SIAM Review, Volume 63, Issue 3, Page 525-555, January 2021.
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. The average MFPT is the expected capture time assuming a uniform distribution of starting points for the random walk. We develop a hybrid asymptotic-numerical approach to predicting optimal configurations of $m$ small stationary circular absorbing traps that minimize the average MFPT in near-disk and elliptical domains. For a general class of near-disk domains, we illustrate through several specific examples how simple, yet highly accurate, numerical methods can be used to implement the asymptotic theory. From the derivation of a new explicit formula for the Neumann Green's function and its regular part for the ellipse, a numerical approach based on our asymptotic theory is used to investigate how the spatial distribution of the optimal trap locations changes as the aspect ratio of an ellipse of fixed area is varied. The results from the hybrid theory for the ellipse are compared with full PDE numerical results computed from the closest point method [S. Iyaniwura et al., Multiscale Model. Simul., to appear]. For long and thin ellipses, it is shown that the optimal trap pattern for $m=2,\ldots,5$ identical traps is collinear along the semimajor axis of the ellipse. For such essentially 1-D patterns, a thin-domain asymptotic analysis is formulated and implemented to accurately predict the optimal locations of collinear trap patterns and the corresponding optimal average MFPT.

中文翻译：

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带小吸收阱的二维近盘和椭圆域中平均首次通过时间的优化

SIAM 评论，第 63 卷，第 3 期，第 525-555 页，2021 年 1 月。
在包含小吸收陷阱的有界二维域中确定布朗粒子的平均首次通过时间 (MFPT) 是生物物理应用的一个基本问题。平均 MFPT 是预期的捕获时间，假设随机游走的起点分布均匀。我们开发了一种混合渐近数值方法来预测 $m$ 小型固定圆形吸收陷阱的最佳配置，这些陷阱将近盘和椭圆域中的平均 MFPT 最小化。对于一类一般的近盘域，我们通过几个具体的例子来说明如何使用简单但高度准确的数值方法来实现渐近理论。从 Neumann Green 函数及其椭圆规则部分的新显式公式的推导，基于我们的渐近理论的数值方法用于研究最佳陷阱位置的空间分布如何随着固定区域椭圆纵横比的变化而变化。将椭圆混合理论的结果与最近点法计算的完整 PDE 数值结果进行比较 [S. Iyaniwura 等人，多尺度模型。同时，出现]。对于细长椭圆，表明 $m=2,\ldots,5$ 相同陷阱的最佳陷阱模式沿椭圆的半长轴共线。对于这种本质上是一维的模式，制定并实施了薄域渐近分析，以准确预测共线陷阱模式的最佳位置和相应的最佳平均 MFPT。