Christine A. Shoemaker

Dynamically dimensioned search algorithm for computationally efficient watershed model calibration

[1] A new global optimization algorithm, dynamically dimensioned search (DDS), is introduced for automatic calibration of watershed simulation models. DDS is designed for calibration problems with many parameters, requires no algorithm parameter tuning, and automatically scales the search to find good solutions within the maximum number of user-specified function (or model) evaluations. As a result, DDS is ideally suited for computationally expensive optimization problems such as distributed watershed model calibration. DDS performance is compared to the shuffled complex evolution (SCE) algorithm for multiple optimization test functions as well as real and synthetic SWAT2000 model automatic calibration formulations. Algorithms are compared for optimization problems ranging from 6 to 30 dimensions, and each problem is solved in 1000 to 10,000 total function evaluations per optimization trial. Results are presented so that future modelers can assess algorithm performance at a computational scale relevant to their modeling case study. In all four of the computationally expensive real SWAT2000 calibration formulations considered here (14, 14, 26, and 30 calibration parameters), results show DDS to be more efficient and effective than SCE. In two cases, DDS requires only 15–20% of the number of model evaluations used by SCE in order to find equally good values of the objective function. Overall, the results also show that DDS rapidly converges to good calibration solutions and easily avoids poor local optima. The simplicity of the DDS algorithm allows for easy recoding and subsequent adoption into any watershed modeling application framework.

Constrained Global Optimization of Expensive Black Box Functions Using Radial Basis Functions

abstractWe present a new strategy for the constrained global optimization of expensive black box functions using response surface models. A response surface model is simply a multivariate approximation of a continuous black box function which is used as a surrogate model for optimization in situations where function evaluations are computationally expensive. Prior global optimization methods that utilize response surface models were limited to box-constrained problems, but the new method can easily incorporate general nonlinear constraints. In the proposed method, which we refer to as the Constrained Optimization using Response Surfaces (CORS) Method, the next point for costly function evaluation is chosen to be the one that minimizes the current response surface model subject to the given constraints and to additional constraints that the point be of some distance from previously evaluated points. The distance requirement is allowed to cycle, starting from a high value (global search) and ending with a low value (local search). The purpose of the constraint is to drive the method towards unexplored regions of the domain and to prevent the premature convergence of the method to some point which may not even be a local minimizer of the black box function. The new method can be shown to converge to the global minimizer of any continuous function on a compact set regardless of the response surface model that is used. Finally, we considered two particular implementations of the CORS method which utilize a radial basis function model (CORS-RBF) and applied it on the box-constrained Dixon–Szegö test functions and on a simple nonlinearly constrained test function. The results indicate that the CORS-RBF algorithms are competitive with existing global optimization algorithms for costly functions on the box-constrained test problems. The results also show that the CORS-RBF algorithms are better than other algorithms for constrained global optimization on the nonlinearly constrained test problem.

Dynamic optimal control for groundwater remediation with flexible management periods

A successive approximation linear quadratic regulator (SALQR) method with management periods is combined with a finite element groundwater flow and transport simulation model to determine optimal time-varying groundwater pump-and-treat reclamation policies. Management periods are groups of simulation time steps during which the pumping policy remains constant. In an example problem, management periods reduced the total computational demand, as measured by the CPU time, by as much as 85% compared to the time needed for the SALQR solution without management periods. Conversely, the optimal costs increased as the number of times that the control can change is reduced. With two simulation periods per management period, the optimal cost increased by less than 1% compared to the optimal cost with no management periods, yet the computational work was reduced by a third. The optimal policies, including the number and locations of wells, changed significantly with the number of management periods. Complexity analysis revealed that the SALQR algorithm with management periods can significantly reduce the computational requirements for nonsteady optimization of groundwater reclamation and other management applications.

