Abstract
Matrices with random entries are encountered in finite element/difference formulations of a broad range of mechanics problems. Monte Carlo simulation, the only general method for solving this class of problems, is usual impractical when dealing with realistic problems.
A new method is presented for solving stochastic problems with random matrices that is based on the representation of the entries of random matrices by stochastic reduced order models (SROMs) and surrogate models. SROMs are random elements with finite numbers of samples that are selected from the samples of target random elements in an optimal manner. Surrogate models are approximations for quantities of interest with known expressions. Numerical examples are used to illustrate the implementation and the performance of the SROM method. The examples include inverses and eigenvalues/eigenvectors of random matrices.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Adhikari S (2004) Complex modes in stochastic systems. Adv Vib Eng 3(1):1–11
Ghosh D, Ghanem RG, Red-Horse J (2005) Analysis of eigenvalues and model interaction of stochastic systems. AIAA J 43(10):2196–2201
Grigoriu M (1992) A solution of the random eigenvalue problem by crossing theory. J Sound Vib 158(1):58–80
Grigoriu M (2002) Stochastic calculus. Applications in science and engineering. Birkhäuser, Boston
Grigoriu M (2012a) Stochastic systems. Uncertainty quantification and propagation. Springer series in reliability engineering. Springer, London/Heidelberg/New York/Dordrecht. ISBN 978-1-4471-2326-9, ISBN 978-1-4471-2327-9 (eBook)
Grigoriu M (2012b) A method for solving stochastic equations by reduced order models and local approximations. J Comput Phys 231(19):6495–6513
Rahman S (2007) Stochastic dynamic systems with complex-valued eigensolutions. Int J Numer Methods Eng 71:963–986
Warner J, Grigoriu M, Aquino W (2013, in press) Stochastic reduced order models for random vectors. Applications to random eigenvalue problems. Probab Eng Mech 31:1–11
Acknowledgements
The work reported in this paper has been supported by the National Science Foundation under grand CMMI-0969150. This support is gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Grigoriu, M. (2015). A Novel Method for Solving Random Eigenvalue Problems. In: Cimellaro, G., Nagarajaiah, S., Kunnath, S. (eds) Computational Methods, Seismic Protection, Hybrid Testing and Resilience in Earthquake Engineering. Geotechnical, Geological and Earthquake Engineering, vol 33. Springer, Cham. https://doi.org/10.1007/978-3-319-06394-2_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-06394-2_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-06393-5
Online ISBN: 978-3-319-06394-2
eBook Packages: Earth and Environmental ScienceEarth and Environmental Science (R0)