1 Hybrid methods for solving large-scale parameter estimation problems Carlos A. Quintero 1 Miguel Argáez 1 Hector Klie 2 Leticia Velázquez 1 Mary Wheeler 2 1 Department of Mathematical Sciences University of Texas at El Paso 13th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Verified Numerical Computations SCAN'2008 El Paso, Texas, USA October 2, ICES-The Center for Subsurface Modeling The University of Texas at Austin

2 Outline Introduction Problem Formulation Parameter Estimation Optimization Framework Global Search Surrogate Model Local Search Numerical Results Conclusion and Future Work

3 Goal We present a hybrid optimization approach for solving global optimization problems, in particular automated parameter estimation models. The hybrid approach is based on the coupling of the Simultaneous Perturbation Stochastic Approximation (SPSA) and a Newton-Krylov Interior-Point method (NKIP) via a surrogate model. We implemented the hybrid approach on several test cases.

4 Problem Formulation where the global solution x* is such that We are interested in problem (1) that have many local minima. We consider the global optimization problem in the form:

5 Types of local minima x global minimum local minima

6 Hybrid Optimization Framework Global Method: SPSA Surrogate Model Local Method: NKIP

7 Global Method: SPSA Stochastic steepest descent direction algorithm (James Spall, 1998) Advantage SPSA gives region(s) where the function value is low, and this allows to conjecture in which region(s) is the global solution. Uses only two objective function evaluations at each iteration to obtain the update of the parameters. Disadvantages Slow method Do not take into account equality/inequality constraints

8 Global Search: SPSA SPSA performs random simultaneous perturbations of all model parameters to generate a descent direction at each iteration This process may be performed by starting with different initial guesses (multistart). Multistart increases the chances for finding a global solution, and yields to find a vast sampling of the parameter space.

9 SPSA: Pseudocode

10 Observations SPSA gives region(s) where the function value is low, and this allows to conjecture in which region(s) is the global solution. This give us a motivation to apply a local method in the region(s) found by SPSA.

11 Local Method Advantages Fast Method: Newton Type Methods Interior-Point Methods allow to add equality/inequality constraints Disadvantage Needs first/second order information Solution Construct a Surrogate Model using the SPSA function values inside the conjecture region(s)

12 A surrogate model is created by 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. Surrogate Model

13 Why Surrogate Models? Most real problems require thousands or millions of objective and constraint function evaluations, and often the associated high cost and time requirements render this infeasible.

14 RBF is typically parameterized by two sets of parameters: the center c which defines its position, and shape r that determines its width or form. An RBF interpolation algorithm (Orr,1996) characterizes the uncertainty parameters: Radial Basis Function (RBF)

15 Our goal is to optimize the surrogate function Where the radial basis functions can be defined as: that are the multiquadric and the Gaussian basis functions, respectively Surrogate Model

16 Hybrid Optimization Scheme (1) Global Search Via SPSA Filtering Surrogate Model Explore Parameter space Multistart (x 0 ) 1 (x 0 ) 2. (x 0 ) k Sampling Target Region

17 Test Case 2 Rastrigin’s Problem:

18 Test Case 2: Rastringin

19 Test Case 2: Rastringin x*=(0,0) global solution f(x*)=0

20 Sampling Data from SPSA Test Case 2: Restringin

21 x*=(0,0) global solution f(x*)=0 Sampling Data from SPSA Test Case 2: Restringin

22 Restringin: Surrogate Model We plot the original model function and the surrogate function:

23 Restringin: Surrogate Model

24 Local Search: NKIP NKIP is a globalized and derivative dependent optimization method based on the global strategy introduced by Miguel Argaéz and Richard Tapia in This method calculates the directions using the conjugate gradient algorithm, and a linesearch is implemented to guarantee a sufficient decrease of the objective function.

25 Local Search: NKIP We consider the optimization problem in the form: where a and b are determined by the sampled points given by SPSA.

26 Hybrid Optimization Scheme (2)

27 Hybrid Optimization Scheme Convergence history for minimizing Rastringin’s function (n=50).

28 where is a nonlinear function and is a set of data observations The objective function can be written as a nonlinear least squares problem: Parameter Estimation Problem The objective function can be written as a nonlinear least squares problem:

29 Our goal is to estimate the permeability based on sensor pressure data. Despite having full information of the pressure field, the inverse problem is highly ill-posed and has multiple local minima. Application

30 Large Scale Two-Phase Flow Problem 3D permeability field

31 Large Scale Two-Phase Flow Problem Figure 2. The true and initial permeability and pressure fields.

32 High Performance Computing We run the simulator IPARS (Integrated Parallel Accurate Reservoir Simulator) on a Linux-based multicore network of workstations. Our goal is to estimate a permeability field that involves 2000 parameters. The field is parameterized by SVD reducing the parameter space to 20 (each parameter represents a scale resolution level). We choose several initials points by adding a percentage of noise to the true singular values: 5%, 10% and 15%

33 Numerical Results Noise5%10%15% Initial starting points by SPSA3020 Range of f(x0)8.56 to to to SPSA Iterations Range of fspsa(x)0.568 to to to Best f spsa (x) Data points used in the surrogate model NKIP Iterations f nkip (x) % Gain by Hybrid Approach16135

34 We combine SPSA and NKIP strategies into a Hybrid Scheme that exploit the best of these two approaches for a given problem in order to achieve maximum efficiency and robustness. The hybrid approach finds a good estimate of the global solution for all the test cases. As the noise level was increased, the hybrid approach presented more difficulties in reproducing the true permeability field. However, the estimation is very accurate in all cases. Conclusions

35 Improve a parallel version of the multi-start technique Parallelize the conjugate gradient of NKIP Add equality constraints to the minimization of the surrogate model Add more Global Optimization Techniques (Simulated Annealing, Evolutionary Algorithms) Future Work

36 Thank you very much for your attention