Difference between revisions of "SIAM Student Chapter Seminar"

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__NOTOC__
 
__NOTOC__
  
*'''When:''' Every other Friday at 1:30 pm
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*'''When:''' 3:30 pm
*'''Where:''' B333 Van Vleck Hall
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*'''Where:''' Zoom
 
*'''Organizers:''' [http://www.math.wisc.edu/~xshen/ Xiao Shen]
 
*'''Organizers:''' [http://www.math.wisc.edu/~xshen/ Xiao Shen]
 
*'''Faculty advisers:''' [http://www.math.wisc.edu/~jeanluc/ Jean-Luc Thiffeault], [http://pages.cs.wisc.edu/~swright/ Steve Wright]  
 
*'''Faculty advisers:''' [http://www.math.wisc.edu/~jeanluc/ Jean-Luc Thiffeault], [http://pages.cs.wisc.edu/~swright/ Steve Wright]  
*'''To join the SIAM Chapter mailing list:''' email [join-siam-chapter@lists.wisc.edu].
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*'''To join the SIAM Chapter mailing list:''' email [mailto:siam-chapter+join@g-groups.wisc.edu siam-chapter+join@g-groups.wisc.edu].
  
 
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<br>
  
== Spring 2020  ==
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== Fall 2020  ==
  
 
{| cellpadding="8"
 
{| cellpadding="8"
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!align="left" | title
 
!align="left" | title
 
|-
 
|-
|Jan 31
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|9/29
|[https://lorenzonajt.github.io/ Lorenzo Najt] (Math)
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|Yu Feng (Math)
|''[[#Jan 31, Lorenzo Najt (Math)|Ensemble methods for measuring gerrymandering: Algorithmic problems and inferential challenges]]''
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|''[[#9/29, Yu Feng (Math)|Phase separation in the advective Cahn--Hilliard equation]]''
 
|-
 
|-
 
|-
 
|-
|Feb 14
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|10/14
|[https://www.math.wisc.edu/~pollyyu/ Polly Yu] (Math)
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|Dongyu Chen (WPI)
|''[[#Feb 14, Polly Yu (Math)|Algebra, Dynamics, and Chemistry with Delay Differential Equations]]''
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|''[[#10/14, Yuchen Dong (WPI)|A Half-order Numerical Scheme for Nonlinear SDEs with one-sided Lipschitz Drift and H\:{o}lder Continuous Diffusion Coefficients]]''
|-
 
|-
 
|Feb 21
 
|Gage Bonner (Physics)
 
|''[[#Feb 21, Gage Bonner (Physics)|Growth of history-dependent random sequences]]''
 
 
|-
 
|-
 
|-
 
|-
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== Abstracts ==
 
== Abstracts ==
  
=== Jan 31, Lorenzo Najt (Math) ===
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=== 9/29, Yu Feng (Math) ===
'''Ensemble methods for measuring gerrymandering: Algorithmic problems and inferential challenges'''
+
'''Phase separation in the advective Cahn--Hilliard equation'''
 
 
We will review some recent work regarding measuring gerrymandering by sampling from the space of maps, including two methods used in a recent amicus brief to the supreme court. This discussion will highlight some of the computational challenges of this approach, including some complexity-theory lower bounds and bottlenecks in Markov chains. We will examine the robustness of these statistical methods through their connection to phase transitions in the self-avoiding walk model, as well as their dependence on artifacts of discretization. This talk is largely based on https://arxiv.org/abs/1908.08881
 
  
=== Feb 14, Polly Yu (Math) ===
+
The Cahn--Hilliard equation is a classic model of phase separation in binary mixtures that exhibits spontaneous coarsening of the phases. We study the Cahn--Hilliard equation with an imposed advection term in order to model the stirring and eventual mixing of the phases. The main result is that if the imposed advection is sufficiently mixing then no phase separation occurs, and the solution instead converges exponentially to a homogeneous mixed state. The mixing effectiveness of the imposed drift is quantified in terms of the dissipation time of the associated advection-hyperdiffusion equation, and we produce examples of velocity fields with a small dissipation time. We also study the relationship between this quantity and the dissipation time of the standard advection-diffusion equation.
'''Algebra, Dynamics, and Chemistry with Delay Differential Equations'''
 
  
Delay differential equations (DDEs) can exhibit more complicated behavior than their ODE counterparts. What is stable in the ODE setting could exhibit oscillation in DDE. Where do delay equations show up anyway? In this talk, we’ll introduce DDEs, and how (sort-of-)linear algebra gives information about the stability of DDEs.
 
  
 +
=== 10/14, Yuchen Dong (WPI) ===
 +
'''A Half-order Numerical Scheme for Nonlinear SDEs with one-sided Lipschitz Drift and Hölder Continuous Diffusion Coefficients'''
  
=== Feb 21, Gage Bonner (Physics) ===
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We consider positivity-preserving explicit schemes for one-dimensional nonlinear stochastic differential
''' Growth of history-dependent random sequences'''
+
equations. The drift coefficients satisfy the one-sided Lipschitz condition, and the diffusion coefficients
 +
are Hölder continuous. To control the fast growth of moments of solutions, we introduce several explicit
 +
schemes including the tamed and truncated Euler schemes. The fundamental idea is to guarantee the
 +
non-negativity of solutions. The proofs rely on the boundedness for negative moments and exponential of
 +
negative moments. We present several numerical schemes for a modified Cox-Ingersoll-Ross model and a
 +
two-factor Heston model and demonstrate their half-order convergence rate.
  
Unlike discrete Markov chains, history-dependent random sequences are sequences of random variables whose "next" term depends on all others seen previously. For this reason, they can be difficult to analyze. I will discuss some simple and fun cases where the long-term behavior of the sequence can be computed explicitly in expectation.
 
  
  

Latest revision as of 21:24, 12 October 2020



Fall 2020

date speaker title
9/29 Yu Feng (Math) Phase separation in the advective Cahn--Hilliard equation
10/14 Dongyu Chen (WPI) A Half-order Numerical Scheme for Nonlinear SDEs with one-sided Lipschitz Drift and H\:{o}lder Continuous Diffusion Coefficients

Abstracts

9/29, Yu Feng (Math)

Phase separation in the advective Cahn--Hilliard equation

The Cahn--Hilliard equation is a classic model of phase separation in binary mixtures that exhibits spontaneous coarsening of the phases. We study the Cahn--Hilliard equation with an imposed advection term in order to model the stirring and eventual mixing of the phases. The main result is that if the imposed advection is sufficiently mixing then no phase separation occurs, and the solution instead converges exponentially to a homogeneous mixed state. The mixing effectiveness of the imposed drift is quantified in terms of the dissipation time of the associated advection-hyperdiffusion equation, and we produce examples of velocity fields with a small dissipation time. We also study the relationship between this quantity and the dissipation time of the standard advection-diffusion equation.


10/14, Yuchen Dong (WPI)

A Half-order Numerical Scheme for Nonlinear SDEs with one-sided Lipschitz Drift and Hölder Continuous Diffusion Coefficients

We consider positivity-preserving explicit schemes for one-dimensional nonlinear stochastic differential equations. The drift coefficients satisfy the one-sided Lipschitz condition, and the diffusion coefficients are Hölder continuous. To control the fast growth of moments of solutions, we introduce several explicit schemes including the tamed and truncated Euler schemes. The fundamental idea is to guarantee the non-negativity of solutions. The proofs rely on the boundedness for negative moments and exponential of negative moments. We present several numerical schemes for a modified Cox-Ingersoll-Ross model and a two-factor Heston model and demonstrate their half-order convergence rate.



Past Semesters