Keywords: optimization, simulation, derivative-free, Gaussian process, hybrid.

Simulators are a useful tool for the assessment water management scenarios. Moreover, simulation can be paired with optimization in order to design and control systems at minimum cost. However, this pairing of optimizer and simulator introduces a number of challenges to obtaining reasonable solutions. For example, the objective function and constraints rely on output from the simulator, and the simulator often requires the numerical solution to a system of nonlinear partial differential equations. Various assumptions can be made to simplify either the objective function or the physical system so that gradient-based methods apply, however the incorporation of realistic objection functions can be accomplished given the availability of derivative-free optimization methods. In this paper, we describe a means of hybridizing two such approaches in order to best solve the problem of hydraulic capture (HC).

We consider the plume containment problem, HC, that was proposed in the literature specifically for benchmarking purposes [1]. The hydrological setting is an unconfined aquifer and the objective function and constraints are nonlinear and discontinuous. Perhaps the most challenging aspect of this problem is that the final number of wells is unspecified. In order to apply continuous optimization approaches, the search begins with a set of N candidate wells and then wells may be subsequently removed from the design until an optimal cost is found. This makes the problem discontinuous with a large decrease in cost associated with the de-installation of a candidate well. Another challenge of the hydraulic capture problem is finding a balance between limiting the computational cost of finding a solution and finding the best possible solution. Many of the derivative-free optimization methods are inexpensive, but they are also local methods which can be highly dependent on initial iterates. Moreover, the single search methods are prone to getting stuck in local minima.

The hybrid approach we describe in this paper addresses both the stated challenges of hydraulic capture. Hybridizing multiple methods allows us to combine the beneficial elements of each method and in turn more efficiently search the design space. Specifically, we propose an algorithm which combines statistical emulation via a treed Gaussian process (TGP) [2] with a pattern search (PS) optimization [3]. The addition of a TGP adds a global flavor to the local PS method without significantly increasing computational cost. Specifically, we apply the hybrid so that the mixed integer HC formulation is solved as a set of nonlinear problems. We also show that the hybrid successfully finds the solution regardless of starting point whereas the success of some traditional optimization methods is highly dependent on starting point.

1

Mayer, Kelley, Miller, "Optimal design for problems involving flow and transport in saturated porous media", Adv. in Water Resources, 12:1233-1256, 2002. doi:10.1016/S0309-1708(02)00054-4

2

Gramacy, Lee, "Bayesian treed Gaussian process models with an application to computer modeling", J. Amer. Statist. Assoc., 2008. doi:10.1198/016214508000000689

3

Gray, Kolda, "APPSPACK 4.0: Asynchronous parallel pattern search for derivative-free optimization", Technical Report SAND2004-6391, Sandia National Labs, Aug. 2004.

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £78 +P&P)