Data-driven combined state and parameter reduction for inverse problems
- PDF / 482,961 Bytes
- 22 Pages / 439.642 x 666.49 pts Page_size
- 44 Downloads / 225 Views
Data-driven combined state and parameter reduction for inverse problems Christian Himpe1 · Mario Ohlberger1
Received: 15 January 2014 / Accepted: 30 April 2015 © Springer Science+Business Media New York 2015
Abstract In this contribution we present an accelerated optimization-based approach for combined state and parameter reduction of a parametrized forward model, which is used to construct a surrogate model in a Bayesian inverse problem setting. Following the ideas presented in Lieberman et al. (SIAM J. Sci. Comput. 32(5), 2523–2542, 2010), our approach is based on a generalized data-driven optimization functional in the construction process of the reduced order model and the usage of a MonteCarlo basis enrichment strategy that results in an additional speed-up of the overall method. In principal, the model reduction procedure is based on the offline construction of appropriate low-dimensional state and parameter spaces and an online inversion step using the resulting surrogate model that is obtained through projection of the underlying forward model onto the reduced spaces. The generalizations and enhancements presented in this work are shown to decrease overall computational time and thus allow an application to large-scale problems. Numerical experiments for a generic model and a fMRI connectivity model are presented in order to compare the computational efficiency of our improved method with the original approach. Keywords Model reduction · Model order reduction · Combined reduction · Optimization · Greedy Mathematics Subject Classifications (2010) 37N40 · 93A15 Communicated by: Karsten Urban Christian Himpe
[email protected] Mario Ohlberger [email protected] 1
Institute for Computational and Applied Mathematics, University of M¨unster, Einsteinstrasse 62, 48149 M¨unster, Germany
C. Himpe, M. Ohlberger
1 Introduction Many physical, chemical, technical, environmental, or bio-medical applications require the solution of inverse problems for parameter estimation and identification. This is in particular the case for complex dynamical systems where only experimental data is accessible via measurements. In neurosciences, a particular application is e.g. the extraction of effective connectivity in neuronal networks from measured data, such as data from electroencephalography (EEG) or functional magnetic resonance imaging (fMRI). If a network with many states (nodes) is considered, the corresponding inverse problem, for deducing the connectivity from the network’s output, is often only accessible in reasonable computational time if model reduction is applied to the underlying forward model. Moreover, the measured data is subject to statistic errors such that it is reasonable to apply a Bayesian inference approach which tries to identify a distribution on the associated parameters, rather than computing deterministic parameter values. In the context of connectivity analysis in neurosciences such an inversion approach has been established by Friston and his collaborators in recent
Data Loading...