A Stochastic Radial Basis Function Method for the Global Optimization of Expensive Functions

We introduce a new framework for the global optimization of computationally expensive multimodal functions when derivatives are unavailable. The proposed Stochastic Response Surface (SRS) Method iteratively utilizes a response surface model to approximate the expensive function and identifies a promising point for function evaluation from a set of randomly generated points, called candidate points. Assuming some mild technical conditions, SRS converges to the global minimum in a probabilistic sense. We also propose Metric SRS (MSRS), which is a special case of SRS where the function evaluation point in each iteration is chosen to be the best candidate point according to two criteria: the estimated function value obtained from the response surface model, and the minimum distance from previously evaluated points. We develop a global optimization version and a multistart local optimization version of MSRS. In the numerical experiments, we used a radial basis function (RBF) model for MSRS and the resulting algorithms, Global MSRBF and Multistart Local MSRBF, were compared to 6 alternative global optimization methods, including a multistart derivative-based local optimization method. Multiple trials of all algorithms were compared on 17 multimodal test problems and on a 12-dimensional groundwater bioremediation application involving partial differential equations. The results indicate that Multistart Local MSRBF is the best on most of the higher dimensional problems, including the groundwater problem. It is also at least as good as the other algorithms on most of the lower dimensional problems. Global MSRBF is competitive with the other alternatives on most of the lower dimensional test problems and also on the groundwater problem. These results suggest that MSRBF is a promising approach for the global optimization of expensive functions.

Numerical solution of continuous-state dynamic programs using linear and spline interpolation

This paper demonstrates that the computational effort required to develop numerical solutions to continuous-state dynamic programs can be reduced significantly when cubic piecewise polynomial functions, rather than tensor product linear interpolants, are used to approximate the value function. Tensor product cubic splines, represented in either piecewise polynomial or B-spline form, and multivariate Hermite polynomials are considered. Computational savings are possible because of the improved accuracy of higher-order functions and because the smoothness of higher-order functions allows efficient quasi-Newton methods to be used to compute optimal decisions. The use of the more efficient piecewise polynomial form of the spline was slightly superior to the use of Hermite polynomials for the test problem and easier to program. In comparison to linear interpolation, use of splines in piecewise polynomial form reduced the CPU time to obtain results of equivalent accuracy by a factor of 250–330 for a stochastic 4-dimensional water supply reservoir problem with a smooth objective function, and factors ranging from 25–400 for a sequence of 2-, 3-, 4-, and 5-dimensional problems. As a result, a problem that required two hours to solve with linear interpolation was solved in a less than a minute with spline interpolation with no loss of accuracy.

Optimal time-varying pumping rates for groundwater remediation: Application of a constrained optimal control algorithm

A numerically efficient procedure is presented for computing optimal time-varying pumping rates for remediation of contaminated groundwater described by two-dimensional numerical models. The management model combines a pollutant transport model with a constrained optimal control algorithm. The transport model simulates the unsteady fluid flow and transient contamination dispersion-advection in a two-dimensional confined aquifer. A Galerkins finite element method coupled with a fully implicit time difference scheme is applied to solve the groundwater flow and contaminant transport equations. The constrained optimal control algorithm employs a hyperbolic penalty function. Several sample problems covering 5-15 years of remediation are given to illustrate the capability of the management model to solve a groundwater quality control problem with time-varying pumping policy and water quality constraints. In the example, the optimal constant pumping rates are 75% more expensive than the optimal time-varying pumping rates, a result that supports the need to develop numerically efficient optimal control-finite element algorithms for groundwater remediation. 28 refs., 7 figs., 10 tabs.

ORBIT: Optimization by Radial Basis Function Interpolation in Trust-Regions

We present a new derivative-free algorithm, ORBIT, for unconstrained local optimization of computationally expensive functions. A trust-region framework using interpolating Radial Basis Function (RBF) models is employed. The RBF models considered often allow ORBIT to interpolate nonlinear functions using fewer function evaluations than the polynomial models considered by present techniques. Approximation guarantees are obtained by ensuring that a subset of the interpolation points is sufficiently poised for linear interpolation. The RBF property of conditional positive definiteness yields a natural method for adding additional points. We present numerical results on test problems to motivate the use of ORBIT when only a relatively small number of expensive function evaluations are available. Results on two very different application problems, calibration of a watershed model and optimization of a PDE-based bioremediation plan, are also encouraging and support ORBITs effectiveness on blackbox functions for which no special mathematical structure is known or available.

