Bi-fidelity reduced polynomial chaos expansion for uncertainty quantification

Abstract

A ubiquitous challenge in design space exploration or uncertainty quantification of complex engineering problems is the minimization of computational cost. A useful tool to ease the burden of solving such systems is model reduction. This work considers a stochastic model reduction method (SMR), in the context of polynomial chaos expansions, where low-fidelity (LF) samples are leveraged to form a stochastic reduced basis. The reduced basis enables the construction of a bi-fidelity (BF) estimate of a quantity of interest from a small number of high-fidelity (HF) samples. A successful BF estimate approximates the quantity of interest with accuracy comparable to the HF model and computational expense close to the LF model. We develop new error bounds for the SMR approach and present a procedure to practically utilize these bounds in order to assess the appropriateness of a given pair of LF and HF models for BF estimation. The effectiveness of the SMR approach, and the utility of the error bound are presented in three numerical examples.

A Appendix

$\varvec{\varXi }$ :

Vector of random variables

$\varvec{{u}}$ :

Vector valued QoI

$\varvec{{c}}_j$ :

PC expansion coefficients

$\psi _{j}$ :

Polynomial basis functions

$\varvec{\delta }_P$ :

PC expansion truncation error

p :

PC total order

$\varvec{{C}}$ :

Matrix of PC coefficients

$\varvec{\xi }_i$ :

ith realization of random variable vector $\varvec{\varXi }$

$\varvec{{U}}$ :

Matrix of QoI $\varvec{{u}}$ samples

N :

Number of samples in $\varvec{{U}}$, $\varvec{{L}}$ and $\varvec{{H}}$

$\varvec{\varPsi }$ :

Measurement matrix for PC regression

$\varvec{{u}}^L$ :

LF QoI

$\varvec{{u}}^H$ :

HF QoI

m :

LF QoI dimension

M :

HF QoI dimension

$\varvec{{L}}$ :

Matrix of LF data

$\varvec{{H}}$ :

Matrix of HF data

$\varvec{{c}}_j^L$ :

PC expansion coefficients from LF data

$\eta _{i}$ :

Reduced basis functions

r :

Number of basis functions in SMR reduced basis; number of HF samples in MID

$\varvec{{c}}_{j}^B$ :

BF coefficients

${\varvec{\delta }}_{r}$ :

Total error in BF estimate

$\varvec{{u}}^B$ :

SMR BF QoI estimate

$\varvec{{C}}^B$ :

SMR Matrix of BF coefficients

$\varvec{{H}}_n$ :

SMR Limited number of n HF samples

$\varvec{\eta }_n$ :

Reduced basis measurement matrix for n samples

n :

Number of HF samples in SMR estimate

$\varvec{\eta }_N$ :

Reduced basis measurement matrix for N samples

${\widehat{\varvec{{H}}}}$ :

SMR BF estimate of $\varvec{{H}}$

${\bar{\varvec{{C}}}}^L$ :

Matrix of MID interpolation coefficients

${\bar{\varvec{{L}}}}^L$ :

MID BF estimate of $\varvec{{L}}$

$\varvec{{H}}_r$ :

MID limited number of r HF samples

${\bar{\varvec{{H}}}}^L$ :

MID BF estimate of $\varvec{{H}}$