The reconstruction of the structure of biological tissue using electromyographic (EMG) data is a non-invasive imaging method with diverse medical applications. Mathematically, this process is an inverse problem. Furthermore, EMG data are highly sensitive to changes in the electrical conductivity that describes the structure of the tissue. Modeling the inevitable measurement error as a stochastic quantity leads to a Bayesian approach. Solving the discretized Bayesian inverse problem means drawing samples from the posterior distribution of parameters, e.g., the conductivity, given measurement data. Using, e.g., a Metropolis–Hastings algorithm for this purpose involves solving the forward problem for different parameter combinations which requires a high computational effort. Low-rank tensor formats can reduce this effort by providing a data-sparse representation of all occurring linear systems of equations simultaneously and allow for their efficient solution. The application of Bayes’ theorem proves the well-posedness of the Bayesian inverse problem. The derivation and proof of a low-rank representation of the forward problem allow for the precomputation of all solutions of this problem under certain assumptions, resulting in an efficient and theory-based sampling algorithm. Numerical experiments support the theoretical results, but also indicate that a high number of samples is needed to obtain reliable estimates for the parameters. The Metropolis–Hastings sampling algorithm, using the precomputed forward solution in a tensor format, draws this high number of samples and therefore enables solving problems which are infeasible using classical methods.

使用肌电图 (EMG) 数据重建生物组织结构是一种具有多种医学应用的非侵入性成像方法。在数学上，这个过程是一个逆问题。此外，EMG 数据对描述组织结构的电导率变化高度敏感。将不可避免的测量误差建模为随机量导致了贝叶斯方法。解决离散贝叶斯逆问题意味着从参数的后验分布中抽取样本，例如电导率，给定测量数据。例如，为此目的使用 Metropolis-Hastings 算法涉及解决不同参数组合的前向问题，这需要大量的计算工作。低秩张量格式可以通过同时提供所有出现的线性方程组的数据稀疏表示来减少这种工作，并允许它们的有效解决方案。贝叶斯定理的应用证明了贝叶斯逆问题的适定性。前向问题的低秩表示的推导和证明允许在某些假设下预先计算该问题的所有解决方案，从而产生有效且基于理论的采样算法。数值实验支持理论结果，但也表明需要大量样本才能获得对参数的可靠估计。Metropolis-Hastings 采样算法，使用张量格式的预先计算的前向解，