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.


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

Matthew Dunlop
Ka Chun Lam
Shiwei Lan
Pengfei Liu

Grad Students

Max Budninskiy
Utkan 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


     

January 24 2018

Andrew Stuart
     

January 31, 2018


     

February 14, 2018


     

February 21, 2018

Thomas Hou
     

February 28, 2018


     

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.


Meetings and Workshops