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Optimization of the Mean First Passage Time in Near-Disk and Elliptical Domains in 2-D with Small Absorbing Traps
[article]

2020
*
arXiv
*
pre-print

*The*determination

*of*

*the*

*mean*

*first*

*passage*

*time*(MFPT) for a Brownian particle

*in*a bounded

*2*-

*D*

*domain*containing

*small*

*absorbing*

*traps*is a fundamental problem

*with*biophysical applications. ... We develop a hybrid asymptotic-numerical approach to predict

*optimal*configurations

*of*m

*small*stationary circular

*absorbing*

*traps*that minimize

*the*average MFPT

*in*

*near*-

*disk*

*and*

*elliptical*

*domains*. ...

*the*average

*mean*

*first*capture

*time*

*of*

*the*particle. ...

##
###
Simulation and Optimization of Mean First Passage Time Problems in 2-D using Numerical Embedded Methods and Perturbation Theory
[article]

2019
*
arXiv
*
pre-print

We develop novel numerical methods

arXiv:1911.07842v1
fatcat:p7parohoi5go7kwvbcs4mtrple
*and*perturbation approaches to determine*the**mean**first**passage**time*(MFPT) for a Brownian particle to be captured by either*small*stationary or mobile*traps*inside a ... bounded*2*-*D*confining*domain*. ...*The*authors thank Justin Tzou for discussions that lead to*the**time*-relaxation algorithm for moving*trap*problems. ...##
###
Global Optimization of the Mean First Passage Time for Narrow Capture Problems in Elliptic Domains
[article]

2022
*
arXiv
*
pre-print

*In*a recent paper , an explicit size-

*and*

*trap*location-dependent expansion

*of*

*the*average

*mean*

*first*

*passage*

*time*(MFPT)

*in*a 2D

*elliptic*

*domain*was derived. ... Further, an asymptotic formula

*the*for

*the*average MFPT

*in*

*elliptic*

*domains*

*with*N circular

*traps*

*of*arbitrary sizes is derived,

*and*sample

*optimal*configurations involving non-equal

*traps*are computed ... Acknowledgements A.C. is thankful to NSERC

*of*Canada for research support through

*the*Discovery grant RGPIN-2019-05570. ...

##
###
Mean first passage time for a small rotating trap inside a reflective disk
[article]

2014
*
arXiv
*
pre-print

We compute

arXiv:1405.2302v2
fatcat:l3a62xxvjbbk3kynxpfnldcpm4
*the**mean**first**passage**time*(MFPT) for a Brownian particle inside a two-dimensional*disk**with*reflective boundaries*and*a*small*interior*trap*that is rotating at a constant angular velocity. ...*In**the*regime 1 ≪ω≪O(ε^-1)*the**optimal**trap*path approaches*the*boundary*of**the**disk*. However as ω is further increased to O(ε^-1),*the**optimal**trap*path "jumps" closer to*the*origin. ... Kolokolnikov is supported by NSERC discovery*and*NSERC accelerator grants. We thank Michael Ward for useful discussions*and*suggestions. ...##
###
Mathematical modeling and numerical computation of narrow escape problems

2012
*
Physical Review E
*

*The*narrow escape problem refers to

*the*problem

*of*calculating

*the*

*mean*

*first*

*passage*

*time*(MFPT) needed for an average Brownian particle to leave a

*domain*

*with*an insulating boundary containing N

*small*... This

*mean*

*first*

*passage*

*time*satisfies

*the*Poisson partial differential equation (PDE) subject to a mixed Dirichlet-Neumann boundary condition on

*the*

*domain*boundary,

*with*

*the*Dirichlet condition corresponding ... Acknowledgements

*The*authors are grateful to NSERC for research support through Discovery grants (A.C.

*and*M.J.W.)

*and*a USRA fellowship (A.R.). ...

