Bayesian inversion for electromyography using low-rank tensor formats

Inverse Problems, vol. 37, pp. 055003

Abstract

Abstract 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.

Authors 4

1. Anna Rörich corresponding

   University of Stuttgart

   Affiliation as printed

   Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Allmandring 5b, 70569 Stuttgart, Germany

2. Tim A. Werthmann corresponding Aachen Institut für Geometrie und Praktische Mathematik

   RWTH Aachen University

   Affiliation as printed

   Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany

3. Dominik Göddeke corresponding

   University of Stuttgart · Stuttgart Center for Simulation Science

   Affiliation as printed

   Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Allmandring 5b, 70569 Stuttgart, Germany

   Stuttgart Center for Simulation Science, University of Stuttgart, Pfaffenwaldring 5a, 70569 Stuttgart, Germany

4. Lars Grasedyck corresponding Aachen Institut für Geometrie und Praktische Mathematik

   RWTH Aachen University

   Affiliation as printed

   Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany