(278b) Guaranteed Relaxations and Bounds on the Solution Sets of Parametric ODEs Via Implicit Linear Multistep Methods | AIChE

(278b) Guaranteed Relaxations and Bounds on the Solution Sets of Parametric ODEs Via Implicit Linear Multistep Methods


Wilhelm, M. - Presenter, University of Connecticut
Stuber, M., University of Connecticut
Valid enclosures of the set of reachable solutions of parametric ordinary differential equation systems (pODEs) play a central role in numerous applications ranging from set-based fault detection [1] to robust model predictive control [2]. These enclosures may be computed by a number of approaches including interval arithmetic [3], interval-Taylor arithmetic [4], differential-inequality approach [5], a discretized-and-relax scheme [6], and the use of a Taylor-McCormick models [7]. In each case, these approaches rely on a series of sequential calculations using explicit expressions for intermediate terms. In a global optimization context, these enclosures are used to construct relaxations required to compute valid lower bounds on the optimal solution value [8].

In this paper, we extend our prior work on calculating state relaxations of pODEs by means of parametric implicit linear multistep methods, first presented in [9]. This contribution consists of three substantial modifications to the prior approach. First, explicit accounting of the truncation error is conducted, resulting in bounds and relaxations that are guaranteed to enclose the entire solution set rather than a discrete approximation of it. Secondly, the mean-value forms of the residual are used to construct relaxations with the property that they are non-necessarily expansive with respect to time. We evaluate the role of different approaches to computing these bounds and relaxations using direct application of parametric implicit function theories [10,11]. As part of our third modification, we consider propagating these bounds and relaxations via parallelepiped-based enclosure via a novel semi-explicit update. In either case, these relaxations may be computed in an expedient fashion via a block-sequential approach.

Time complexity and tightness of the relaxations are compared with the existing higher-order enclosure [12], Lohner’s method [13], and Hermite-Obreshkoff method [14]. An analysis of the linear stability of these set-valued methods and estimation order is discussed along with comparisons to existing methods. The utility of these method for computing bounds and relaxations is demonstrated via a number of literatures examples. Additionally, we provide initial numerical results on using these relaxations and bounds in global optimization using a software extension to the EAGO software package in Julia [15, 16].

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[9] ME Wilhelm, AV Le, MD Stuber. Global Optimization of Stiff Dynamical Systems, AIChE Journal, 65 (12).

[10] Stuber, M.D., Scott, J.K., and P.I. Barton. Convex and Concave Relaxations of Implicit Functions. Optimization Methods and Software. 30(3), 424-460, 2014.

[11] MD Stuber, Evaluation of process systems operating envelopes, Massachusetts Institute of Technology.

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[15] Bezanson, J. et al. Julia: A fresh approach to numerical computing. SIAM Review. 59, 65-98, 2017.

[16] Wilhelm, M. and Stuber, M. Easy Advanced Global Optimization (EAGO): An Open-Source Platform for Robust and Global Optimization in Julia. AIChE Annual Meeting 2017.