Karhunen–Loève decomposition of random fields based on a hierarchical matrix approach

Abstract

SUMMARYThe simulation of the behavior of structures with uncertain properties is a challenging issue, because it requires suitable probabilistic models and adequate numerical tools. Nowadays, it is possible to perform probabilistic investigations of the structural performance, which take into account a space‐variant uncertainty characterization of the structures. Given a structural solver and the probabilistic models, the reliability analysis of the structural response depends on the continuous random fields approximation, which is carried out by means of a finite set of random variables. The paper analyzes the main aspects of discretization in the case of 2D problems. The combination of the well‐known Karhunen–Loève series expansion, the finite element method and the hierarchical matrices approach is proposed in the paper. Copyright © 2013 John Wiley & Sons, Ltd.

Efficient uncertainty quantification in stochastic finite element analysis based on functional principal components

“…Moreover, it is worth highlighting again that we focus on providing a finite representation of the response function u (x, θ ) rather than of the input quantities. On the contrary, many literature works are devoted to model input quantities [1,14,32].…”

“…The only random parameter is the Young modulus E; in the literature, a strong effort is devoted to the choice of suitable probabilistic laws for describing the uncertain properties of the random field E (see, e.g., [1,3]). Nevertheless, we focus here on the adequacy and the assessment of the proposed algorithm; hence, we model the field E in three simple but realistic ways, increasing the overall variability at each step.…”

“…3, we adopt the functional principal component analysis (FPCA) to express the displacement function u(x, θ ), which is the functional counterpart of the classical principal component analysis (PCA). PCA treats data as multivariate vectors rather than functions, and it is a widespread technique in the context of numerical analysis, where it is equivalently referred as proper orthogonal decomposition (POD) [3,10,30] or as Karhunen-Loeve decomposition [1,20].…”

“…Let us denote by θ the collection of random input parameters endowed with a probability law L(θ ); coherently, we denote by u(x, θ ) the solution of problem (1). At each iteration of the MC algorithm, a value θ * is drawn from L(θ ).…”

Monte Carlo Finite Volume Element Methods for the Convection-Diffusion Equation with a Random Diffusion Coefficient

“…We solve (41) by discretizing as a matrix, and the discretization corresponds to a piecewise constant FEM approximation to (41), and then MATLAB's function eig() is used. For other numerical methods, see [22][23][24]. The spectral contribution of to the KL expansion is plotted in Figure 4, which shows the decay of the eigenvalues with different choice of parameter ].…”