Bi-fidelity conditional value-at-risk estimation by dimensionally decomposed generalized polynomial chaos expansion

Abstract

Digital twin models allow us to continuously assess the possible risk of damage and failure of a complex system. Yet high-fidelity digital twin models can be computationally expensive, making quick-turnaround assessment challenging. Toward this goal, this article proposes a novel bi-fidelity method for estimating the conditional value-at-risk (CVaR) for nonlinear systems subject to dependent and high-dimensional inputs. For models that can be evaluated fast, a method that integrates the dimensionally decomposed generalized polynomial chaos expansion (DD-GPCE) approximation with a standard sampling-based CVaR estimation is proposed. For expensive-to-evaluate models, a new bi-fidelity method is proposed that couples the DD-GPCE with a Fourier-polynomial expansion of the mapping between the stochastic low-fidelity and high-fidelity output data to ensure computational efficiency. The method employs measure-consistent orthonormal polynomials in the random variable of the low-fidelity output to approximate the high-fidelity output. Numerical results for a structural mechanics truss with 36-dimensional (dependent random variable) inputs indicate that the DD-GPCE method provides very accurate CVaR estimates that require much lower computational effort than standard GPCE approximations. A second example considers the realistic problem of estimating the risk of damage to a fiber-reinforced composite laminate. The high-fidelity model is a finite element simulation that is prohibitively expensive for risk analysis, such as CVaR computation. Here, the novel bi-fidelity method can accurately estimate CVaR as it includes low-fidelity models in the estimation procedure and uses only a few high-fidelity model evaluations to significantly increase accuracy.

Appendix: Three step process to construct measure-consistent orthonormal polynomials

This appendix summarizes a process to generate the multivariate orthonormal polynomial basis of GPCE in Sect. 2.3. The orthonormal polynomial functions are consistent with an arbitrary, non-product-type probability measure $f_{\textbf{X}}(\textbf{x})\text {d}\textbf{x}$ of $\textbf{X}$ and determined by the following three steps.

• 1. Given $m \in \mathbb {N}_0$, create an $L_{N,m}$-dimensional column vector

  $$\begin{aligned} \textbf{M}_m(\textbf{x})=(\textbf{x}^{\textbf{j}^{(1)}},\ldots ,\textbf{x}^{\textbf{j}^{(L_{N,m})}})^{\intercal }, \end{aligned}$$

  of monomials whose elements are the monomials $\textbf{x}^{\textbf{j}}$ for $|\textbf{j}|\le m$ arranged in the aforementioned order. It is referred to as the monomial vector in $\textbf{x}=(x_1,\ldots ,x_N)^\intercal$ of degree at most m.

• 2. Construct an $L_{N,m} \times L_{N,m}$ monomial moment matrix of $\textbf{M}_m(\textbf{X})$, defined as

  $$ {\textbf{G}}_m:= \mathop {\mathrm {\mathbb {E}}}\limits [\textbf{M}_m(\textbf{X})\textbf{M}_{m}^{\intercal }(\textbf{X})]=\int _{\mathbb {{\bar{A}}}^{N}}\textbf{M}_{m}(\textbf{x})\textbf{M}_{m}^{\intercal }(\textbf{x}) f_{\textbf{X}}(\textbf{x})\text {d}.$$

  For an arbitrary PDF $f_{\textbf{X}}(\textbf{x})$, $\textbf{G}_m$ cannot be determined exactly, but it can be estimated with good accuracy by numerical integration and/or sampling methods (Lee and Rahman 2020).

• 3. Select the $L_{N,m} \times L_{N,m}$ whitening matrix ${\textbf{W}}_m$ from the Cholesky decomposition of the monomial moment matrix ${\textbf{G}}_m$ (Rahman 2018), leading to

  $$\begin{aligned} {\textbf{W}}_{m}^{-1}{\textbf{W}}_{m}^{-\intercal }={\textbf{G}}_{m}. \end{aligned}$$

  Then employ the whitening transformation to generate multivariate orthonormal polynomials from

  $$\begin{aligned} {\varvec{\Psi }}_m(\textbf{x})={\textbf{W}}_m \textbf{M}_m(\textbf{x}). \end{aligned}$$