Kriging

Kriging is a method of interpolation which is based on a statistical correlation between sample points. It assumes that the distance between sample points reflects a spatial correlation which can be used to explain variations in the surface.

Universal Kriging Solution

Consider a design study containing $n$ input parameters that runs for $m$ designs. This design study provides $m$ sample points as input to a surrogate fitting calculation for each response.

$y$ denotes the vector of sample response values involving $n$ parameters.

Figure 1. EQUATION_DISPLAY

y = [y_{j}] = {[y_{1}, \dots, y_{m}]}^{t}, j \in [1, m]

(5143)

$X$ denotes the input matrix to the Kriging computation composed of $n$ parameters over $m$ sample points:

Figure 2. EQUATION_DISPLAY

X = [x_{i, j}] = (\begin{matrix} x_{1, 1} & \begin{matrix} {\begin{matrix} x \end{matrix}}_{2, 1} & \dots \end{matrix} & x_{n, 1} \\ \begin{matrix} x_{1, 2} \\ ⋮ \end{matrix} & \begin{matrix} {\begin{matrix} x \end{matrix}}_{2, 2} & \dots \end{matrix} & \begin{matrix} \begin{matrix} {\begin{matrix} x \end{matrix}}_{n, 2} \end{matrix} \\ ⋮ \end{matrix} \\ x_{1, m} & \begin{matrix} \begin{matrix} x_{2, m} \end{matrix} & \dots \end{matrix} & {\begin{matrix} x \end{matrix}}_{n, m} \end{matrix}), i \in [1, n] j \in [1, m]

(5144)

The Kriging surrogate function $y (x)$ includes a regression term and a stochastic process $z (x)$ .

The surrogate calculation determines the coefficients of the following surrogate function from the discrete sample points to form a continuous function that predicts the response value for the entire design space.

Figure 3. EQUATION_DISPLAY

y (x) = \sum_{h = 1}^{p} β_{h} ϕ_{h} (x) + z (x)

(5145)

where:

$\sum_{h = 1}^{p} β_{h} ϕ_{h} (x)$ is the regression term, which represents the global best fit in the design space.
$ϕ_{h} (x)$ is the polynomial basis function.
$p$ is the number of monomials in the basis function.
$β_{h}$ is the polynomial coefficient, which is determined during the surrogate function calculation.
For a polynomial with 2 input parameters $x_{1}, x_{2}$ , the polynomial basis function is $ϕ_{h} (x) = {1, x_{1}, x_{2}, x_{1} x_{2}, {x_{1}}^{2}, {x_{2}}^{2}}$ and $p = 6$ .
For example, $5 x_{1} + 2 x_{1} x_{2} - 3 {x_{2}}^{2}$ has the coefficient $β_{h} = {0, 5, 0, 2, 0, - 3}$ .
$z (x)$ is a realization of a stochastic process with a mean of zero and sample variance $σ^{2}$ .

The covariance matrix of $z (x)$ is given by:

Figure 4. EQUATION_DISPLAY

C o v [z (x_{j}), z (x_{k})] = σ^{2} R

(5146)

where $R = [R (x_{j}, x_{k})] j, k \in (1, m)$ is the correlation matrix. $x_{j}, x_{k}$ are the input parameter values of two sample points. The dimension of the sample points is $m$ , which comes from the number of design runs. Therefore, the dimension of the correlation matrix is $(m \times m)$ . $R (x_{j}, x_{k})$ is the user-specified fitting function (correlation function).

Fitting Functions

The following fitting functions (correlation functions) are available in Design Manager, where

n

denotes the number of input parameters (design variables).

θ_{i}

is the shape factor for the i-th correlation parameter:

Gaussian:

Figure 5. EQUATION_DISPLAY

$R (x_{j}, x_{k}) = \exp [- \sum_{i = 1}^{n} θ_{i} {\cdot | x_{i, j} - x_{i, k} |}^{2}]$

(5147)
Exponential:

Figure 6. EQUATION_DISPLAY

$R (x_{j}, x_{k}) = \exp [- \sum_{i = 1}^{n} θ_{i} \cdot | x_{i, j} - x_{i, k} |]$

(5148)
Linear:

Figure 7. EQUATION_DISPLAY

$R (x_{j}, x_{k}) = \sum_{i = 1}^{n} (\max {0, 1 - θ_{i} \cdot | x_{j} - x_{k} |})$

(5149)
Spherical:

Figure 8. EQUATION_DISPLAY

$R (x_{j}, x_{k}) = \sum_{i = 1}^{n} (1 - 1.5 ξ_{i} + 0.5 ξ_{i}^{3})$

(5150)

where $ξ_{i} = \min {1, θ_{i} \cdot | x_{i, j} - x_{i, k} |}$ .

Shape Factor and Tuning

In the fitting functions, the shape factors control how much each sample point affects the surrogate approximation. Small shape factors enhance the influence of the sample point, while large values weaken the influence.

In the Kriging model, the values of the polynomial term $\sum_{h = 1}^{p} β_{h} ϕ_{h} (x)$ , $σ^{2}$ ,and $θ = [θ_{1}, θ_{2}, \dots, θ_{n}]$ are determined by maximizing the likelihood function.

Simcenter STAR-CCM+ provides the following tuning options for Kriging:

None: Turns off the automatic tuning. You must adjust the shape factor manually.
Fast Kriging: Sweeps a fixed series of shape factors using the same value for all input parameters and selects the best factor. This method usually gives a sufficient level of cross-validation. You can improve the cross-validation value by manually trying different shape factors in the neighbourhood of the auto-tuned value.
Precise Kriging: Uses a more in-depth optimization method to vary $θ$ s separately for each input parameter. This method usually provides better tuning results than Fast Kriging, but consumes up to 15 times more computational time.
Gaussian Process: Adds an additional noise factor to the predictor and automatically optimizes it with the shape factors for the best results. No manual shape modification is possible with the Gaussian process.