1. A Hybrid Optimization Approach for Automated Parameter Estimation Problems Carlos A. Quintero 1 Miguel Argáez 1, Hector Klie 2, Leticia Velázquez 1 and Mary Wheeler 2 Problem Formulation Contact Information Carlos A. Quintero, Graduate Student The University of Texas at El Paso Department of Mathematical Sciences 500 W. University Avenue El Paso, Texas 79968-0514 Email: caquintero@utep.edu Phone: (915) 747-6858 Fax: (915) 747-6502 Abstract We present a hybrid optimization approach for solving automated parameter estimation problems that is based on the coupling of the Simultaneous Perturbation Stochastic Approximation (SPSA)[Ref.1] and a globalized Newton-Krylov Interior Point algorithm (NKIP) presented by Argáez et al.[Refs.2,3]. The procedure generates a surrogate model on which the NKIP algorithm can be used to find an optimal solution. We implement the hybrid optimization algorithm on a test case, and present some numerical results. where the global solution x* is such that We are interested in problem (1) that may have many local minima. We consider the global optimization problem in the form: Local Search: NKIP Parameter estimation using sensor pressure data This analysis was done to show that despite having full information of the fluid flow pressure field in this small case, the inverse problem is still highly ill- posed and has multiple local minima. This illustrates how challenging parameter estimation can be on realistic reservoir scenarios. Therefore, strategies to regularize and reparameterize the inverse problem need to be employed. Our goal is to estimate the permeability based on pressure measurements. We formulate the parameter estimation as a nonlinear least squares problem: where x * is the true values of the pressure and F is a positive diagonal matrix. We run a single-phase simulation for a given permeability field K of size 10x10 and some given default parameters. The simulation returns the pressure field at 100 different pressure observations points. Test Case We find the surrogate model f s (x) using an interpolation method with the data,, provided by SPSA. This can be performed in different ways, e.g., radial basis functions, kriging, regression analysis, or using artificial neural networks. In our test case, we optimize the surrogate function Surrogate Model Hybrid Optimization Scheme Filtering Data Global Search Via SPSA Target Regions Surrogate Model Explore Parameter space Interpolate Response surface No Optimal Solution found for Original model? Optimal Solution found for Original model? Local Search Via NKIP Local Search Via NKIP Stop Yes Optimal Solution The initial stage of the framework consist of a multiscale treatment of the permeability field through successive wavelet transformations. The coarsest grid which, represents a highly constrained parameter spaced, is sampled with the aid of a SPSA that detects the most promising search regions. SPSA is a global derivative free optimization method that uses only objective function values. In contrast with most algorithms which requires the gradient of the objective function that is often difficult or impossible to obtain. SPSA is efficient in high-dimensional problems in providing a good approximate solution using few function values [Ref. 5]. We use multistart on to increase the chances for finding a global solution, and find several target regions of the parameter space. Global Search: SPSA Future Work References 1 Department of Mathematical Sciences The University of Texas at El Paso 2 ICES-The Center for Subsurface Modeling The University of Texas at Austin [1] J. C. Spall. Introduction to stochastic search and optimization: Estimation, simulation and control. John Wiley & Sons, Inc., New Jersey, 2003. [2] M. Argáez, R. Sáenz, and L. Velázquez. A trust–region interior–point algorithm for nonnegative constrained minimization. Technical report, Department of Mathematics, The University of Texas at El Paso, 2004. [3] M. Argáez and R.A. Tapia. On the global convergence of a modified augmented Lagrangian linesearch interior-point Newton method for nonlinear programming. J. Optim. Theory Appl., 114:1–25, 2002. [4]Mark J. L. Orr. Matlab Functions for Radial Basis Function Networks, 1999. [5] H. Klie and W. Bangerth and M. F. Wheeler and M. Parashar and V. Matossian. Parallel well location optimization using stochastic algorithms on the grid computational framework. 9th European Conf. on the Mathematics of Oil Recovery (ECMOR), August, 2004. [6] A.J. Keane and P.B. Nair. Computational Approaches for Aerospace Design: The Pursuit of Excellence. Wiley, England, 2005. The scheme is to use SPSA as the sampling device to perform a global search of the parameter space and switch to NKIP to perform the local search via a surrogate model. Further research and numerical experimentation are needed to demonstrate the effectiveness of the hybrid optimization scheme being proposed, especially for solving large application problems of interest of the Army. Acknowledgments The authors were supported by DOD-PET Project EQMKYG 003. where a and b are determined by the sampled points given by SPSA. NKIP is a globalized and derivative dependent optimization method. This method calculates the directions using the conjugate gradient algorithm, and a linesearch is implemented to guarantee a sufficient decrease of the objective function as described in Argáez and Tapia [Ref.4]. This algorithm has been developed for obtaining an optimal solution for large scale and degenerate problems. Using the surrogate model as a target function, we use NKIP to find a local solution of the model in few iterations steps. We consider the optimization problem in the form: Numerical Results We run the test case in a Dell Laptop using Matlab 7.1. We use 5 starting random points and allow 2000 iterations for SPSA. We obtain 10167 search points and we filter 105 points to create the surrogate model. Next we use NKIP to find a local solution in 103 iterations and the optimal function value obtained is 0.00353. We now plot the permeability and pressures calculated from the hybrid approach versus true data. where the multiquadric basis functions are given by The interpolation algorithm [Ref.4] characterizes the uncertainty parameters