CMX is a new 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.

## FacultyVenkat ChandrasekaranMathieu Desbrun Thomas Hou Houman Owhadi Peter Schröder Andrew Stuart Joel Tropp ## Von Karman InstructorsFranca HoffmannKa Chun Lam ## Postdoctoral ResearchersMatthew DunlopKa Chun Lam Shiwei Lan Pengfei Liu ## Grad StudentsMax BudninskiyUtkan Candogan JiaJie Chen De Huang Nikola Kovachki Dzhelil Rufat Florian Schaefer Yong Shen Soh Armeen Taeb Pengchuan Zhang |
## Lunch Seminars(Will be held at 12 noon in Annenberg 213, unless otherwise specified.)
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).
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. ## 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.
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. ## Meetings and Workshops |