An efficient reduced‐order method for stochastic eigenvalue analysis

Zhibao Zheng · Michael Beer · Udo Nackenhorst

11 citations0/0 files storedInternational Journal for Numerical Methods in Engineering

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Probabilistic and Robust Engineering Design Matrix Theory and Algorithms Model Reduction and Neural Networks

Abstract

Abstract This article presents an efficient numerical algorithm to compute eigenvalues of stochastic problems. The proposed method represents stochastic eigenvectors by a sum of the products of unknown random variables and deterministic vectors. Stochastic eigenproblems are thus decoupled into deterministic and stochastic analyses. Deterministic vectors are computed efficiently via a few number of deterministic eigenvalue problems. Corresponding random variables and stochastic eigenvalues are solved by a reduced‐order stochastic eigenvalue problem that is built by deterministic vectors. The computational effort and storage of the proposed algorithm increase slightly as the stochastic dimension increases. It can solve high‐dimensional stochastic problems with low computational effort, thus the proposed method avoids the curse of dimensionality with great success. Numerical examples compared to existing methods are given to demonstrate the good accuracy and high efficiency of the proposed method.

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References

1991Stochastic Finite Elements: A Spectral Approach 2010Adaptive sparse polynomial chaos expansion based on least angle regression 1993The stochastic finite element method: basic perturbation technique and computer implementation 2014Numerical methods for the discretization of random fields by means of the Karhunen–Loève expansion 2008A polynomial dimensional decomposition for stochastic computing 2007Efficient characterization of the random eigenvalue problem in a polynomial chaos decomposition 2017Simulation of multi-dimensional random fields by Karhunen–Loève expansion 2006Random matrix eigenvalue problems in structural dynamics

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