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 an 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 controlling the approximation error (in expectation over the data) of the posterior distribution. We also present sampling strategies that 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-$\mathcal{B}^2_{11}$ prior.

中文翻译：

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贝叶斯逆问题的无数据似然信息降维

识别低维信息参量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一条可行的途径。本文介绍了一种新颖的基于梯度的降维方法，该方法中，已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（即在观察数据之前）检测到问题的昂贵的低维结构。该策略对于多个反演问题特别重要，因为可以重复使用相同的知悉子空间。所提出的方法允许控制后验分布的近似误差（预期在数据上）。我们还提出了利用知情子空间从准确的后验分布中有效抽取样本的采样策略。在两个数值示例上成功地说明了该方法：具有高斯过程先验的基于PDE的逆问题以及具有Poisson数据和先验Besov-$ \ mathcal {B} ^ 2_ {11} $的层析成像问题。