##
###
Diffusion on a Sphere with Localized Traps: Mean First Passage Time, Eigenvalue Asymptotics, and Fekete Points

2009
*
SIAM Journal on Applied Mathematics
*

Here, we attack this problem by calculating asymptotic results for

doi:10.1137/080733280
fatcat:wzitzqbktvejddbtzlp44fx3ey
*the**mean**first**passage**time*for a diffusing particle confined to*the*surface*of*a sphere,*in**the*presence*of*N partially*absorbing**traps*...*The*rate at which*the**small*diffusing molecule becomes captured by one*of**the**traps*is determined by asymptotically calculating*the*principal eigenvalue for*the*Laplace operator on*the*sphere*with**small*...*In**the*context*of**the**mean**first**passage**time*problem,*the*constant χ*in*(3. ...##
###
Optimization of First Passage Times by Multiple Cooperating Mobile Traps

2017
*
Multiscale Modeling & simulation
*

We study

doi:10.1137/16m1060169
fatcat:imehoidb6bfwjh4mi4usmcyrpu
*the**mean*capture*time**of*an unbiased random walker by multiple*absorbing*mobile*traps**in*bounded*domains**of*one*and*two spatial dimensions. ...*In*two dimensions, we consider two*small**traps*rotating*with**the*same angular velocity ω inside a unit*disk*,*and*characterize*the**optimal*positions (radii*of*rotation*and*relative phase)*of**the*two*traps*... RGPIN-33798*and*NSERC Accelerator Grant No. RG-PAS/461907. We gratefully acknowledge*the*anonymous reviewers whose feedback greatly improved*the*presentation*of*this article. ...##
###
Mean First Passage Time for a Small Rotating Trap inside a Reflective Disk

2015
*
Multiscale Modeling & simulation
*

We compute

doi:10.1137/140968604
fatcat:rftgu25prfhcziiukz6dudklr4
*the**mean**first**passage**time*(MFPT) for a Brownian particle inside a two-dimensional*disk**with*reflective boundaries*and*a*small*interior*trap*that is rotating at a constant angular velocity. ...*In*Figure 1 (a), we illustrate a scenario*in*which a*trap*is located at x = 1/*2*on a*domain**with*reflecting boundaries at x = 0*and*x = 1. ...*The*authors thank Michael Ward for useful discussions*and*suggestions. ...##
###
An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part I: Two-Dimensional Domains

2010
*
Multiscale Modeling & simulation
*

*The*

*mean*

*first*

*passage*

*time*(MFPT) is calculated for a Brownian particle

*in*a bounded two-dimensional

*domain*that contains N

*small*nonoverlapping

*absorbing*windows on its boundary. ...

*In*

*the*asymptotic limit where

*the*

*absorbing*patches have

*small*measure,

*the*method

*of*matched asymptotic expansions is used to calculate

*the*MFPT

*in*an arbitrary two-dimensional

*domain*

*with*a smooth boundary ... Cyril Muratov

*of*NJIT for exhibiting

*the*connection between our results

*and*

*the*dilute fraction limit

*of*homogenization theory discussed

*in*section 3. ...

##
###
Asymptotic Methods For PDE Problems In Fluid Mechanics and Related Systems With Strong Localized Perturbations In Two-Dimensional Domains
[chapter]

2010
*
Asymptotic Methods in Fluid Mechanics: Survey and Recent Advances
*

bodies, eigenvalue problems

doi:10.1007/978-3-7091-0408-8_2
fatcat:gbxmxmkozvgt5dzmkhl2mcvvu4
*in*perforated*domains*,*the*calculation*of**the**mean**first**passage**time*for Brownian motion*in*a*domain**with**small**traps*, etc. ... Denoting*the*trajectory*of**the*Brownian particle by X(t),*the**mean**first**passage**time*(MFPT) v(x) is defined as*the*expectation value*of**the**time*τ taken for*the*Brownian particle to become*absorbed*somewhere ... a cell-signaling problem*in*mathematical biology (cf. [52] ),*the*narrow escape problem from a sphere that has*small**absorbing*windows on its boundary,*and**the**mean**first**passage**time**in*a three-dimensional ...##
###
An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part II: The Sphere

