A Metaheuristic Adaptive Cubature Based Algorithm to Find Bayesian Optimal Designs for Nonlinear Models

G-optimal designs for hierarchical linear models: an equivalence theorem and a nature-inspired meta-heuristic algorithm

Hierarchical linear models are widely used in many research disciplines and estimation issues for such models are generally well addressed. Design issues are relatively much less discussed for hierarchical linear models but there is an increasing interest as these models grow in popularity. This paper discusses the G-optimality for predicting individual parameters in such models and establishes an equivalence theorem for confirming the G-optimality of an approximate design. Because the criterion is non-differentiable and requires solving multiple nested optimization problems, it is much harder to find and study G-optimal designs analytically. We propose a nature-inspired meta-heuristic algorithm called competitive swarm optimizer (CSO) to generate G-optimal designs for linear mixed models with different means and covariance structures. We further demonstrate that CSO is flexible and generally effective for finding the widely used locally D-optimal designs for nonlinear models with multiple interacting factors and some of the random effects are correlated. Our numerical results for a few examples suggest that G and D-optimal designs may be equivalent and we establish that D and G-optimal designs for hierarchical linear models are equivalent when the models have only a random intercept only. The challenging mathematical question of whether their equivalence applies more generally to other hierarchical models remains elusive.

Metaheuristics for pharmacometrics

Metaheuristics is a powerful optimization tool that is increasingly used across disciplines to tackle general purpose optimization problems. Nature-inspired metaheuristic algorithms is a subclass of metaheuristic algorithms and have been shown to be particularly flexible and useful in solving complicated optimization problems in computer science and engineering. A common practice with metaheuristics is to hybridize it with another suitably chosen algorithm for enhanced performance. This paper reviews metaheuristic algorithms and demonstrates some of its utility in tackling pharmacometric problems. Specifically, we provide three applications using one of its most celebrated members, particle swarm optimization (PSO), and show that PSO can effectively estimate parameters in complicated nonlinear mixed-effects models and to gain insights into statistical identifiability issues in a complex compartment model. In the third application, we demonstrate how to hybridize PSO with sparse grid, which is an often-used technique to evaluate high dimensional integrals, to search for D -efficient designs for estimating parameters in nonlinear mixed-effects models with a count outcome. We also show the proposed hybrid algorithm outperforms its competitors when sparse grid is replaced by its competitor, adaptive gaussian quadrature to approximate the integral, or when PSO is replaced by three notable nature-inspired metaheuristic algorithms.

Constructing robust and efficient experimental designs in groundwater modeling using a Galerkin method, proper orthogonal decomposition, and metaheuristic algorithms

Estimating parameters accurately in groundwater models for aquifers is challenging because the models are non-explicit solutions of complex partial differential equations. Modern research methods, such as Monte Carlo methods and metaheuristic algorithms, for searching an efficient design to estimate model parameters require hundreds, if not thousands of model calls, making the computational cost prohibitive. One method to circumvent the problem and gain valuable insight on the behavior of groundwater is to first apply a Galerkin method and convert the system of partial differential equations governing the flow to a discrete problem and then use a Proper Orthogonal Decomposition to project the high-dimensional model space of the original groundwater model to create a reduced groundwater model with much lower dimensions. The reduced model can be solved several orders of magnitude faster than the full model and able to provide an accurate estimate of the full model. The task is still challenging because the optimization problem is non-convex, non-differentiable and there are continuous variables and integer-valued variables to optimize. Following convention, heuristic algorithms and a combination is used search to find efficient designs for the reduced groundwater model using various optimality criteria. The main goals are to introduce new design criteria and the concept of design efficiency for experimental design research in hydrology. The two criteria have good utility but interestingly, do not seem to have been implemented in hydrology. In addition, design efficiency is introduced. Design efficiency is a method to assess how robust a design is under a change of criteria. The latter is an important issue because the design criterion may be subjectively selected and it is well known that an optimal design can perform poorly under another criterion. It is thus desirable that the implemented design has relatively high efficiencies under a few criteria. As applications, two heuristic algorithms are used to find optimal designs for a small synthetic aquifer design problem and a design problem for a large-scale groundwater model and assess their robustness properties to other optimality criteria. The results show the proof of concept is workable for finding a more informed and efficient model-based design for a water resource study.