Browsing by Author "Göddeke, Dominik (Prof. Dr.)"

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Open Access
A Bayesian approach to parameter reconstruction from surface electromyographic signals
(2021) Rörich, Anna; Göddeke, Dominik (Prof. Dr.)
Applying a Bayesian approach to infer the electrical conductivity of a body or body part from surface electromyographic (EMG) signals yields a non-invasive and radiation-free imaging technique. Further, measuring the surface EMG signals that stem from voluntary muscle contractions, there is no need to apply external electrical stimuli to the body. The electrical conductivity provides structural information of the corresponding tissue that is used to estimate whether the tissue has isotropic or anisotropic properties and which is the preferred conducting direction, if applicable. Additionally, changes in the magnitude of the electrical conductivity indicate changes in the tissue material. Together, these properties of the electrical conductivity provide medical images of the examined body part. This imaging process results in an inverse and mathematically ill-posed problem. Including a stochastic model of the inevitable measurement error into the mathematical problem description, the whole system is embedded into a probabilistic framework. Thus, instead of estimating the structure of the examined body part, the probability distribution of the parameters describing the tissue structure given surface EMG measurements, the so-called posterior distribution, is estimated. This Bayesian approach to inverse problems not only yields more information about the quantities of interest than classical regularization approaches, but also has a regularizing effect on the ill-posed problem. Indeed, the Bayesian inverse problem of inferring the tissue structure from surface EMG measurements is proven to be well-posed. This yields the convergence of the inversion algorithm and allows establishing error bounds and thus quantifying the uncertainties in the solution of the inverse EMG problem. Numerically, Markov chain Monte Carlo methods are used to explore the posterior distribution. Accelerations of these sampling methods are achieved by deriving a data-sparse representation of the discretized forward model for all conceivable discretizations of the parameters describing the tissue structure. The resulting approach is not only mathematically well-founded, but also faster by orders of magnitude. Finally, the proposed sampling algorithms are applied to several use cases that are related to clinical applications.
Open Access
Efficient simulation of challenging PDE problems on CPU and GPU clusters
(2021) Schirwon, Malte; Göddeke, Dominik (Prof. Dr.)
The main contribution of this dissertation is to show how efficient parallelization techniques for numerical simulations of partial differential equations (PDEs) can be developed and which aspects have to be considered in order to obtain the best possible performance. For this purpose, the target platforms range from high-performance workstations to small clusters and up to supercomputers. In particular, we focus on platforms accelerated by graphics cards. We emphasize that the efficient numerical simulation of PDE problems comprises and combines, in novel ways, aspects from numerical analysis, numerical methods (algorithmics, data structures and other areas more related to computer science) and hardware details. Many models in science, engineering and economics are based on systems of PDEs. The choice of modeling techniques, the implementation of numerical solution techniques, as well as the chosen target platform limit the accuracy and the duration of the simulation. Increasing the accuracy and/or reducing the duration of the simulation is usually not possible without efficient software. Based on three application scenarios, we adapt already existing methodologies and algorithms to the target platforms or change the way they are implemented in order to achieve optimal efficiency. As a guiding scheme, we consider the challenging case of unstructured data and schemes. The first application is the wave propagation in optical fibers. We present an MPI-parallel implementation that is particularly suitable for small clusters. %Here, we change the numerical method and the implementation technique to increase efficiency and decrease runtime. The second application scenario is the flow in porous media. Based on both applications, we develop implementation techniques that increase their efficiency. Furthermore, we present an adapted version of a neighborhood algorithm that further increases the efficiency for current graphics cards. The increased efficiency and reduced runtime allows to perform more complex simulations. %For example, higher resolutions can be simulated or more physical parameters can be included. One of theses applications is considered to be the third application, which is seismic wave propagation and waveform inversion. The feasibility of developing efficient implementations for computationally powerful target platforms permits us to consider the inversion of seismic waves in viscoelastic materials. In particular, we present an inversion scheme that also allows us to determine the damping parameters of the viscoelastic material. In addition, regularization methods and a modified solver method are presented, which can be used for a more efficient solution of such problems.
Open Access
Mesh refinement for parallel-adaptive FEM : theory and implementation
(2019) Alkämper, Martin; Göddeke, Dominik (Prof. Dr.)
We investigate parallel adaptive grid refinement and focus in particular on hierarchically adaptive, parallel and conforming simplicial grids that use Newest Vertex Bisection (NVB) as their refinement strategy. One challenge of NVB is its applicability to arbitrary simplex grids, which is not possible with the current compatibility condition. We define a novel, more natural weak compatibility condition for the initial grid and show that using this condition the iterative refinement algorithm terminates using NVB. We design an algorithm to relabel an arbitrary d-dimensional simplicial grid to fulfil this weak compatibility condition. The algorithm is of complexity O(n), where n is the number of elements in the grid. We also consider NVB on partitioned grids for parallel computing. Another challenge is that refinement may propagate over partition boundaries. This is resolved by adding an outer loop to the refinement algorithm, that requires global communication. We prove that the amount of global communication needed and the number of outer iterations in the refinement propagation to reach a conforming situation is bounded. We extend the grid manager DUNE-ALUGrid to provide parallel, adaptive, conforming 2d grids. Furthermore we develop the software package DUNE-ACFem which is able to conveniently describe mathematical problems within efficient C++ code. We demonstrate the utility of DUNE-ACFEM and DUNE-ALUGrid at the problem of noise removal on images with adaptive finite elements.