2010
*
Multiscale Modeling & simulation
*

*The*

*mean*

*first*

*passage*

*time*(MFPT) is calculated for a Brownian particle

*in*a spherical

*domain*

*in*R 3 that contains N

*small*nonoverlapping

*absorbing*windows, or

*traps*, on its boundary. ... Denoting

*the*trajectory

*of*

*the*Brownian particle by X(t),

*the*

*mean*

*first*

*passage*

*time*(MFPT) v(x) is defined as

*the*expectation value

*of*

*the*

*time*τ taken for

*the*Brownian particle to become

*absorbed*somewhere ... We would like to thank Sheldon Richards

*and*Professor Raymond Spiteri

*of*

*the*Department

*of*Computer Science at

*the*University

*of*Saskatchewan for providing Table 4 .

*2*. ...

##
###
Trapping of Planar Brownian Motion: Full First Passage Time Distributions by Kinetic Monte-Carlo, Asymptotic and Boundary Integral Methods
[article]

2021
*
arXiv
*
pre-print

*In*contrast to problems posed

*in*finite

*domains*, simple moments

*of*

*the*distribution, such as

*the*

*mean*(MFPT)

*and*variance, are not defined

*and*it is necessary to obtain

*the*full arrival statistics. ... To validate

*the*results

*of*this theory,

*and*to obtain

*the*arrival

*time*statistics

*in*very general configurations

*of*

*absorbers*, we introduce an efficient Kinetic Monte Carlo (KMC) method that describes ... Preceding works on

*the*theory

*of*

*first*arrival

*times*to

*small*

*absorbing*sites have largely focussed on determining moments such as

*the*

*mean*

*first*

*passage*

*time*(MFPT) [11, 15, 18-20, 33, 34]

*and*

*in*some ...

##
###
Time scale of diffusion in molecular and cellular biology

2014
*
Journal of Physics A: Mathematical and Theoretical
*

*The*narrow escape

*time*NET

*in*diffusion theory is

*the*

*mean*

*first*

*passage*

*time*(MFPT)

*of*a Brownian trajectory to a

*small*

*absorbing*part

*of*an otherwise reflecting boundary

*of*a bounded

*domain*. ...

*The*coarse-grained

*time*scale

*of*cellular function is determined from molecular diffusion by

*the*

*mean*

*first*

*passage*

*time*

*of*molecular Brownian motion to a

*small*targets or through narrow

*passages*. ...

##
###
Geometrical structure of Laplacian eigenfunctions
[article]

2013
*
arXiv
*
pre-print

We summarize

arXiv:1206.1278v2
fatcat:pmeiikacdrhjxjqjpoh44tvaii
*the*properties*of*eigenvalues*and*eigenfunctions*of**the*Laplace operator*in*bounded Euclidean*domains**with*Dirichlet, Neumann or Robin boundary condition. ...*The*main focus is put onto multiple intricate relations between*the*shape*of*a*domain**and**the*geometrical structure*of*eigenfunctions. ...*The*authors thank H. Obuse*and*A. Barnett for providing their images*of*eigenfunctions*and*allowing us to reproduce them. We acknowledge fruitful discussions*with*Y. G. Sinai*and*A. L. Delitsyn. ...##
###
Surveying Diffusion in Complex Geometries. An Essay
[article]

2009
*
arXiv
*
pre-print

We look into

arXiv:0909.1588v1
fatcat:2wdehn7m75f3fjfodvfbvn77cu
*the*role*of*geometrical complexity at different levels, from boundary microroughness to hierarchical structure*and*connectivity*of**the*whole diffusion-confining*domain*. ... Here,*the*geometrical complexity, entangled*with**the*stochastic character*of*diffusive motion, results*in*numerous fascinating*and*sometimes unexpected effects like diffusion screening or localization. ... Plapp for careful proofreading*of**the*manuscript*and*numerous advices. ...
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