Model-based design of experiments (MBDoE) for parameter precision can greatly accelerate the development of predictive mechanistic models. A particular focus has been on sequential MBDoE, where (possibly dynamic) experiments are designed one-at-a-time using gradient-based techniques. For nonlinear models, the resulting optimisation problems are typically nonconvex, and thus prone to converging to local optima. Numerical failure caused by singular information matrices is also commonplace. Such computational challenges hinder the design of experimental campaigns comprising multiple parallel runs.
Effort-based methods overcome these challenges by discretising the experimental space into a finite set of candidate experiments. They determine the fraction of the total number of experiments (the “effort”) associated with each candidate, with the objective of selecting the combination of experiments that will generate the highest information content. Treating the efforts as continuous variables leads to a convex formulation that can be solved efficiently using convex optimisation techniques. However, the optimised efforts typically assume fractional values, which need to be rounded to integer values in a subsequent step. This a posteriori effort rounding can result in large suboptimality, especially for campaigns with a small number of experiments.
As part of the research project, a discrete-effort methodology has been developed for the exact design of experiment campaigns comprising a finite number of experiments. Integrality constraints on the number of experiment replicates are enforced directly through pure integer programming, with a convex objective function and linear constraints. The experimental design space is discretised using low-discrepancy sampling, and the Fisher information matrix is computed for each candidate experiment. The resulting optimisation problem considers one of the standard experiment design criteria (e.g., D-optimality, A-optimality) applied to the overall information matrix of the campaign.
Two prototype implementations of the proposed approach are under development: the first one as a new experiment design solver withing the gPROMS modelling framework, the second in Python by leveraging a pre-existing package developed at Imperial College London. Several optimisation techniques were evaluated for the solution of the pure integer program; these include the classical outer-approximation algorithm used in mixed-integer nonlinear optimisation, as well as more advanced algorithms that attempt to capture the curvature of the objective function. Outer-approximation proved to be the most reliable option due to significant number of optimisation variables involved, and several modifications were implemented to tailor the solution algorithm to the considered optimisation problem and effectively speed up the computations.
The application of the proposed methodology on selected case studies has shown that experiment campaigns obtained by solving the exact design case lead to better objective function values than those obtained by the combination of continuous-effort formulations and apportionment, and the benefit is particularly relevant when considering low number of experiments. The method prioritises experiments with exceptional information content, whereas continuous-effort designs attempt to include as many candidates with nonzero effort as possible. In addition, the increase in computational effort with respect to continuous-effort methods is often negligible.
Figure 1. Results from the application of the methodology to a case study related to a bio-chemical process, showing the discretisation of the experimental design space and the selected experiments that maximise the information content of the campaign.