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2021 | OriginalPaper | Chapter

A Sequential Sensor Selection Strategy for Hyper-Parameterized Linear Bayesian Inverse Problems

Authors : Nicole Aretz-Nellesen, Peng Chen, Martin A. Grepl, Karen Veroy

Published in: Numerical Mathematics and Advanced Applications ENUMATH 2019

Publisher: Springer International Publishing

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Abstract

We consider optimal sensor placement for hyper-parameterized linear Bayesian inverse problems, where the hyper-parameter characterizes nonlinear flexibilities in the forward model, and is considered for a range of possible values. This model variability needs to be taken into account for the experimental design to guarantee that the Bayesian inverse solution is uniformly informative. In this work we link the numerical stability of the maximum a posterior point and A-optimal experimental design to an observability coefficient that directly describes the influence of the chosen sensors. We propose an algorithm that iteratively chooses the sensor locations to improve this coefficient and thereby decrease the eigenvalues of the posterior covariance matrix. This algorithm exploits the structure of the solution manifold in the hyper-parameter domain via a reduced basis surrogate solution for computational efficiency. We illustrate our results with a steady-state thermal conduction problem.

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Footnotes
1
The extension to the infinite-dimensional setting poses additional challenges that will be discussed in a future work.
 
2
A general finite-dimensional space can be considered via an affine transformation, c.f. [2, 5].
 
3
For conciseness, we refer the reader to [3] for a definition of these properties.
 
4
We can readily generalize this setting to non-coercive problems by employing a Petrov-Galerkin formulation. A stability analysis similar to [3] will be explored in a future publication.
 
5
Possibilities for target values are highly dependent on the library \({ \mathcal {L} }\). In practice, β 0 should be chosen by carefully monitoring the changes in β.
 
Literature
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Metadata
Title
A Sequential Sensor Selection Strategy for Hyper-Parameterized Linear Bayesian Inverse Problems
Authors
Nicole Aretz-Nellesen
Peng Chen
Martin A. Grepl
Karen Veroy
Copyright Year
2021
DOI
https://doi.org/10.1007/978-3-030-55874-1_48

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