Hybrid methods for solving large-scale parameter estimation problems

1. Hybrid methods for solving large-scale parameter estimation problems 2 ICES-The Center for Subsurface Modeling The University of Texas at Austin Carlos A. Quintero1 Miguel Argáez1 Hector Klie2 Leticia Velázquez1 Mary Wheeler2 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 , 2008

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

3. 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. Goal

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

5. Types of local minima local minima global minimum x

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). • Multistartincreases 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. Surrogate Model • 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.

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. Radial Basis Function (RBF) • 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:

15. Surrogate Model • 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

16. Global Search Via SPSA Hybrid Optimization Scheme (1) Explore Parameter space Multistart (x0)1 (x0)2 . . . (x0)k . . . Target Region Filtering Sampling + Surrogate Model

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. Test Case 2: Restringin • Sampling Data from SPSA

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

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 2002. 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. Parameter Estimation Problem The objective function can be written as a nonlinear least squares problem: The objective function can be written as a nonlinear least squares problem: The objective function can be written as a nonlinear least squares problem: where is a nonlinear function and is a set of data observations

29. Application • 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.

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

34. Conclusions • 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.

35. Future Work 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)

36. Thank you very much for your attention 36