Integration based profile likelihood calculation for PDE constrained parameter estimation problems
Integration based profile likelihood calculation for PDE constrained parameter estimation problems
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DOI：
10.1088/0266-5611/32/12/125009
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发表时间：
2016-12-01
期刊：
影响因子：
2.1
通讯作者：
Kaltenbacher, B.
中科院分区：
文献类型：
--
作者：
Boiger, R.;Hasenauer, J.;Kaltenbacher, B.
偏微分方程（PDE）模型在工程和自然科学中被广泛用于描述时空过程。所考虑的过程的参数往往是未知的，必须从实验数据估计。由于部分观测值和测量噪声，这些参数估计值具有不确定性。这种不确定性可以使用轮廓似然度来评估，这是一种可靠但计算密集的方法。在本文中，我们提出了集成为基础的方法（陈和Jennrich 2002年J. COMPUT。Graph. Stat. 11 714-32）并将其适用于具有PDE约束的逆问题。虽然现有的方法的轮廓似然计算的参数估计问题的PDE约束依赖于重复优化，所提出的方法利用了一个动态系统的发展沿着的似然配置文件。我们推导出动力系统的未约估计问题，证明收敛性和研究的PDE情况下的集成为基础的方法的性能。为了评估所提出的方法，我们比较它与一个简单的反应扩散模型的细胞图案化过程的最先进的算法。我们观察到一个良好的准确性的方法，以及显着的速度相比，建立的方法。基于积分的轮廓计算有利于严格的不确定性分析计算要求偏微分方程约束的参数估计问题。
Partial differential equation (PDE) models are widely used in engineering and natural sciences to describe spatio-temporal processes. The parameters of the considered processes are often unknown and have to be estimated from experimental data. Due to partial observations and measurement noise, these parameter estimates are subject to uncertainty. This uncertainty can be assessed using profile likelihoods, a reliable but computationally intensive approach. In this paper, we present the integration based approach for the profile likelihood calculation developed by (Chen and Jennrich 2002 J. Comput. Graph. Stat. 11 714-32) and adapt it to inverse problems with PDE constraints. While existing methods for profile likelihood calculation in parameter estimation problems with PDE constraints rely on repeated optimization, the proposed approach exploits a dynamical system evolving along the likelihood profile. We derive the dynamical system for the unreduced estimation problem, prove convergence and study the properties of the integration based approach for the PDE case. To evaluate the proposed method, we compare it with state-of-the-art algorithms for a simple reaction-diffusion model for a cellular patterning process. We observe a good accuracy of the method as well as a significant speed up as compared to established methods. Integration based profile calculation facilitates rigorous uncertainty analysis for computationally demanding parameter estimation problems with PDE constraints.