使用低秩张量格式的肌电贝叶斯反演
Bayesian inversion for electromyography using low-rank tensor formats
摘要：
利用肌电数据重建生物组织结构是一种具有广泛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，肌电图数据对描述组织结构的电导率的变化高度敏感。将不可避免的测量误差建模为随机量导致贝叶斯方法。求解离散化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，给出测量数据。例如，用于此目的的Metropolis-Hastings算法涉及求解不同参数组合的正向问题，这需要高计算量。低秩张量格式可以通过同时提供所有发生的线性方程组的数据稀疏表示来减少这种努力，并允许它们的有效解。应用贝叶斯定理证明了贝叶斯反问题的适定性。推导和证明的低秩表示的前向问题允许预先计算的所有解决方案，这个问题在一定的假设下，从而在一个有效的和基于理论的采样算法。数值实验支持理论结果，但也表明，一个高的样本个数是需要获得可靠的估计参数。Metropolis-Hastings采样算法使用张量格式的预先计算的前向解，绘制了如此高的样本个数，因此能够解决使用经典方法不可行的问题。
英文摘要：
The reconstruction of the structure of biological tissue using electromyographic data is a non-invasive imaging method with diverse medical applications. Mathematically, this process is an inverse problem. Furthermore, electromyographic 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 Bayes-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 Bayes-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.
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