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

Abstract

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.

Equivalence theorems for c and DA‐optimality for linear mixed effects models with applications to multitreatment group assignments in health care

“…In addition, Masoudi et al (2019) applied ICA and found Bayesian optimal designs and, both Zhang et al (2020) and Liu et al (2021) modified the CSO algorithm and found optimal designs for different types of regression models with random effects. A most recent application using PSO and Genetic Algorithm is described in Ushijima et al (2021), where they found efficient designs for groundwater models after the complex models were simplified using a Galerkin method and a Proper Orthogonal Decomposition.…”

Applications of nature-inspired metaheuristic algorithms for tackling optimization problems across disciplines

“…In addition to numerous applications of metaheuristics in engineering and computer science, metaheuristics has also found applications ranging from addressing substantiability issues 85 to land use 19 and agriculture 58 . See also 86 , who used metaheuristic algorithms to design placements of the groundwater wells in the Los Angeles Basin.…”

Model-driven optimal experimental design for calibrating cardiac electrophysiology models

“…Optimal experimental design (OED) is a method of designing experiments which uses an objective approach to elicit data containing "optimal information" for model calibration (Lindley et al, 1956;Kiefer, 1959;Smucker et al, 2018). It is extremely popular in physical sciences such as geosciences (Ushijima and Yeh, 2013;Seidler et al, 2016;Ushijima et al, 2021), mechanical engineering (Wickham et al, 1995;Gupta and Dhingra, 2013), chemical engineering (Rodriguez-Fernandez et al, 2007;Gottu Mukkula and Paulen, 2017;Schenkendorf et al, 2018), etc. In this study, we apply the techniques of OED to cardiac cellular electrophysiology to expedite the development of AP models suitable for both basic research and safety-critical applications.…”