AI 中文总结
针对定量光声层析成像和电阻抗断层成像这两个非线性成像逆问题，基于两阶段推进方法开发高效贝叶斯框架，通过为辅助变量制定回归问题并避免MCMC采样，以较低计算成本实现准确重建和可靠不确定性估计。
AI 中文摘要
贝叶斯方法为估计非线性逆问题中的参数并量化估计中的不确定性提供了自然框架。然而，当此类非线性逆问题的正向模型由偏微分方程给出时，贝叶斯推断通常借助MCMC方法进行，由于每次MCMC迭代都需求解偏微分方程，这些方法计算成本高，对大规模成像问题往往不实用。本文针对定量光声层析成像（QPAT）和电阻抗断层成像（EIT）这两个非线性成像逆问题，基于最近提出的两阶段推进方法，开发了一种计算高效的贝叶斯框架。首先为辅助变量制定贝叶斯回归问题，其后验可闭式获得，再通过确定性重建映射推进以得到未知参数的后验，避免MCMC采样。给出了严格的测度论依据来解释诱导后验为贝叶斯后验，并推导了QPAT和EIT的后验收缩率。数值结果表明，该方法以较低计算成本提供了准确重建和可靠的不确定性估计。
英文摘要
Bayesian methods provide a natural framework for estimating a parameter in non-linear inverse problems and quantifying uncertainty in the estimation. However, when the forward model for such non-linear inverse problems is given by some Partial Differential Equation (PDE), Bayesian inference is typically carried out by resorting to MCMC methods. Since each MCMC iteration requires solving a PDE, these methods become computationally expensive and are often impractical for large-scale imaging problems. In this work, we develop a computationally efficient Bayesian framework for two such nonlinear imaging inverse problems: Quantitative Photoacoustic Tomography (QPAT) and Electrical Impedance Tomography (EIT). Building on a recently proposed two-stage pushforward methodology, we first formulate a Bayesian regression problem for an auxiliary variable whose posterior is available in closed form. This posterior is then pushed forward through a deterministic reconstruction map to obtain a posterior on the unknown parameter, avoiding MCMC sampling. We give a rigorous measure-theoretic justification to interpret the induced posterior as a Bayesian posterior and derive posterior contraction rates for both QPAT and EIT. Numerical results show that the proposed method provides accurate reconstructions and reliable uncertainty estimates at a arguably lower computational cost than standard Bayesian approaches.