Local function approximation in evolutionary algorithms for the optimization of costly functions

We develop an approach for the optimization of continuous costly functions that uses a space-filling experimental design and local function approximation to reduce the number of function evaluations in an evolutionary algorithm. Our approach is to estimate the objective function value of an offspring by fitting a function approximation model over the k nearest previously evaluated points, where k=(d+1)(d+2)/2 and d is the dimension of the problem. The estimated function values are used to screen offspring to identify the most promising ones for function evaluation. To fit function approximation models, a symmetric Latin hypercube design (SLHD) is used to determine initial points for function evaluation. We compared the performance of an evolution strategy (ES) with local quadratic approximation, an ES with local cubic radial basis function (RBF) interpolation, an ES whose initial parent population comes from an SLHD, and a conventional ES. These algorithms were applied to a twelve-dimensional (12-D) groundwater bioremediation problem involving a complex nonlinear finite-element simulation model. The performances of these algorithms were also compared on the Dixon-Szego test functions and on the ten-dimensional (10-D) Rastrigin and Ackley test functions. All comparisons involve analysis of variance (ANOVA) and the computation of simultaneous confidence intervals. The results indicate that ES algorithms with local approximation were significantly better than conventional ES algorithms and ES algorithms initialized by SLHDs on all Dixon-Szego test functions except for Goldstein-Price. However, for the more difficult 10-D and 12-D functions, only the cubic RBF approach was successful in improving the performance of an ES. Moreover, the results also suggest that the cubic RBF approach is superior to the quadratic approximation approach on all test functions and the difference in performance is statistically significant for all test functions with dimension d/spl ges/4.

Scalable thread scheduling and global power management for heterogeneous many-core architectures

Future many-core microprocessors are likely to be heterogeneous, by design or due to variability and defects. The latter type of heterogeneity is especially challenging due to its unpredictability. To minimize the performance and power impact of these hardware imperfections, the runtime thread scheduler and global power manager must be nimble enough to handle such random heterogeneity. With hundreds of cores expected on a single die in the future, these algorithms must provide high power-performance efficiency, yet remain scalable with low runtime overhead. This paper presents a range of scheduling and power management algorithms and performs a detailed evaluation of their effectiveness and scalability on heterogeneous many-core architectures with up to 256 cores. We also conduct a limit study on the potential benefits of coordinating scheduling and power management and demonstrate that coordination yields little benefit. We highlight the scalability limitations of previously proposed thread scheduling algorithms that were designed for small-scale chip multiprocessors and propose a Hierarchical Hungarian Scheduling Algorithm that dramatically reduces the scheduling overhead without loss of accuracy. Finally, we show that the high computational requirements of prior global power management algorithms based on linear programming make them infeasible for many-core chips, and that an algorithm that we call Steepest Drop achieves orders of magnitude lower execution time without sacrificing power-performance efficiency.

Improved Strategies for Radial basis Function Methods for Global Optimization

We propose some strategies that can be shown to improve the performance of the radial basis function (RBF) method by Gutmann [J. Global optim. 19(3), 201–227 (2001a)] (Gutmann-RBF) and the RBF method by Regis and Shoemaker [J. Global optim. 31, 153–171 (2005)] (CORS–RBF) on some test problems when they are initialized by symmetric Latin hypercube designs (SLHDs). Both methods are designed for the global optimization of computationally expensive functions with multiple local optima. We demonstrate how the original implementation of Gutmann-RBF can sometimes converge slowly to the global minimum on some test problems because of its failure to do local search. We then propose Controlled Gutmann-RBF (CG-RBF), which is a modification of Gutmann-RBF where the function evaluation point in each iteration is restricted to a subregion of the domain centered around a global minimizer of the current RBF model. By varying the size of this subregion in different iterations, we ensure a better balance between local and global search. Moreover, we propose a complete restart strategy for CG-RBF and CORS-RBF whenever the algorithm fails to make any substantial progress after some threshold number of consecutive iterations. Computational experiments on the seven Dixon and Szegö [Towards Global optimization, pp. 1–13. North-Holland, Amsterdam (1978)] test problems and on nine Schoen [J. Global optim. 3, 133–137 (1993)] test problems indicate that the proposed strategies yield significantly better performance on some problems. The results also indicate that, for some fixed setting of the restart parameters, the two modified RBF algorithms, namely CG-RBF-Restart and CORS-RBF-Restart, are comparable on the test problems considered. Finally, we examine the sensitivity of CG-RBF-Restart and CORS-RBF-Restart to the restart parameters.