A maximum cost-performance sampling strategy for multi-fidelity PC-Kriging

Abstract

To reduce the computational cost of uncertainty propagation, multi-fidelity polynomial chaos approaches have been developed by fusing a few expensive high-fidelity data points and many less expensive lower-fidelity data points to build a stochastic metamodel. However, previous studies mainly focused on multi-model fusion. Systematically allocating sample points from multi-fidelity models to ensure both the accuracy and efficiency of the metamodel still remain challenging. To address this issue, a new maximum cost performance (MCP) sequential sampling strategy considering both the sample cost and accuracy improvement is proposed based on the recently developed multi-fidelity PC-Kriging (MF-PCK) approach. With the proposed sampling strategy, the input location with the largest prediction error is identified as the new input sample point, and then, the multi-fidelity model with the largest CP index is selected for evaluation to reduce the computational cost as much as possible. Furthermore, a sample density function is introduced to avoid the clustering of samples, which can prevent wastage of sample points and the singularity problem. The effectiveness and relative advantage of the proposed multi-fidelity sampling strategy in terms of efficiency is demonstrated by comparative studies using several numerical examples for uncertainty propagation and an airfoil robust optimization problem.

H,H p, T p, V p and V d for Hyperparameter Estimation

$$ \mathrm{H}=\left[\begin{array}{l}{\Phi}^1\left({\mathbf{D}}_1\left(\boldsymbol{\upxi} \right)\right)\kern7.50em 0\kern0.5em \begin{array}{cc}\kern7.2em \cdots & \kern5em 0\end{array}\\ {}\begin{array}{ccc}{\rho}^1{\Phi}^1\left({\mathbf{D}}_2\left(\boldsymbol{\upxi} \right)\right)& \kern5.75em {\Phi}^2\left({\mathbf{D}}_2\left(\boldsymbol{\upxi} \right)\right)& \begin{array}{cc}\kern4.8em 0& \kern5.25em 0\end{array}\end{array}\\ {}\begin{array}{l}\begin{array}{ccc}{\rho}^1{\rho}^2{\Phi}^1\left({\mathbf{D}}_3\left(\boldsymbol{\upxi} \right)\right)& \kern5em {\rho}^2{\Phi}^2\left({\mathbf{D}}_3\left(\boldsymbol{\upxi} \right)\right)& \begin{array}{cc}\kern2.25em {\Phi}^3\left({\mathbf{D}}_3\left(\boldsymbol{\upxi} \right)\right)& \kern3.2em \vdots \end{array}\end{array}\\ {}\begin{array}{ccc}\vdots & \kern9.5em \vdots & \begin{array}{cc}\kern6.5em \vdots & \kern4.7em \vdots \end{array}\end{array}\\ {}\begin{array}{ccc}\left(\prod \limits_{i=1}^{s-1}{\rho}^i\right){\Phi}^1\left({\mathbf{D}}_s\left(\boldsymbol{\upxi} \right)\right)& \left(\prod \limits_{i=2}^{s-1}{\rho}^i\right){\Phi}^2\left({\mathbf{D}}_s\left(\boldsymbol{\upxi} \right)\right)& \kern5.25em \begin{array}{cc}\cdots & \kern3em {\Phi}^s\left({\mathbf{D}}_s\left(\boldsymbol{\upxi} \right)\right)\end{array}\end{array}\end{array}\end{array}\right] $$

(22)

where Φ^m(D_j(ξ)) is a matrix of size n_j × (P + 1) (j, m = 1,2, …,s), formulated as:

$$ {\Phi}^m\left({\mathbf{D}}_j\left(\boldsymbol{\upxi} \right)\right)=\left[{\Phi}_0^m{\mathbf{D}}_j\left(\boldsymbol{\upxi} \right),{\Phi}_1^m{\mathbf{D}}_j\left(\boldsymbol{\upxi} \right),\dots, {\Phi}_P^m{\mathbf{D}}_j\left(\boldsymbol{\upxi} \right)\right] $$

(23)

