EXTRACTING COMPLEX NONLINEAR RESPONSE SURFACES FROM DETERMINISTIC MODELS WITH MULTIPLE INPUTS

D. Shahsavani*, A. Grimvall
Department of Mathematics, Linköping university, Sweden
dasha@mai.liu.se

Applications of computer code models can become computationally very expensive if they require model runs for many different sets of inputs. Accordingly, it is often of great interest to derive computationally cheaper surrogate models that can mimic the output of the original model. The currently used techniques for estimating response surfaces for deterministic models with multiple inputs are usually based on design criteria that tend to favour regular fractional factorial designs or an almost uniform coverage of the experimental region (Bursztyn and Steinberg [1]). Moreover, it is common that a global parametric or nonparametric model is fitted to the computed responses for all design points (Iooss et al. [2]). Here, we develop a local approximation technique that is particularly appropriate for single-output deterministic models for which the curvature or roughness of the response surface varies markedly over the experimental region. In addition, we illustrate the performance of our method by deriving surrogate models of the INCA-N model (Whitehead et al. [3]), which is a hydro-geochemical model of the flows of water and nitrogen through a river basin.
A suitable experimental design for deriving surrogate models should be space-filling and have a particularly good coverage of regions where the response surface is so rough or nonlinear that it is difficult to extrapolate observed model outputs to outputs for previously untried inputs. We propose a sequential design in which a p-dimensional experimental region D = {x; ai  xi  bi, i = 1, … , p} is partitioned into cuboid-shaped sub-regions, and the corners and centre of D are taken as an initial set of design points (figure 1-(i)). New sub-regions are formed by estimating the roughness of the response surface in each of the existing sub-regions and then splitting the one having the maximum roughness into two halves. New design points are formed by taking the centre and corners of the new sub-regions (figure 1- (ii) and (iii)).

Figure 1. Design points selected in the first three steps of our sequential algorithm.

The roughness of the response surface in a given sub-region Di is estimated by employing a simple local regression technique. First, a second-order polynomial

is fitted to the model outputs for all design points in a minimal neighborhood of Di. Then, we compute

where V(Di) denotes the volume of Di. Finally, the sub-region having the maximum R-value is split into two halves along the axis for which  achieves its maximum value.
The behaviour of the proposed design algorithm can be illustrated by the results obtained when the INCA-N model was used to examine how the average annual loss of nitrogen from an agricultural soil is related to the denitrification rate and the maximum nitrate uptake rate by the crop grown on that soil. The left diagram in figure 2 shows a response surface obtained by running the INCA-N model for a dense grid of design points. The right diagram shows the design points that were selected when a surrogate model was derived. In particular, it can be seen that the design points are concentrated to sub-regions where the response is non-linear. 

Figure 2. Response surface for nitrogen losses computed with the INCA-N model (left). Design points selected by our sequential design algorithm (right).

Further computer experiments involving the INCA-N model showed that high efficiency surrogate models could be derived for the relationship between the average annual loss of nitrogen and various subsets of model parameters controlling the turnover of nitrogen. This makes such surrogate models attractive for sensitivity and uncertainty analyses that can require thousands of model runs. Figure 3 shows the efficiency achieved for models with five and seven parameters, respectively. In particular, it can be seen that, even though the model under consideration was moderately nonlinear, our design was superior to grid designs with the same number of design points.

Figure 3. Efficiency of surrogate models derived from INCA-N simulations using our sequential design and regular grid designs. Model output: average annual nitrogen loss from an agricultural soil. Model inputs: parameters controlling the turnover of nitrogen.

Acknowledgements:

The authors are grateful for financial support from the Ministry of Science, Research and Technology of I.R. Iran, and the Swedish National Research Council.

References

[1] Bursztyn D, Steinberg DM: Comparison of designs for computer experiments. J Stat Plann Infer 136, 1103-1119(2006)

[2] Iooss B, Van Dorpe F, Devictor N: Response surfaces and sensitivity analyses for an environmental model of dose calculations. Reliability Engineering and System Safety 91, 1241-1251(2006)

[3] Whitehead PG, Wilson EJ, Butterfield D: A semi-distributed nitrogen model for multiple source assessment in catchments (INCA): Model structure and process equations. Sci. Tot. Environ. 210/211, 547-558(1998)