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Data-free likelihood-informed dimension reduction of Bayesian inverse problems
Inverse Problems ( IF 2.1 ) Pub Date : 2021-03-18 , DOI: 10.1088/1361-6420/abeafb
Tiangang Cui 1 , Olivier Zahm 2
Affiliation  

Identifying a low-dimensional informed parameter subspace offers a viable path to alleviating the dimensionality challenge in the sampled-based solution to large-scale Bayesian inverse problems. This paper introduces a novel gradient-based dimension reduction method in which the informed subspace does not depend on the data. This permits online–offline computational strategy where the expensive low-dimensional structure of the problem is detected in an offline phase, meaning before observing the data. This strategy is particularly relevant for multiple inversion problems as the same informed subspace can be reused. The proposed approach allows to control the approximation error (in expectation over the data) of the posterior distribution. We also present sampling strategies which exploit the informed subspace to draw efficiently samples from the exact posterior distribution. The method is successfully illustrated on two numerical examples: a PDE-based inverse problem with a Gaussian process prior and a tomography problem with Poisson data and a Besov-${\mathcal{B}}_{11}^{2}$ prior.



中文翻译:

贝叶斯逆问题的无数据似然信息降维

识别低维的知情参数子空间为缓解大规模贝叶斯逆问题的基于采样的解决方案中的维数挑战提供了可行的途径。本文介绍了一种新的基于梯度的降维方法,其中通知子空间不依赖于数据。这允许在线-离线计算策略,其中在离线阶段检测到问题的昂贵的低维结构,这意味着在观察数据之前。该策略与多个反演问题特别相关,因为可以重复使用相同的知情子空间。所提出的方法允许控制后验分布的近似误差(对数据的期望)。我们还提出了利用知情子空间从精确的后验分布中有效地抽取样本的采样策略。该方法在两个数值例子中得到了成功说明:一个基于 PDE 的反问题,一个高斯过程先验,一个断层扫描问题,一个泊松数据和一个 Besov-${\mathcal{B}}_{11}^{2}$ 事先的。

更新日期:2021-03-18
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