Mirzaev Inom, Byrne Erin C, Bortz David M
Department of Applied Mathematics, University of Colorado, Boulder, CO 80309-0526.
The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760.
Inverse Probl. 2016;32(9). doi: 10.1088/0266-5611/32/9/095005. Epub 2016 Jul 15.
We investigate the inverse problem of identifying a conditional probability measure in measure-dependent evolution equations arising in size-structured population modeling. We formulate the inverse problem as a least squares problem for the probability measure estimation. Using the Prohorov metric framework, we prove existence and consistency of the least squares estimates and outline a discretization scheme for approximating a conditional probability measure. For this scheme, we prove general method stability. The work is motivated by Partial Differential Equation (PDE) models of flocculation for which the shape of the post-fragmentation conditional probability measure greatly impacts the solution dynamics. To illustrate our methodology, we apply the theory to a particular PDE model that arises in the study of population dynamics for flocculating bacterial aggregates in suspension, and provide numerical evidence for the utility of the approach.
我们研究了在大小结构种群建模中出现的依赖测度的演化方程中识别条件概率测度的反问题。我们将反问题表述为概率测度估计的最小二乘问题。利用普罗霍罗夫度量框架，我们证明了最小二乘估计的存在性和一致性，并概述了一种用于逼近条件概率测度的离散化方案。对于该方案，我们证明了一般方法的稳定性。这项工作的动机来自于絮凝的偏微分方程（PDE）模型，其中破碎后条件概率测度的形状对解的动力学有很大影响。为了说明我们的方法，我们将该理论应用于悬浮液中絮凝细菌聚集体的种群动力学研究中出现的一个特定的PDE模型，并为该方法的实用性提供了数值证据。
