Multi-fidelity information fusion based on prediction of kriging

Abstract

In this paper, a novel kriging-based multi-fidelity method is proposed. Firstly, the model uncertainty of low-fidelity and high-fidelity models is quantified. On the other hand, the prediction uncertainty of kriging-based surrogate models(SM) is confirmed by its mean square error. After that, the integral uncertainty is acquired by math modeling. Meanwhile, the SMs are constructed through data from low-fidelity and high-fidelity models. Eventually, the low-fidelity (LF) and high-fidelity (HF) SMs with integral uncertainty are obtained and a proposed fusion algorithm is implemented. The fusion algorithm refers to the Kalman filter’s idea of optimal estimation to utilize the independent information from different models synthetically. Through several mathematical examples implemented, the fused SM is certified that its variance is decreased and the fused results tend to the true value. In addition, an engineering example about autonomous underwater vehicles’ hull design is provided to prove the feasibility of this proposed multi-fidelity method in practice. In the future, it will be a helpful tool to deal with reliability optimization of black-box problems and potentially applied in multidisciplinary design optimization.

Appendices

Appendix A

Consider the approximation of a function f(x). The design variable x is a vector with n dimensions. A stochastic process F(x) is defined to realize the deterministic response of f(x) as follows:

$$ F\left(\boldsymbol{x}\right)=\mu +Z\left(\boldsymbol{x}\right) $$

(A1)

μ is defined as constant and Z(x) is defined as a stochastic process. Z(x) has the following stochastic behaviors:

$$ \begin{array}{l}\kern4.1em E\left[Z\left(\boldsymbol{x}\right)\right]=0\\ {}Cov\left[Z\left(\boldsymbol{x}\right),\kern0.5em Z\left(\boldsymbol{x}\hbox{'}\right)\right]={\sigma}^2R\left(\varTheta, \kern0.5em \boldsymbol{x},\kern0.5em \boldsymbol{x}\hbox{'}\right)\\ {}\kern1.9em R\left(\varTheta, \kern0.5em \boldsymbol{x},\kern0.5em \boldsymbol{x}\hbox{'}\right)={\displaystyle \prod_{j=1}^n{R}_j\left({\theta}_j,\kern0.5em {x}_j-{x}_j\hbox{'}\right)}\end{array} $$

(A2)

σ ² is the process variance for the response and R(θ, x, w) is the correlation model between any two points x and x '. Θ = {θ ₁, θ ₂, ⋯ θ _n } is a set of parameters which determines the gradient of R(Θ, x, x '). This paper uses the Gaussian correlation function, which is defined as

$$ R\left({\theta}_j,\kern0.5em {x}_j,\kern0.5em {x}_j\hbox{'}\right)= \exp \left(-{\theta}_j\left|{x}_j-{x}_j\hbox{'}\right|{}^2\right) $$

(A3)

Next, assume that there are N sample points x ⁽¹⁾, x ⁽²⁾, ⋯, x ^(N) given by true function f(x). The Kriging model realizes all the given points as follows:

$$ \begin{array}{l}f\left({\boldsymbol{x}}^{(i)}\right)=F\left({\boldsymbol{x}}^{(i)}\right)\\ {}\kern3.05em =\mu +Z\left({\boldsymbol{x}}^{(i)}\right)\end{array} $$

(A4)

In Kriging method, these parameters μ, σ ², Θ are obtained by maximum likelihood estimation (MLE) (Forrester and Keane 2009). Here the estimated values are given:

$$ \begin{array}{l}\kern2em \widehat{\mu}=\frac{{\mathbf{1}}^{\mathrm{T}}{\boldsymbol{R}}^{-1}\boldsymbol{f}}{{\mathbf{1}}^{\mathrm{T}}{\boldsymbol{R}}^{-1}\mathbf{1}}\\ {}\kern1.6em {\widehat{\sigma}}^2=\frac{{\left(\boldsymbol{f}-\mathbf{1}\widehat{\mu}\right)}^{\mathrm{T}}{\boldsymbol{R}}^{-1}\left(\boldsymbol{f}-\mathbf{1}\widehat{\mu}\right)}{N}\\ {}Ln\left(\varTheta \right)=-\frac{S}{2} \ln \left(2\pi \right)-\frac{S}{2} \ln {\widehat{\sigma}}^2-\frac{1}{2} \ln \left|\boldsymbol{R}\right|\end{array} $$

(A5)

Where f = [f(x ⁽¹⁾), f(x ⁽²⁾), ⋯, f(x ^(N)) ]^T, R is a N × N correlations matrix whose element in the the j-th column of the i-th line is defined as R(Θ, x ⁽ⁱ⁾, x ^(j)).

