Parametric Interval Newton Methods

Ensuring that a process meets all performance specifications robustly while operating at steady-state, taking into account disturbance uncertainty in the inputs, is a primary objective of any design engineer.  Due to system complexity and the limitations of experimentation, such robustness guarantees can often only be estimated using a model-based approach.  However, process models are inherently inaccurate since model equations may contain structural uncertainty and/or uncertainty in the model parameters.  Often times at the design stage, a perfect model is assumed and structural uncertainty is disregarded.  It is important, however, to consider uncertainty in the model parameters.  Uncertainty in the inputs and uncertainty in the model parameters will be regarded in this study as parametric uncertainty.

At the core of all steady-state process simulations is the necessity to solve systems of nonlinear algebraic equations, typically of the form h(x) = 0, with h:ℝ^nx→ℝ^nx.  Taking into account parametric uncertainty, model equations inherit parameter-dependence.  Therefore, the solution(s) of steady-state process models of the parameter-dependent form h(x,p) = 0, with h:ℝ^nxxℝ^np→ℝ^nx, are implicit functions, x(p), of the uncertainty parameter p, that takes values from some uncertainty interval P.  Classical (real-number) iterative methods, such as Newton's method, have been the workhorse for solving systems of nonlinear algebraic equations for many years, and will be extended to the parametric case here.  Computational round-off errors associated with real-number iterative methods prompted the development of Newton methods utilizing Interval Analysis, known as interval Newton methods, to calculate rigorous bounds on the solutions.  The extension to parameter-dependent problems has not been fully developed with a limited number of novel works in [1,2,4-6] with some mention in [3,7], mainly resulting from the application to sensitivity analysis.

In [9], the authors introduced the interval Newton method with the Gauss-Seidel implementation within a generalized bisection framework, as applied to process simulation.  Their main goal was “to find any and all solutions to a system of nonlinear algebraic equations lying within the variable bounds” [9].  The “feasibility study” in [9] provides us with a proof-of-concept for the application of interval methods to process simulation problems.  However, the parametric case was not considered.  We have extended this idea to consider the parametric case, in what will be called parametric interval Newton methods and parameterized generalized bisection.

Three varieties of interval Newton-type methods, with varying implementations, have been extended to tackle parametric nonlinear systems of equations encountered in process simulation.  The three methods are the parametric interval Newton method with a Gauss-Seidel implementation, the parametric interval successive substitution method, and the parametric Krawczyk method with a componentwise implementation.  It is claimed without proof in [1] that the well-known and useful Krawczyk exclusion/inclusion tests are applicable to parameter-dependent problems.  We prove that the inclusion/exclusion properties hold for parametric interval Newton-type methods, in general.  Each method ultimately provides us with rigorous bounds (enclosures) on the steady-state solutions for all realizations of uncertainty with varying tightness.  That is, each method provides us with conservative estimates, of varying degree, on the worst-case scenarios of process unit operation.  Another useful result is that given certain conditions and properties of h, the exact range of the implicit function solution x(p), over the uncertainty interval P, can be calculated using an interval Newton-type method.  Thus, we can calculate the range over which x(p) varies, over the domain P, with certainty.

We show that as we partition P into subsets, the enclosures calculated by the parametric interval Newton-type methods become tighter.  Furthermore, as P is partitioned to a degenerate interval (real-number) P = [p*, p*], the parametric interval Newton methods converge to (tight) interval enclosures of the solution to h(x,p*)  =  0, which is simply reducible to the non-parametric system h(x) = 0, where the method in [9] is then applicable.  Along with the inclusion/exclusion properties of interval Newton-type methods, these properties provide us with a natural way to locate and enclose parametric solutions to within some specified tolerance.  This procedure is the primary modification of the generalized bisection procedure to yield the parameterized generalized bisection procedure, applicable to the considered problem.

The methods were prototyped in MATLAB® with the commercially available INTLAB toolbox [8] and tested on some small models.  One such model is of a proprietary Chevron® Compact Separator, designed for use in deep-sea oil recovery applications.  The methods discussed calculate rigorous enclosures on the parameter-dependent process variables within 10% overestimation.  This study provides us with a proof-of-concept that parametric interval Newton methods can calculate rigorous bounds on system variables over an entire range of parametric uncertainty.  This information in turn, will allow design engineers to determine whether a process robustly meets performance specifications taking into account uncertainty in the model parameters and in the inputs.

REFERENCES

[1]         David M. Gay, Perturbation bounds for nonlinear equations, SIAM Journal on Numerical Analysis, 18 (1981), pp. 654-663.

[2]          ______, Computing perturbation bounds for nonlinear algebraic equations, SIAM Journal on Numerical Analysis, 20 (1983), pp. 638-651.

[3]          Eldon Hansen and G. William Walster, Global Optimization Using Interval Analysis, Marcel Dekker, New York, second ed., 2004.

[4]          L. V. Kolev and I. P. Nenov, Cheap and tight bounds on the solution set of perturbed systems of nonlinear equations, Reliable Computing, 7 (2001), pp. 399-408.

[5]          Arnold Neumaier, The enclosure of solutions of parameter dependent systems, Reliability in Computing, Academic Press, Inc., San Diego, CA, 1988, pp. 269-286.

[6]          ______, Rigorous sensitivity analysis for parameter-dependent systems of equations, Journal of Mathematical Analysis and Applications, 144 (1989), pp. 16-25.

[7]          ______, Interval Methods for Systems of Equations, Cambridge University Press, Cambridge, 1990.

[8]          S.M. Rump, INTLAB - INTerval LABoratory, in Developments in Reliable Computing, Tibor Csendes, ed., Kluwer Academic Publishers, Dordrecht, 1999, pp. 77-104.

[9]          C. A. Schnepper and M. A. Stadtherr, Robust process simulation using interval methods, Computers in Chemical Engineering, 20(1996), pp. 187-199.