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On spectral representation method and Karhunen–Loève expansion in modelling construction material properties

Chen E. J., Ding L., Liu Y., Ma X., Skibniewski M. J.

Archives of Civil and Mechanical Engineering

2018

Vol. 18, no. 3

768--783

Randomness in construction material properties (e.g. Young's modulus) can be simulated by stationary random processes or random fields. To check the stationarity of commonly used techniques, three random process generation methods were considered: Xn(t), Yn(t), and Zn(t). Methods Xn(t) and Yn(t) are based on a truncation of the spectral representation method with the first n terms. Xn(t) has random amplitudes while Yn(t) has random harmonics phases. Method Zn(t) is based on the Karhunen–Loève expansion with the first n terms as well. The effects of the truncation technique on the mean-square error, covariance function, and scale of fluctuation were examined in this study; these three methods were shown to have biased estimations of variance with finite n. Modified forms for those methods were proposed to ensure the truncated processes were still zero-mean, unit-variance, and had a controllable scale of fluctuation; in particular, the modified form of Karhunen–Loève expansion was shown to be stationary in variance. As a result, the modified forms for those three methods are advantageous in simulating statistically homogenous material properties. The effectiveness of the modified forms was demonstrated by a numerical example.

A dynamic bi-orthogonal field equation approach to efficient Bayesian inversion

Tagade P. M., Choi H. L.

International Journal of Applied Mathematics and Computer Science

2017

Vol. 27, no. 2

229--243

This paper proposes a novel computationally efficient stochastic spectral projection based approach to Bayesian inversion of a computer simulator with high dimensional parametric and model structure uncertainty. The proposed method is based on the decomposition of the solution into its mean and a random field using a generic Karhunen–Loève expansion. The random field is represented as a convolution of separable Hilbert spaces in stochastic and spatial dimensions that are spectrally represented using respective orthogonal bases. In particular, the present paper investigates generalized polynomial chaos bases for the stochastic dimension and eigenfunction bases for the spatial dimension. Dynamic orthogonality is used to derive closed-form equations for the time evolution of mean, spatial and the stochastic fields. The resultant system of equations consists of a partial differential equation (PDE) that defines the dynamic evolution of the mean, a set of PDEs to define the time evolution of eigenfunction bases, while a set of ordinary differential equations (ODEs) define dynamics of the stochastic field. This system of dynamic evolution equations efficiently propagates the prior parametric uncertainty to the system response. The resulting bi-orthogonal expansion of the system response is used to reformulate the Bayesian inference for efficient exploration of the posterior distribution. The efficacy of the proposed method is investigated for calibration of a 2D transient diffusion simulator with an uncertain source location and diffusivity. The computational efficiency of the method is demonstrated against a Monte Carlo method and a generalized polynomial chaos approach.