Finally, minimize the mean squared error (MSE)

$$ {\widehat{s}}^2\left(\boldsymbol{x}\right)=Var\left[\widehat{f}\left(\boldsymbol{x}\right)-F\left(\boldsymbol{x}\right)\right] $$

(A6)

Meanwhile meet the following unbiasedness constraint:

$$ E\left[\widehat{f}\left(\boldsymbol{x}\right)\right]=E\left[F\left(\boldsymbol{x}\right)\right] $$

(A7)

The best linear unbiased predictor (BLUP) $\widehat{f}\left(\boldsymbol{x}\right)$ results in the form as

$$ \widehat{f}\left(\boldsymbol{x}\right)=\widehat{\mu}+{\boldsymbol{r}}^T\left(\boldsymbol{x}\right){\boldsymbol{R}}^{-1}\left(\boldsymbol{f}-\mathbf{1}\widehat{\mu}\right) $$

(A8)

Where r(x) is a N-dimensional vector. The i-th element of r(x) is R(Θ, x, x ⁽ⁱ⁾) where x is the any location to be estimated. The final form of MSE is:

$$ {\widehat{s}}^2\left(\boldsymbol{x}\right)={\widehat{\sigma}}^2\left[1-{\boldsymbol{r}}^{\mathrm{T}}\left(\boldsymbol{x}\right){\boldsymbol{R}}^{-1}\boldsymbol{r}\left(\boldsymbol{x}\right)+\frac{{\left(1-{\mathbf{1}}^{\mathrm{T}}{\boldsymbol{R}}^{-1}\boldsymbol{r}\left(\boldsymbol{x}\right)\right)}^2}{{\mathbf{1}}^{\mathrm{T}}{\boldsymbol{R}}^{-1}\mathbf{1}}\right] $$

(A9)

Appendix B

Six-hump camel-back function (SC)

$$ \begin{array}{l}SC\left({x}_1\kern0.5em ,{x}_2\right)=4{x}_1^2-2.1{x}_1^4+\frac{x_1^6}{3}+{x}_1{x}_2-4{x}_2^2+4{x}_2^4\\ {}{y}_h=SC\left({x}_1\kern0.5em ,{x}_2\right)\kern1em {y}_l=SC\left(0.7{x}_1\kern0.5em ,0.7{x}_2\right)+{x}_1{x}_2-15\kern1em {x}_{1,2}\in \left[-2,2\right]\end{array} $$

Branin function (BR)

$$ \begin{array}{l}BR\left({x}_1\kern0.5em ,{x}_2\right)=10+{\left[{x}_2-5.1\times \frac{x_1^2}{4{\pi}^2}+\frac{5{x}_1}{\pi }-6\right]}^2+10 \cos \left({x}_1\right)\left[1-\frac{1}{8\pi}\right]\\ {}{y}_h=BR\left({x}_1\kern0.5em ,{x}_2\right)-22.5{x}_2\kern1em {y}_l=BR\left(0.7{x}_1\kern0.5em ,0.7{x}_2\right)-15.75{x}_2+20{\left(0.9+{x}_1\right)}^2-50\kern1em \\ {}{x}_1\in \left[-5,10\right],\kern0.5em {x}_2\in \left[0,15\right]\end{array} $$

Booth function (BT)

$$ \begin{array}{l}BT\left({x}_1\kern0.5em ,{x}_2\right)={\left({x}_1+2{x}_2-7\right)}^2+{\left(2{x}_1+{x}_2-5\right)}^2\\ {}{y}_h=BT\left({x}_1\kern0.5em ,{x}_2\right)\kern1em {y}_l=BT\left(0.4{x}_1\kern0.5em ,{x}_2\right)+1.7{x}_1{x}_2-{x}_1+2{x}_2\kern1em {x}_{1,2}\in \left[-10,10\right]\end{array} $$

Bohachevsky function (BC)

$$ \begin{array}{l}BC\left({x}_1\kern0.5em ,{x}_2\right)={x}_1^2+2{x}_2^2-0.3 \cos \left(3\pi {x}_1\right)-0.4 \cos \left(4\pi {x}_2\right)+0.7\\ {}{y}_h=BC\left({x}_1\kern0.5em ,{x}_2\right)\kern1em {y}_l=BC\left(0.7{x}_1\kern0.5em ,{x}_2\right)+{x}_1{x}_2-12\kern1em {x}_{1,2}\in \left[-100,100\right]\end{array} $$

Himmelblau function (HM)

$$ \begin{array}{l}HM\left({x}_1\kern0.5em ,{x}_2\right)={\left({x}_1^2+{x}_2-11\right)}^2+{\left({x}_2^2+{x}_1-7\right)}^2\\ {}{y}_h=HM\left({x}_1\kern0.1em ,{x}_2\right)\kern.3em {y}_l=HM\left(0.5{x}_1\kern0.1em ,0.8{x}_2\right)+{x}_2^3-{\left({x}_1+1\right)}^2\kern.3em {x}_{1,2}\in \left[-3,3\right]\end{array} $$