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The reconstruction of the structure of biological tissue usingelectromyographic data is a non-invasive imaging method with diverse medicalapplications. Mathematically, this process is an inverse problem. Furthermore,electromyographic data are highly sensitive to changes in the electricalconductivity that describes the structure of the tissue. Modeling theinevitable measurement error as a stochastic quantity leads to a Bayesianapproach. Solving the discretized Bayes-inverse problem means drawing samplesfrom the posterior distribution of parameters, e.g., the conductivity, givenmeasurement data. Using, e.g., a Metropolis-Hastings algorithm for this purposeinvolves solving the forward problem for different parameter combinations whichrequires a high computational effort. Low-rank tensor formats can reduce thiseffort by providing a data-sparse representation of all occurring linearsystems of equations simultaneously and allow for their efficient solution. Theapplication of Bayes' theorem proves the well-posedness of the Bayes-inverseproblem. The derivation and proof of a low-rank representation of the forwardproblem allow for the precomputation of all solutions of this problem undercertain assumptions, resulting in an efficient and theory-based samplingalgorithm. Numerical experiments support the theoretical results, but alsoindicate that a high number of samples is needed to obtain reliable estimatesfor the parameters. The Metropolis-Hastings sampling algorithm, using theprecomputed forward solution in a tensor format, draws this high number ofsamples and therefore enables solving problems which are infeasible usingclassical methods.

Bayesian inversion for electromyography using low-rank tensor formats