For the hierarchical-fidelity case,

$$ {\mathbf{H}}_p=\left(\left(\prod \limits_{i=1}^{s-1}{\rho}^i\right){\boldsymbol{\Phi}}^1\left(\boldsymbol{\upxi} \left({\mathbf{x}}_{pre}\right)\right),\dots, {\rho}^{s-1}{\boldsymbol{\Phi}}^{s-1}\left(\boldsymbol{\upxi} \left({\mathbf{x}}_{pre}\right)\right),{\boldsymbol{\Phi}}^s\left(\boldsymbol{\upxi} \left({\mathbf{x}}_{pre}\right)\right)\right) $$

(24)

$$ {\mathbf{T}}_p={\left({t}_1{\left({\mathbf{x}}_{pre},{\mathbf{D}}_1\right)}^{\mathrm{T}},\dots, {t}_s{\left({\mathbf{x}}_{pre},{\mathbf{D}}_s\right)}^{\mathrm{T}}\right)}^{\mathrm{T}} $$

(25)

$$ {t}_1\left({\mathbf{x}}_{pre},{\mathbf{D}}_1\right)=\left(\prod \limits_{i=1}^{s-1}{\rho}^i\right){\sigma}_1^2{R}_1{\left({\mathbf{x}}_{pre},{\mathbf{D}}_1\right)}^{\mathrm{T}} $$

(26)

$$ {t}_m\left({\mathbf{x}}_{pre},{\mathbf{D}}_m\right)={\rho}^{m-1}{t}_{m-1}\left({\mathbf{x}}_{pre},{\mathbf{D}}_m\right)+\left(\prod \limits_{i=t}^{s-1}{\rho}^i\right){\sigma}_m^2{R}_m{\left({\mathbf{x}}_{pre},{\mathbf{D}}_m\right)}^{\mathrm{T}},m=2,\dots, s $$

(27)

$$ {\mathbf{V}}_p={\left(\prod \limits_{i=1}^{s-1}{\rho}^i\right)}^2{\sigma}_1^2{R}_1\left({\mathbf{x}}_{pre},{\mathbf{x}}_{pre}\right)+\dots +{\left({\rho}^{s-1}\right)}^2{\sigma}_{s-1}^2{R}_{s-1}\left({\mathbf{x}}_{pre},{\mathbf{x}}_{pre}\right)+{\sigma}_s^2{R}_s\left({\mathbf{x}}_{pre},{\mathbf{x}}_{pre}\right) $$

(28)

$$ {\mathbf{V}}_d=\left(\begin{array}{ccc}{V}_{1,1}& \cdots & {V}_{1,s}\\ {}\vdots & \ddots & \vdots \\ {}{V}_{s,1}& \vdots & {V}_{s,s}\end{array}\right) $$

(29)

For the m^th diagonal block,

$$ {V}_{m,m}={\sigma}_m^2{R}_m\left({\mathbf{D}}_m\right)+{\sigma}_{m-1}^2{\left({\rho}^{m-1}\right)}^2{R}_{m-1}\left({\mathbf{D}}_{m-1}\right)+\dots +{\sigma}_1^2{\left(\prod \limits_{i=1}^{m-1}{\rho}^i\right)}^2{R}_1\left({\mathbf{D}}_1\right) $$

(30)

For the off-diagonal block,

$$ {V}_{m,{m}^{\prime }}=\underset{1\le t<{t}^{\prime}\le s}{\left(\prod \limits_{i=t}^{t^{\prime }-1}{\rho}^i\right)}\left({\sigma}_m^2{R}_m\left({\mathbf{D}}_m,{\mathbf{D}}_{m^{\prime }}\right)+\dots +{\sigma}_1^2{\left(\prod \limits_{i=1}^{m-1}{\rho}^i\right)}^2{R}_1\left({\mathbf{D}}_1,{\mathbf{D}}_{m^{\prime }}\right)\right) $$

(31)

