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 Monte-Carlo 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.
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Friston, K.J., Harrison, L., Penny, W.: Dynamic causal modelling. Neuroimage 19(4), 1273–1302 (2003)
Moran, R.J., Kiebel, S.J., Stephan, K.E., Reilly, R.B., Daunizeau, J., Friston, K.J.: A neural mass model of spectral responses in electrophysiology. NeuroImage 37(3), 706–720 (2007)
Stephan, K.E., Friston, K.J.: Models of effective connectivity in neural systems. In: Jirsa, V.K., McIntosh, A.R. (eds.) Handbook of Brain Connectivity, Understanding Complex Systems, pp 303–327. Springer, Berlin Heidelberg (2007)
Stuart, A.W.: Inverse problems: A Bayesian perspective. Acta Numer. 19(1), 451–559 (2010)
Biegler, L., Biros, G., Ghattas, O., Heinkenschloss, M., Keyes, D., Mallick, B., Tenorio, L., Waanders, B., Willcox, K., Marzouk, Y.: Large-scale inverse problems and quantification of uncertainty. Wiley Series in Computational Statistics. Wiley (2011)
Weile, S., Michielssen, E., Grimme, E., Gallivan, K.: A method for generating rational interpolant reduced order models of two- parameter linear systems. Appl. Math. Lett. 12(5), 93–102 (1999)
Feng, L., Benner, P.: A robust algorithm for parametric model order reduction. In: PAMM, vol. 7(1), pp. 1021501–1021502 (2007)
Baur, U., Benner, P.: Parametrische Modellreduktion mit dünnen Gittern. In: GMA-Fachausschuss 1.30, Modellbildung, Identifizierung und Simulation in der Automatisierungstechnik, Salzburg (2008)
Lohmann, B., Eid, R.: Efficient order reduction of parametric and nonlinear models by superposition of locally reduced models. In: Methoden und Anwendungen der Regelungstechnik. Erlangen-Münchener Workshops 2007 und 2008. Shaker Verlag, Aachen (2009)
Eid, R., Castañé-Selga, R., Panzer, H., Wolf, T., Lohmann, B.: Stability-preserving parametric model reduction by matrix interpolation. Math. Comp. Model. Dyn. 17(4), 319–335 (2011)
Amsallem, D., Farhat, C.: An online method for interpolating linear parametric reduced-order models. SIAM J. Sci. Comput. 33(5), 2169–2198 (2011)
Haasdonk, B., Ohlberger, M.: Reduced basis method for finite volume approximations of parametrized linear evolution equations. M2AN 42(2), 277–302 (2008)
Haasdonk, B.: Convergence rates of the POD-greedy method. M2AN 47(3), 859–873 (2013)
Haasdonk, B., Ohlberger, M.: Efficient reduced models and a posteriori error estimation for parametrized dynamical systems by offline/online decomposition. Math. Comp. Model. Dyn. 17(2), 145–161 (2011)
Nguyen, N.C., Rozza, G., Huynh, D.B.P., Patera, A.T.: Reduced basis approximation and a posteriori error estimation for parametrized parabolic PDEs: application to real-time Bayesian parameter estimation. In: Large-Scale Inverse Problems and Quantification of Uncertainty, Wiley Ser. Comput. Stat., pp. 151–178. Wiley (2010)
Himpe, C., Ohlberger, M.: Cross-Gramian based combined state and parameter reduction for large-scale control systems. Math. Problem Eng. 2014, 1–13 (2014)
Lieberman, C., Willcox, K., Ghattas, O.: Parameter and state model reduction for large-scale statistical inverse problems. SIAM J. Sci. Comput. 32(5), 2523–2542 (2010)
Bui-Thanh, T., Willcox, K., Ghattas, O., van Bloemen Waanders, B.: Goal-oriented, model-constrained optimization for reduction of large-scale systems. J. Comput. Phys. 224(2), 880–896 (2007)
Bui-Thanh, T., Willcox, K., Ghattas, O.: Model reduction for large-scale systems with high-dimensional parametric input space. SIAM J. Sci. Comput. 30(6), 3270–3288 (2008)
Bui-Thanh, T., Burstedde, C., Ghattas, O., Martin, J., Stadler, G., Wilcox, L.C.: Extreme-scale UQ for Bayesian inverse problems governed by PDEs. In: Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis, pp. 1–11 (2012)
Bashir, O., Willcox, K., Ghattas, O., van Bloemen Waanders, B., Hill, J.: Hessian-based model reduction for large-scale systems with initial-condition inputs. Int. J. Numer. Meth. Engng. 73(6), 844–868 (2008)
Flath, H.P., Wilcox, L.C., Akcelik, V., Hill, J., van Bloemen Waanders, B., Ghattas, O.: Fast algorithms for Bayesian uncertainty quantification in large-scale linear inverse problems based on low-rank partial Hessian approximations. SIAM J. Sci. Comput. 33(1), 407–432 (2011)
Lieberman, C., Willcox, K.: Goal-oriented inference: Approach, linear theory, and application to advection diffusion. SIAM J. Sci. Comput. 34(4), A1880–A1904 (2012)
Lieberman, C., Van Bloemen Waanders, B: Hessian-based model reduction approach to solving large-scale source inversion problems. In: CSRI Summer Proceedings 2007, pp. 37–48 (2007)
Martin, A., Grepl, Krcher, M.: Reduced basis a posteriori error bounds for parametrized linear-quadratic elliptic optimal control problems. Comptes Rendus Mathematique 349(15–16), 873–877 (2011)
Himpe, C.: {optmor} - optimization-based model order reduction (Version 1.2). doi:10.5281/zenodo.17796 (2014)
Octave Community. GNU Octave 3.8. http://www.gnu.org/software/octave (2014)
MATLAB. The MathWorks Inc., Natick, Massachusetts (2014)
Kamrani, E., Foroushani, A., Vaziripour, M., Sawan, M.: Detecting the stable, observable and controllable states of the human brain dynamics. OJMI 2(4), 128–136 (2012)
Galbally, D., Fidkowski, K., Willcox, K., Ghattas, O.: Non-linear model reduction for uncertainty quantification in large-scale inverse problems. Int. J. Numer. Meth. Engng. 81(12), 1581–1608 (2010)
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Communicated by: Karsten Urban
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Himpe, C., Ohlberger, M. Data-driven combined state and parameter reduction for inverse problems. Adv Comput Math 41, 1343–1364 (2015). https://doi.org/10.1007/s10444-015-9420-5
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DOI: https://doi.org/10.1007/s10444-015-9420-5