Improving Approximations using Sequential Sampling

Overview

To improve the accuracy of the approximation, you can use a sequential sampling process that iteratively adds more sampling points to the approximation, up to a maximum number of points and iterations that you specify. Depending upon the number of iterations and the number of points, multiple points may be sampled during each iteration. The sequential sampling strategy starts with the initial set of points that you specified to initialize the approximation and then iteratively adds additional points to this set using a strategy that is appropriate for the approximation technique.

Kriging approximation uses the entropy approach (described below) to obtain the set of sample points during each iteration and uses the correlation matrix to space the new sample locations. A generic method, maximin scaled distance approach (also described below) is used for all other approximations including EBF and RBF. Because strict orthogonality requirements are enforced on the sample set used to fit Chebyshev polynomials, sequential sampling is not available for Orthogonal Polynomial models.

To avoid regions of failed runs in the design space, all failed sample locations are stored, and a penalty function based on the distance of the new point from the failed points is added to the entropy and minimum scaled distance.

Entropy Approach

Kriging approximation uses the entropy approach to obtain the set of sample points during each iteration and uses the correlation matrix to space the new sample locations.

Given an existing set of n samples, and m potential sample locations, the entropy of the total sample set is given by:

| R_{m + n} | | J_{m + n}^{T} R_{m + n}^{- 1} J_{m + n} |

where R _m₊_n is the correlation matrix with m + n points and J_m₊_n is an (m + n) vector of 1s (Koehler & Owen, 1996, and Jin et al., 2002). Isight chooses m new sample points that maximize the entropy value of the combined sample set. The theta values for the correlation matrix are kept constant during the search for the sample locations. Once new sample locations are identified, the approximation initialization is restarted; the theta values from the previous sampling iteration are used as the starting values while searching for the optimum thetas.

As an example, consider a Kriging approximation that has been created with 50 sample points and a maximum of 15 points are to be added in three iterations. During the first iteration, Isight uses the theta values obtained from the initial approximation to construct the correlation matrix. An optimization problem is then set up to locate the first additional point by maximizing the entropy. The objective function for a new point proposed by the optimization algorithm is computed as follows: the initial correlation matrix is augmented with one more rows and columns corresponding to the proposed point; the entropy is then calculated using the equation listed above. Next, once the maximum entropy has been identified by the optimization algorithm, this point is added permanently to the correlation matrix. To add the second additional point in this iteration, this augmented matrix is used for entropy computation (with n = 20 and m = 2). The next three points are added in a similar manner. Once all the five points of the first iteration have been identified, the Kriging approximation is re-fit to obtain the maximum likelihood estimates of theta for the total (55) points. The theta values obtained are kept the same while constructing the correlation matrix to add each of the five points in the second iteration. At the end of the second iteration, a new ML estimate of theta is obtained for a total (60) points. The third iteration proceeds in a similar fashion. At the end of each iteration, the error in the approximation is evaluated; sequential sampling stops when the error is within the limit specified.

Maximin Scaled Distance Approach

New points are selected that maximize the minimum scaled distance between any two points in the combined sample set.

Given an existing set of n samples X_A, additional points X_c are added such that

\max_{X_{c}} [\min_{1 \leq i \leq n, l \leq j \leq m + n}^{X_{c} \neq X_{A j}} {‖ X_{c i} - X_{A j} ‖}_{m}]

where || X_i – X_j||m = (X_i – X_j)T W (X_i – X_j) and W is the transformation matrix (Jin et al., 2002)

The transformation matrix is the identity matrix for RBF and all other approximations except EBF. For EBF, the transformation matrix is available from the approximation fit data. Similar to Kriging, this transformation matrix is used as the starting point in the next iteration to find the best EBF fit.

Using the same example described in Entropy Approach, let us suppose that the maximin approach is used instead. Sequential sampling iterations proceed in a manner that is similar to the entropy approach with one crucial difference: the objective function is now the minimum distance to an existing sample point. A new sample point is available when the optimization algorithm generates a point that is farthest to all existing points. The additional point is added to the set of existing points to identify the next additional point. The approximation is re-fit at the end of each sequential sampling iteration and its error evaluated.

Sequential Sampling References

Koehler, J.R., and A.B. Owen, “Computer Experiments,” in Ghosh, S. and Rao, C.R., eds, Handbook of Statistics, Elsevier Science, New York, vol. 13, pp. 261–308, 1996.

Jin, R., W. Chen, and A. Sudjianto, “On Sequential Sampling for Global Metamodeling in Engineering Design,” ASME 2002 Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Montreal, Canada, September 29–October 2, 2002, DETC2002/DAC-34092.