Assigning a Response Surface to the Objective Function

A response surface model, also called a proxy model, is a representation of a real system or its simulation model. A proxy model becomes very useful when the direct

evaluation of the system or simulation model is either impossible or too expensive and time consuming (for more details see Section 2.10 of Chapter 2). In most engineering optimization applications, including history matching, response surface models are usually constructed with polynomial regression techniques. Interpolation methods such as kriging, thin spline and neural network are also used. Polynomial regression is straight forward to implement, and the constructed surface is accurate if the problem is not strongly non-linear and when the parameter space is not too extensive. Interpolation methods have the advantage of honouring all the training data and also handle scattered data. They tend to smooth out some changes in the response surface, however, including noise and under-fitting. More complex proxy models (e.g. artificial neural networks) may provide an accurate proxy to the data, but the required number of training and testing points is often very large and therefore, they offer limited compensation when we desire to save computing time.

The ‘divide and conquer’ method takes advantage of a polynomial regression in the form of a multi-dimensional second order function with linear, quadratic and interacting terms for parameters to assigned a surface to objective function. Higher order terms are assumed to be small or lost in the system noise. Mathematically this equation is parameter counters, and n_d is the number of parameters. In practice the above equation utilizes the normalized sampled parameters to avoid distortion effects. In the SHM workflow we also tend to sample the parameters on a log₁₀ scale and hence the parameters are represented in this form. Therefore, a linear scaling transformation is applied on parameter values to map them onto the domain of [-1,1], on a log₁₀.scale.

2

max i min i mid

i i =1, 2, ..., n_d (3.12)

where θ’ is the normalized parameter vector, and θimin

, θimid

, and θimax

are the minimum, middle and maximum of the range of the i^th element of parameter vector θ, respectively.

The singular value decomposition technique with least square regression method (Rao 1973, Paige 1985, Montgomery 2000, Press et al. 2007) is used to derive the coefficients and to construct the polynomial misfit from a number of experimental simulations where the full misfits are calculated from Equation 2.3 or 2.4 of Chapter 2 .

Typically, the coefficients of the response surface are derived using the initial ensemble of the models used in the neighbourhood algorithm run (in the full SHM inversion of the problem) but we may do so at any time during the NA execution. After the coefficients of the polynomial misfit (Equation 3.10) are computed, the negligible coefficients are identified. First the interacting coefficients are ranked from smallest to largest, then starting with the smallest, they are discarded, each in turn, from the polynomial. They are set to zero in f(θ) (i.e. Equation 3.10). As this is done, the correlation factor, R², between the calculated misfit values by the polynomial model and the true misfit values (calculated J(θ) by Equation 2.2) is measured.

When the R² drops below 0.95, the process of discarding the insignificant coefficients is stopped and a threshold is established as the smallest interacting coefficient in the polynomial. Then by interrogating the remaining interaction terms, the independent groups of the parameters that can be searched separately are characterized. Figure 3.6 illustrates the sequential steps involved in the parameter space decoupling and misfit function decomposition.

Figure 3.6: The sequential steps involved in the misfit function decomposition and decoupling the parameter space.

Separated parameter groups are defined in the following way: starting with parameter 1 as the focus (e.g. P1 in Figure 3.7), and considering it as the first element of 'Group One', then initially there are two types of parameters: i) those that do not explicitly interact with parameter 1 (with b1i smaller than the threshold), and ii) those that interact directly with parameter 1 (with b1i larger than the threshold), which are placed in this group, (e.g. 'Group One' consist of P1 P7 P8 in Figure 3.7). However, the interrogation of this group is not yet complete. If any element (parameter) of this group other than parameter 1, has interaction with one of the parameters outside the group, that parameter is then moved into this group (i.e. b_1i is small but b_1j and b_ij are large), (e.g.

now P2 and P6 are added to 'Group One' in Figure 3.7). Thereby, parameters interacting implicitly with parameter 1 are captured as well, (e.g. eventually 'Group One' consists of P1 P7 P8 P2 P6 in Figure 3.7). This step finishes when all interactions are found and no more parameters could be moved into this group.

models 0 2004006008001000120014001600

misfit 0 2004006008001000120014001600

misfit 0 2004006008001000120014001600

misfit 0 2004006008001000120014001600

misfit

Figure 3.7: The way that interactions between parameters are found when the individual parameter groups are decoupled (as an example for 10 parameters P1 to P10). Apart from the directly interacting parameters, the parameters that interact implicitly (in red circles) are combined in one group.

The same procedure is repeated iteratively to identify additional groups but each time starting with one of the parameters that is still outside of a group. The subdivided parameter groups are thus found and are then considered to be mutually interacting within but independent of others. The resulting groups then form the sub-volumes of the problem parameter space, based on which the misfit function is then decomposed to the sub-misfits, accordingly.

3.2.3 Parallel-SHM Method

Once the independent sub-volumes of parameters are identified NA can be used to search them to find the minimum of the misfit. Here, all parameters are updated simultaneously but the sub-volumes are modified based on the misfit as it is decomposed. Because actual misfits of sub-volumes cannot be calculated, the misfits are estimated using Equation 3.9 to determine the fraction of the total calculated misfit that should be used for each sub-domain, i.e. each ji in Equation 3.9. The Neighbourhood Algorithm was adapted appropriately to parallelise the search of sub-spaces separately but simultaneously, and thus to update the full parameter vector, θ, during each iteration. The updated parameters are passed to the original SHM workflow for all remaining steps. Figure 3.8 summarizes the Parallel-SHM workflow. The core

P1

P2

P4

P5

( P3 P10) P3

P6

P7

P9

P10 P8

( P1 P7 P8 P2 P6 )

( P4 P5 P9 ) Resulting Groups Interactions

P1 P7 P8 P2 P6 P8

P3P10

P4 P9 P5 P9

loop inside the large rectangle in the schematic shows the SHM loop, the small boxes at the top show additional analyses of dividing the parameter space and decomposition of the misfit, and subsequent parallel searching of the parameter sub-spaces.

Figure 3.8: The core loop inside the large blue rectangle shows the SHM process. The red rectangles show additional analyses used in decomposition of the misfit and subsequent parallel search of the parameter sub-volumes, i.e. Parallel-SHM method.