We present a flexible method for computing Bayesian optimal experimental designs (BOEDs) for inverse problems with intractable posteriors. The approach is applicable to a wide range of BOED problems and can accommodate various optimality criteria, prior distributions and noise models. The key to our approach is the construction of a transport-map-based surrogate to the joint probability law of the design, observational and inference random variables. This order-preserving transport map is constructed using tensor trains and can be used to efficiently sample from (and evaluate approximate densities of) conditional distributions that are required in the evaluation of many commonly-used optimality criteria. The algorithm is also extended to sequential data acquisition problems, where experiments can be performed in sequence to update the state of knowledge about the unknown parameters. The sequential BOED problem is made computationally feasible by preconditioning the approximation of the joint density at the current stage using transport maps constructed at previous stages. The flexibility of our approach in finding optimal designs is illustrated with some numerical examples inspired by disease modeling and the reconstruction of subsurface structures in aquifers.

翻译：我们提出了一种灵活的方法，用于计算逆问题中具有难以处理后验分布的贝叶斯最优实验设计。该方法适用于广泛的贝叶斯最优实验设计问题，并能容纳各种最优性准则、先验分布和噪声模型。该方法的核心在于构建一个基于传输映射的代理模型，用以近似实验设计、观测变量和推理随机变量的联合概率分布。这种保序传输映射采用张量列构建，可用于高效采样（并评估近似密度）条件分布，这些条件分布是评估许多常用最优性准则所必需的。该算法还扩展至序贯数据采集问题，其中可依次进行实验以更新对未知参数的知识状态。通过利用先前阶段构建的传输映射对当前阶段联合密度的近似进行预处理，使得序贯贝叶斯最优实验设计问题在计算上变得可行。我们以疾病建模和含水层地下结构重建为背景的数值示例，展示了该方法在寻找最优实验设计方面的灵活性。