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


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.

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  1. The GPCE in (6) should not be confused with that of (Xiu and Karniadakis 2002). The GPCE, presented here, is meant for an arbitrary dependent probability distribution of random input. In contrast, the existing PCE, whether classical (Wiener 1938) or generalized (Xiu and Karniadakis 2002), still requires independent random inputs.


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This material is based on research sponsored by the Air Force Research Lab (AFRL) and Defense Advanced Research Projects Agency (DARPA) under agreement number FA8650-21-2-7126. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon.

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The MATLAB code files of the Examples 1–2 are provided in the following Github repository: Unfortunately, the source codes implementing a set of built-in Abaqus functions for FEA, used in Example 2, cannot be shared due to the external code license. Readers further interested in the codes are encouraged to contact the authors by e-mail.


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

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. 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. 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. 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

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

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Lee, D., Kramer, B. Bi-fidelity conditional value-at-risk estimation by dimensionally decomposed generalized polynomial chaos expansion. Struct Multidisc Optim 66, 33 (2023).

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  • Risk measures
  • Conditional value-at-risk
  • Generalized polynomial chaos expansion
  • Dimensionally decomposed GPCE
  • Bi-fidelity modeling