[1] An optimal experimental design algorithm is developed to select locations for a network of observation wells that provide maximum information about unknown groundwater pumping in a confined, anisotropic aquifer. The design uses a maximal information criterion that chooses, among competing designs, the design that maximizes the sum of squared sensitivities while conforming to specified design constraints. The formulated optimization problem is non-convex and contains integer variables necessitating a combinatorial search. Given a realistic large-scale model, the size of the combinatorial search required can make the problem difficult, if not impossible, to solve using traditional mathematical programming techniques. Genetic algorithms (GAs) can be used to perform the global search ; however, because a GA requires a large number of calls to a groundwater model, the formulated optimization problem still may be infeasible to solve. As a result, proper orthogonal decomposition (POD) is applied to the groundwater model to reduce its dimensionality. Then, the information matrix in the full model space can be searched without solving the full model. Results from a small-scale test case show identical optimal solutions among the GA, integer programming, and exhaustive search methods. This demonstrates the GA's ability to determine the optimal solution. In addition, the results show that a GA with POD model reduction is several orders of magnitude faster in finding the optimal solution than a GA using the full model. The proposed experimental design algorithm is applied to a realistic, two-dimensional, large-scale groundwater problem. The GA converged to a solution for this large-scale problem.Citation: Ushijima, T. T., and W. W.-G. Yeh (2013), Experimental design for estimating unknown groundwater pumping using genetic algorithm and reduced order model, Water Resour. Res., 49,[6688][6689][6690][6691][6692][6693][6694][6695][6696][6697][6698][6699]

We develop an experimental design algorithm to select locations for a network of observation wells that provide the maximum robust information about unknown hydraulic conductivity in a confined, anisotropic aquifer. Since the information that a design provides is dependent on an aquifer's hydraulic conductivity, a robust design is one that provides the maximum information in the worst-case scenario. The design can be formulated as a max-min optimization problem. The problem is generally non-convex, non-differentiable, and contains integer variables. We use a Genetic Algorithm (GA) to perform the combinatorial search. We employ proper orthogonal decomposition (POD) to reduce the dimension of the groundwater model, thereby reducing the computational burden posed by employing a GA. The GA algorithm exhaustively searches for the robust design across a set of hydraulic conductivities and finds an approximate design (called the High Frequency Observation Well Design) through a Monte Carlo-type search. The results

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.

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