We study the Electrical Impedance Tomography Bayesian inverse problem for recovering the conductivity given noisy measurements of the voltage on some boundary surface electrodes. The uncertain conductivity depends linearly on a countable number of uniformly distributed random parameters in a compact interval, with the coefficient functions in the linear expansion decaying at an algebraic rate. We analyze the surrogate Markov Chain Monte Carlo (MCMC) approach for sampling the posterior probability measure, where the multivariate sparse adaptive interpolation, with interpolating points chosen according to a lower index set, is used for approximating the forward map. The forward equation is approximated once before running the MCMC for all the realizations, using interpolation on the finite element (FE) approximation at the parametric interpolating points. When evaluation of the solution is needed for a realization, we only need to compute a polynomial, thus cutting drastically the computation time. We contribute a rigorous error estimate for the MCMC convergence. In particular, we show that there is a nested sequence of interpolating lower index sets for which we can derive an interpolation error estimate in terms of the cardinality of these sets, uniformly for all the parameter realizations. An explicit convergence rate for the MCMC sampling of the posterior expectation of the conductivity is rigorously derived, in terms of the interpolating point number, the accuracy of the FE approximation of the forward equation, and the MCMC sample number. We perform numerical experiments using an adaptive greedy approach to construct the sets of interpolation points. We show the benefits of this approach over the simple MCMC where the forward equation is repeatedly solved for all the samples and the non-adaptive surrogate MCMC with an isotropic index set treating all the random parameters equally.

翻译：本文研究电阻抗层析成像的贝叶斯反问题，旨在根据边界电极上电压的含噪测量数据恢复电导率。不确定的电导率线性依赖于紧致区间内可数个均匀分布的随机参数，且线性展开中的系数函数以代数速率衰减。我们分析了用于后验概率测度采样的替代马尔可夫链蒙特卡洛（MCMC）方法，其中采用根据低指标集选取插值点的多元稀疏自适应插值来近似前向映射。在执行MCMC之前，所有实现的前向方程均通过参数化插值点的有限元（FE）近似进行一次插值逼近。当需要针对某次实现求解方程时，仅需计算多项式，从而大幅缩减计算时间。我们为MCMC收敛性提供了严格的误差估计，特别证明了存在嵌套的低指标集序列，使得能够以这些指标集的基数为变量导出插值误差估计，且该误差对所有参数实现一致成立。基于插值点数、前向方程FE近似的精度以及MCMC采样数，我们严格推导了电导率后验期望的MCMC采样显式收敛速率。数值实验中采用自适应贪婪方法构建插值点集，结果表明该方法相较于对全部样本重复求解前向方程的简单MCMC，以及采用等向指标集（将所有随机参数等同处理）的非自适应替代MCMC具有显著优势。