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Research Seminars: Applied and Computational Mathematics

Fall 2022

Time & Location: Typically talks will be in Gibson Hall 325 at 3:00 PM..
Organizers: Punshon-Smith, Samuel and Buvoli, Tommaso

Archives

 

September 9

Title: Microswimmer in confinements

Hongfei Chen | Tulane University

Abstract:  We consider the active Brownian particle (ABP) model for a two-dimensional microswimmer with fixed speed, whose direction of swimming changes according to a Brownian process. The probability density for the swimmer obeys a Fokker–Planck equation defined on the configuration space, whose structure depends on the swimmer’s shape, center of rotation and domain of swimming. We enforce zero probability flux at the boundaries of configuration space. We consider the dynamics of a swimmer in an infinite channel and a lattice of point obstacles. At first neglecting hydrodynamic interactions, we derive a reduced equation for a swimmer in an infinite channel, in the limit of small rotational diffusivity, and find that the invariant density depends strongly on the swimmer’s precise shape and center of rotation. We also give a formula for the mean reversal time: the expected time taken for a swimmer to completely reverse direction in the channel. Using homogenization theory, we find an expression for the effective longitudinal diffusivity of a swimmer in the channel, and show that it is bounded by the mean reversal time. Finally, we include hydrodynamic interactions with walls, and examine the role of shape. For a swimmer in a periodic lattice of point obstacles, we apply the same framework as in the channel, solve the invariant density and use the homogenization approach to compute effective diffusivity. Moreover, when the swimmer is only a circular or needle diffuser that cannot swim, we solve the cell problem and effective diffusivity asymptotically.

Time: TBA
Location:  Gibson 325

 

September 16

Title: Modeling aspects of vesicle electrodeformation and transport

Adnan Morshed  | Tulane University

Abstract:  Characterization of the mechanical properties of cells and other sub-micron vesicles such as virus and neurotransmitter vesicles are necessary to understand their deformation dynamics. This characterization can be done by submerging the vesicle in a fluid medium and deforming it with controlled electric field exposure known as electrodeformation. Electrodeformation of biological and artificial lipid vesicles is directly influenced by the vesicle and media properties and geometric factors. The problem is compounded when the vesicle is naturally charged which creates electrophoretic forcing on the vesicle membrane. This talk will highlight some of the modeling aspects of electrodeformation and transport of charged vesicles immersed in a fluid media under the influence of a DC electric field. The electric field and fluid-solid interactions are resolved using a hybrid immersed interface-immersed boundary technique. Viability of the width averaged domain conductivity and current as indicators of vesicle deformation and movement will be discussed. Vesicle movement due to electrophoresis is also characterized by the change in local conductivity which can serve as a potential sensor for electrodeformation experiments and in solid-state nanopore applications.

Time: 3:00
Location:  Gibson 325

 

September 30

Title: Using regularity to estimate Lyapunov exponents

Sam Punshon-Smith - Tulane University

Abstract:  I will discuss a new method for estimating Lyapunov exponents for hypoelliptic diffusions from below using quantitative local regularity estimates on the stationary measure of the lift to the projective bundle. For damped driven truncations of nonlinear conservative stochastic partial differential equations (SPDE) in fluctuation dissipation scaling, I will outline a general strategy for proving positivity of the top Lyapunov exponent that can be applied to Galerkin truncations of the stochastic Navier-Stokes equations on T^2. The proof in the Navier-Stokes case involves verifying transitivity properties of a certain matrix Lie algebra associated with the linearized Euler equations using tools from algebraic geometry. The techniques are quite general and can be applied more broadly to Galerkin truncations of many other evolution type SPDE. This work is joint with Jacob Bedrossian and Alex Blumenthal.

Time: 3:00
Location:  Gibson 325

October 7
Fall Break

 

October 14

Title: Discovering governing equations from data using sparse identification and deep learning methods

Joseph Bakarji  | University of Washington

Abstract:  Recent advances in machine learning methods have made it possible to automate the process of scientific discovery. Will we one day be able to discover physical laws directly from data, with as little assumptions about the underlying physical system as possible? I briefly go over the latest attempts to accomplish this goal and focus on my recent work in combining deep learning with sparse identification of differential equations. First, I show how probability distribution function (PDF) equations can be inferred from Monte Carlo simulations for coarse-graining and closure approximations. Second, I present our latest results on discovering dimensionless groups from data, using the Buckingham Pi theorem as a constraint. And third, I go over the deep delay autoencoder algorithm that reconstructs high dimensional systems of differential equations from partial measurements as motivated by Takens' theorem. I finally highlight the limitations of these methods and propose a few directions for future research.

Time: 3:00
Location:  Gibson 325

 

October 21

Title: TBA

Laurel Ohm  | Princeton University

Abstract:  TBA

Time: 3:00
Location:  Gibson 325

 

November 4
Clifford Lectures

 

November  11

Title: TBA

Anna Nelson  |  Duke University

Abstract:  TBA

Time: 3:00
Location:  Gibson 325

 

 November 18/19
Clifford Lectures

 

December  2

Title: TBA

Klas Modin  |  University of Gothenberg Chalmers

Abstract:  TBA

Time: 2:00pm-3:00pm - Note the different time
Location:  Online over Zoom

 

December 9
Last Day Of Class