For the nonhierarchical-fidelity case,

$$ {\mathbf{H}}_p=\left({\rho}^1{\boldsymbol{\Phi}}^1\left(\boldsymbol{\upxi} \left({\mathbf{x}}_{pre}\right)\right),\dots, {\rho}^{s-1}{\boldsymbol{\Phi}}^{s-1}\left(\boldsymbol{\upxi} \left({\mathbf{x}}_{pre}\right)\right),{\boldsymbol{\Phi}}^s\left(\boldsymbol{\upxi} \left({\mathbf{x}}_{pre}\right)\right)\right) $$

(32)

$$ {\mathbf{T}}_p=\left[{\boldsymbol{\rho}}^{\mathrm{T}}\mathbf{E}{e}_1R\left({\mathbf{x}}_{pre},{\mathbf{D}}_1\right),\dots, {\boldsymbol{\rho}}^{\mathrm{T}}\mathbf{E}{e}_{s-1}R\left({\mathbf{x}}_{pre},{\mathbf{D}}_{s-1}\right),{\boldsymbol{\rho}}^{\mathrm{T}}\mathbf{E}\boldsymbol{\rho } R\left({\mathbf{x}}_{pre},{\mathbf{D}}_s\right)+{\sigma}_{\delta}^2{R}_{\delta}\left({\mathbf{x}}_{pre},{\mathbf{D}}_s\right)\right] $$

(33)

$$ {\mathbf{V}}_p={\boldsymbol{\rho}}^{\mathrm{T}}\mathbf{E}\boldsymbol{\rho } R\left({\mathbf{x}}_{pre},{\mathbf{x}}_{pre}\right)+{\sigma}_{\delta}^2{R}_{\delta}\left({\mathbf{x}}_{pre},{\mathbf{x}}_{pre}\right) $$

(34)

$$ {\mathbf{V}}_d=\left[\begin{array}{l}{e}_1^{\mathrm{T}}\mathbf{E}{e}_1R\left({\mathbf{D}}_1,{\mathbf{D}}_1\right)\kern2.25em \cdots \kern0.5em \begin{array}{cc}\kern1.75em {e}_1^{\mathrm{T}}\mathbf{E}{e}_{s-1}R\left({\mathbf{D}}_1,{\mathbf{D}}_{s-1}\right)& \kern4.75em {e}_1^{\mathrm{T}}\mathbf{E}\boldsymbol{\rho } R\left({\mathbf{D}}_1,{\mathbf{D}}_s\right)\end{array}\\ {}\kern3.5em \begin{array}{ccc}\vdots & \kern3.5em \ddots & \begin{array}{cc}\kern10.25em & \kern7.5em \vdots \end{array}\end{array}\\ {}\begin{array}{l}\begin{array}{ccc}{e}_{s-1}^{\mathrm{T}}\mathbf{E}{e}_1R\left({\mathbf{D}}_{s-1},{\mathbf{D}}_1\right)& \cdots & \kern2em \begin{array}{cc}{e}_{s-1}^{\mathrm{T}}\mathbf{E}{e}_{s-1}R\left({\mathbf{D}}_{s-1},{\mathbf{D}}_{s-1}\right)& \kern2.25em {e}_{s-1}^{\mathrm{T}}\mathbf{E}\boldsymbol{\rho } R\left({\mathbf{D}}_{s-1},{\mathbf{D}}_s\right)\end{array}\end{array}\\ {}\begin{array}{ccc}\rho \mathbf{E}{e}_1R\left({\mathbf{D}}_s,{\mathbf{D}}_{s-1}\right)& \kern1.5em \cdots & \kern2em \begin{array}{cc}\rho \mathbf{E}{e}_{s-1}R\left({\mathbf{D}}_s,{\mathbf{D}}_{s-1}\right)& \kern3.75em {\boldsymbol{\rho}}^{\mathrm{T}}\mathbf{E}\boldsymbol{\rho } R\left({\mathbf{D}}_s,{\mathbf{D}}_s\right)\end{array}\end{array}+{\sigma}_{\delta}^2{R}_{\delta}\left({\mathbf{D}}_s,{\mathbf{D}}_s\right)\end{array}\end{array}\right] $$

(35)

where e_i is an (s−1)-dimensional unit column vector, where the i^th element is 1, while the others are 0.