the structure of the signal, which consists of non-sinusoidal oscillations, with time varying amplitude/frequency, and by the heteroscedastic nature of the noise. I will discuss recent progress in solving this kind of problem by combining the cepstrum-based nonlinear time-frequency analysis, manifold learning, and random matrix theory. The medical problems motivating this work will be discussed: (1) the extraction of a fetal ECG signal from a single lead maternal abdominal ECG signal; (2) separating respiratory signal from a non-contact PPG signal. If time permits, the clinical trial results and/or an application to the atrial fibrillation will be discussed.
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. 2018: Meetings & Workshops |