(508e) 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.