(399f) Forward-Backard Contractor in Verified Simulation Using Interval Analysis for Dynamic Optimisation

Many practically important processes are modelled by ordinary differential equations (ODEs). There is an interest in obtaining the global optima for these problems since they often offer substantial improvement compared to locally optimal solutions. Furthermore, in safety critical applications it is important to ensure that the behaviour of the process is within prescribed safe limits.

The deterministic methods for the solution of dynamic systems to global optimality are only able to address low dimensional problems. This is because in the sequential approach the integration has to be stopped early due to the generation of the overestimation in the verified simulation method.

Verified simulation seeks to enclose the trajectory of a dynamic system defined by ordinary differential equations (ODEs) within upper and lower bounds. This task is challenging when uncertainty in the parameters or the initial conditions needs to be accounted for because the method is subject to overestimation due to the dependency and wrapping effect problems (Jackson, 1975). Interval analysis is a key component in the verified simulation method since real amounts can be replaced for interval amounts and thus allowing the representation of uncertainty (Moore et al., 2009).

This work focuses on the reduction of the overestimation by means of the use of a forward-backward contractor (Jaulin et al. 2001) (also known as constraint propagation or HC4 (Benhamou et al. 1999)) in collaboration with a verified method using the Krawczyk and Newton contractors (Perez-Galvan and Bogle, 2014). The contractor is implemented in an interval Taylor series method in a stepwise fashion.

Numerical experiments using the method demonstrate the effectiveness of the method in the reduction of the overestimation. In the examples uncertainty is accounted by adding interval values in the parameters and initial conditions. Significant reduction of the overestimation and longer simulation times were observed after using the forward-backward contractor in the verified simulation method.

References

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L. Jackson. Interval arithmetic error-bounding algorithms. SIAM Journal on Numerical Analysis, 12(2):223-238, 1975.

L. Jaulin, M Kieffer, O. Didrit, and E. Walter. Applied Interval Analysis. Springer, London, 2001.

R. E. Moore, R. B. Kearfort, and M. J. Cloud. Introduction to Interval Analysis. SIAM, Philadelphia, 2009.

C. Perez-Galvan and I. D. L. Bogle. Comparison between interval methods to solve initial value problems in chemical process design. In J. J. Klemes, P. S. Varbanov, and P. Y. Liew, editors, 24th European Symposium on Computer Aided Process Engineering, 33:1405-1410. Elsevier, 2014.