Application of MATLAB and Python optimizers to two case studies involving groundwater flow and contaminant transport modeling

Abstract

One approach for utilizing geoscience models for management or policy analysis is via a simulation-based optimization framework—where an underlying model is linked with an optimization search algorithm. In this regard, MATLAB and Python are high-level programming languages that implement numerous optimization routines, including gradient-based, heuristic, and direct-search optimizers. The ever-expanding number of available algorithms makes it challenging for practitioners to identify optimizers that deliver good performance when applied to problems of interest. Thus, the primary contribution of this paper is to present a series of numerical experiments that investigated the performance of various MATLAB and Python optimizers. The experiments considered two simulation-based optimization case studies involving groundwater flow and contaminant transport. One case study examined the design of a pump-and-treat system for groundwater remediation, while the other considered least-squares calibration of a model of strontium (Sr) transport. Using these case studies, the performance of 12 different MATLAB and Python optimizers was compared. Overall, the Hooke–Jeeves direct search algorithm yielded the best performance in terms of identifying least-cost and best-fit solutions to the design and calibration problems, respectively. The IFFCO (implicit filtering for constrained optimization) direct search algorithm and the dynamically dimensioned search (DDS) heuristic algorithm also consistently yielded good performance and were up to 80% more efficient than Hooke–Jeeves when applied to the pump-and-treat problem. These results provide empirical evidence that, relative to gradient- and population-based alternatives, direct search algorithms and heuristic variants, such as DDS, are good choices for application to simulation-based optimization problems involving groundwater management.

Introduction

A common approach for utilizing geoscience models in a management or policy-analysis context is to incorporate them into a simulation-based optimization framework—where an underlying process-based geoscience model is linked with an optimization search algorithm (Banta et al., 2008, Freeze and Gorelick, 1999). During optimization, a given search algorithm iteratively adjusts various model inputs (i.e., parameters or design variables) in order to minimize an application-specific cost function computed on the basis of model outputs (i.e., response variables). The fundamental output of an optimization algorithm is an “optimal” configuration of design variables along with the corresponding value of the cost function.

Numerous optimization search algorithms have been developed and may be classified according to a variety of alternative taxonomies (e.g., Dréo et al., 2006, Talbi, 2002). For this paper, it is convenient to informally divide the spectrum of optimizers into four categories: (1) gradient-based algorithms that use derivative information to guide the search; (2) heuristic algorithms that do not utilize derivative information but instead follow empirical guidelines and incorporate elements of structured randomness; (3) direct search algorithms that are derivative free but utilize deterministic stencil-based procedures; and (4) hybrid methods that combine aspects of the previous three categories.

Example applications of simulation-based optimization in a geoscience context include the design of pump-and-treat systems (McKinney and Lin, 1996), groundwater supply systems (Chu and Chang, 2010), landfill liners (Bartelt-Hunt et al., 2006), agricultural operations (Gitau et al., 2006), waste load allocation strategies (Burn and Yulianti, 2001), and geothermal reservoirs (Akin et al., 2010). In addition, calibration of geoscience models is typically formulated as a simulation-based optimization problem—where uncertain model parameters are adjusted to minimize differences between simulated outputs and corresponding observational data (Massoudieh et al., 2008).

For a given application, researchers have generally used optimization routines that are embedded within the code of the underlying geoscience model (e.g., Akin et al., 2010, Lim et al., 2010). However, various efforts by commercial entities, research labs, academics, and independent enthusiasts have led to the development of numerous general-purpose optimization libraries using the MATLAB and Python programming languages (Dahl and Vandenberghe, 2008, Hart, 2009, Kelley, 1999, Venkataraman, 2009). These libraries can potentially eliminate the need to implement embedded optimization codes if the associated external optimizers are sufficiently robust and deliver good performance when applied to problems of interest.

To assist practitioners with identifying “best-in-class” MATLAB and Python optimizers for a given simulation-based optimization problem, a series of numerical experiments were performed involving two case studies relevant to the geosciences community and a suite of 12 MATLAB and Python optimizers. The case studies consider optimization from two distinctly different contexts—a pump-and-treat case study emphasizes least-cost remedial system design, while a subsurface strontium transport application considers the best-fit calibration of model parameters.

The remainder of the paper is organized as follows: Section 2 describes the selected case-study simulation-based optimization problems; Section 3 describes the optimizers that were selected for consideration in this study along with the associated numerical experiments; Section 4 presents the results of the numerical experiments and compares the performance of the selected optimizers; and Section 5 provides some concluding remarks.