
Alternating linear scheme in a Bayesian framework for lowrank tensor approximation
Multiway data often naturally occurs in a tensorial format which can be ...
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A parameterdependent smoother for the multigrid method
The solution of parameterdependent linear systems, by classical methods...
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A model reduction approach for inverse problems with operator valued data
We study the efficient numerical solution of linear inverse problems wit...
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Lowrank tensor methods for Markov chains with applications to tumor progression models
Continuoustime Markov chains describing interacting processes exhibit a...
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LowRank Tensor Regression for XRay Tomography
Tomographic imaging is useful for revealing the internal structure of a ...
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A Lowrank Solver for the Stochastic Unsteady NavierStokes Problem
We study a lowrank iterative solver for the unsteady NavierStokes equa...
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Diffusion tensor imaging with deterministic error bounds
Errors in the data and the forward operator of an inverse problem can be...
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Bayesian inversion for electromyography using lowrank tensor formats
The reconstruction of the structure of biological tissue using electromyographic data is a noninvasive 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 Bayesinverse problem means drawing samples from the posterior distribution of parameters, e.g., the conductivity, given measurement data. Using, e.g., a MetropolisHastings algorithm for this purpose involves solving the forward problem for different parameter combinations which requires a high computational effort. Lowrank tensor formats can reduce this effort by providing a datasparse representation of all occurring linear systems of equations simultaneously and allow for their efficient solution. The application of Bayes' theorem proves the wellposedness of the Bayesinverse problem. The derivation and proof of a lowrank representation of the forward problem allow for the precomputation of all solutions of this problem under certain assumptions, resulting in an efficient and theorybased 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 MetropolisHastings 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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