Exploring equivalence domain in nonlinear inverse problems using Covariance Matrix Adaption Evolution Strategy (CMAES) and random sampling
Exploring equivalence domain in nonlinear inverse problems using Covariance Matrix Adaption Evolution Strategy (CMAES) and random sampling
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DOI：
10.1093/gji/ggw063
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发表时间：
2016-05-01
影响因子：
2.8
通讯作者：
Kuvshinov, Alexey V.
中科院分区：
文献类型：
--
作者：
Grayver, Alexander V.;Kuvshinov, Alexey V.
提出了一种求解非线性偏微分方程（PDE）约束反问题等价区域（艾德）的采样方法。为此，我们首先应用最先进的随机优化算法称为协方差矩阵自适应进化策略（CMAES），以确定模型空间的低失配区域。然后对这些区域进行随机抽样，以创建等效模型的集合并量化不确定性。CMAES的目的是探索模型空间的全球性和非常病态的问题是强大的。我们表明，收敛所需的迭代次数增长在一个温和的速度相对于未知数和算法是并行的。我们通过使用广义高斯分布将问题公式化。这使我们能够无缝地使用任意范数作为残差和正则化项。我们表明，各种正规化规范有利于研究不同类别的等价解决方案。我们进一步展示了如何使用CMAES提供的信息，可以大大提高性能的标准大都会黑斯廷斯马尔可夫链蒙特卡罗算法。该方法进行了测试，通过使用个别和联合反演大地电磁，可控源电磁（EM）和全球电磁感应数据。
This paper presents a methodology to sample equivalence domain (ED) in nonlinear partial differential equation (PDE)-constrained inverse problems. For this purpose, we first applied state-of-the-art stochastic optimization algorithm called Covariance Matrix Adaptation Evolution Strategy (CMAES) to identify low-misfit regions of the model space. These regions were then randomly sampled to create an ensemble of equivalent models and quantify uncertainty. CMAES is aimed at exploring model space globally and is robust on very ill-conditioned problems. We show that the number of iterations required to converge grows at a moderate rate with respect to number of unknowns and the algorithm is embarrassingly parallel. We formulated the problem by using the generalized Gaussian distribution. This enabled us to seamlessly use arbitrary norms for residual and regularization terms. We show that various regularization norms facilitate studying different classes of equivalent solutions. We further show how performance of the standard Metropolis-Hastings Markov chain Monte Carlo algorithm can be substantially improved by using information CMAES provides. This methodology was tested by using individual and joint inversions of magneotelluric, controlled-source electromagnetic (EM) and global EM induction data.