Likelihood-informed dimension reduction for nonlinear inverse problems
Likelihood-informed dimension reduction for nonlinear inverse problems
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DOI：
10.1088/0266-5611/30/11/114015
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发表时间：
2014-11-01
期刊：
影响因子：
2.1
通讯作者：
Spantini, A.
中科院分区：
文献类型：
--
作者：
Cui, T.;Martin, J.;Spantini, A.
反问题的固有维度受先验信息、观测数据的精度和数量以及正演算子的平滑特性的影响。从贝叶斯的角度来看，在许多问题中，从前到后的变化可能被限制在参数空间的相对低维子空间内。我们提出了一种降维方法，通过刻画先验和似然对后验分布支持度的相对影响，定义和识别这样一个子空间，称为似然信息子空间(LIS)。这种辨识为使用非线性正演模型和高斯先验的贝叶斯推断提供了新的和更有效的计算方法。特别地，我们将后验分布近似为定义在LIS上的低维后验分布与边缘化为互补子空间的先验分布的乘积。然后，马尔可夫链蒙特卡罗抽样可以在更低的维度上进行，计算效率显著提高。我们还引入了一种Rao-Blackwell化策略，该策略将蒙特卡罗估计的后验期望去随机化，以获得额外的方差减少。我们用两个数值例子证明了我们的方法的有效性：由椭圆型偏微分方程控制的地下水系统渗透率的推断，以及基于全球臭氧监测系统(GOOS)观测的大气遥感问题。
The intrinsic dimensionality of an inverse problem is affected by prior information, the accuracy and number of observations, and the smoothing properties of the forward operator. From a Bayesian perspective, changes from the prior to the posterior may, in many problems, be confined to a relatively low-dimensional subspace of the parameter space. We present a dimension reduction approach that defines and identifies such a subspace, called the 'likelihood-informed subspace' (LIS), by characterizing the relative influences of the prior and the likelihood over the support of the posterior distribution. This identification enables new and more efficient computational methods for Bayesian inference with nonlinear forward models and Gaussian priors. In particular, we approximate the posterior distribution as the product of a lower-dimensional posterior defined on the LIS and the prior distribution marginalized onto the complementary subspace. Markov chain Monte Carlo sampling can then proceed in lower dimensions, with significant gains in computational efficiency. We also introduce a Rao-Blackwellization strategy that de-randomizes Monte Carlo estimates of posterior expectations for additional variance reduction. We demonstrate the efficiency of our methods using two numerical examples: inference of permeability in a groundwater system governed by an elliptic PDE, and an atmospheric remote sensing problem based on Global Ozone Monitoring System (GOMOS) observations.