Inverse problems based on electromyographic (EMG) data are at the core of the "Digital Human" Visionary Example of EXC 2075. The reconstruction of the structure of biological tissue using electromyographic data is a non-invasive imaging method with diverse medical applications. In this cooperation between Anna Rörich (GS SimTech PhD Researcher from the Institute of Applied Analysis and Numerical Simulation) and Dominik Göddeke (SimTech PI from the Institute of Applied Analysis and Numerical Simulation, IANS) with colleagues from RWTH Aachen, the inevitable measurement error is included into the EMG model directly as a stochastic quantity, leading to a so-called Bayesian approach.
The paper https://doi.org/10.1088/1361-6420/abd85a shows under which conditions this notoriously ill-posed problem admits uniquely existing solutions in the weak sense, and also proposes a novel data-driven acceleration technique to actually compute solutions, based on deriving a data-sparse representation for all conceivable discretization of the parameter space of the forward model in the inversion procedure. In practice, the resulting approach is not only mathematically well-founded, but also faster by orders of magnitude.
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.