Parameter Estimation Schemes for Solving ADH Groundwater Models Mark Byers¹ Miguel Argáez¹, Jackie Hallberg² and Leticia Velázquez¹

¹

Department of Mathematical Sciences The University of Texas at El Paso

²

US Army Corps of Engineers Abstract

A hybrid optimization MATLAB code is being developed by the University of Texas at El Paso (UTEP) and University of Texas at Austin that is to be used with the ADH (ADaptive Hydraulics) groundwater model constructed at the Engineering Research Development Center (ERDC).

The hybrid code offers a combined global and local optimization method by coupling Newton-Krylov Interior-Point (NKIP), Global Levenberg Marquardt (GLM) and the Simultaneous Perturbation Stochastic Analysis (SPSA) methods.

The current model problem uses ADH in conjunction with a nonlinear parameter estimation package named PEST.

Our goal is to replace PEST with our hybrid optimization code that has potential application to the parameter estimation problems faced by the ERDC.

Model Problem

ADH is modeling a heat transport problem using measurements from an arid region. The site is a desert landscape with disturbed soil at 768 meters in elevation. During a period of 5 days, various types of measurements that were taken include temperature, velocity, saturation, soil properties and wind speed.

There are m=468 observations and n=7 parameters. The parameters are the specific heat and thermal conductivity of the undisturbed silt, disturbed soil and road surface, and the bulk emissivity of the road surface Heat equation in 3d

ADH

ADH is a state-of-the-art modeling system, developed by the Coastal and Hydraulics Laboratory at ERDC, that is capable of handling both saturated and unsaturated groundwater, overland flow, three dimensional Navier-Stokes flow, and two- or three-dimensional shallow water problems. Applications include the drainage through heterogeneous soil and rainfall/runoff in a simple test basin.

ADH uses the Galerkin Finite Element Method to solve the heat equation problem with local mesh refinement/coarsening to capture steep gradients.

A linear system must be solved in each Newton iteration. ADH offers 4 preconditioners including Point Jacobi and Two-Level Additive Schwarz that can dramatically reduce the number of iterations.

where • • u is the temperature as a function of time and space • u

t

is the rate of change of temperature at a point over time

k

is a material-specific quantity depending on the thermal conductivity, density and heat capacity Arid Region Model Input elevation(m), Latitude, Longitude, GMT-Local 768 3433 2 24.26 250.79 -3 1 9999 2 Press Temp Rh WdSp Dirct Vis Aer Prec C l o u d x x Solar Solar Solar … Day Hour (mb) (C) (%) (m/s) (deg) (km) H mm/hrP t L t M t H Global Direct diffuse 344 0 974.3 6.9 25.8 0.70 126.1 30.9 10 0 1 0 0 0 0 0 0 0.00 999.00 0.00 344 1 974.0 6.0 27.1 0.52 207.9 33.5 10 0 1 0 0 0 0 0 0 0.00 999.00 0.00 344 2 973.9 7.3 23.5 0.98 330.3 33.8 10 0 1 0 0 0 0 0 0 0.00 999.00 0.00 344 3 973.2 7.7 22.0 0.68 332.2 33.9 10 0 1 0 0 0 0 0 0 0.00 999.00 0.00 344 4 973.0 5.8 25.2 0.05 155.7 34.2 10 0 1 0 0 0 0 0 0 0.00 999.00 0.00 …………………………………………………………………………………………..………….

……………………………………………………………………………………………………..

349 19 975.1 11.8 19.5 0.69 133.2 28.2 10 0 1 0 0 0 0 0 0 0.00 999.00 80.02 349 20 975.9 11.2 20.6 0.84 117.9 13.1 10 0 1 0 0 0 0 0 0 0.00 999.00 92.8

349 21 976.0 10.0 22.3 0.47 164.6 18.4 10 0 1 0 0 0 0 0 0 0.00 999.00 111.43 349 22 976.0 6.4 27.0 0.09 223.5 25.4 10 0 1 0 0 0 0 0 0 0.00 999.00 127.70 349 23 976.0 6.6 27.5 0.26 241.8 24.6 10 0 1 0 0 0 0 0 0 0.00 999.00 136.02

Arid Region Landscape

PEST

PEST is a nonlinear parameter estimation package capable of estimating parameters for any computer model.

It solves a nonlinear least squares problem and minimizes the differences between the model's outputs and field measurements such as the calculated and measured temperature. PEST communicates with a model through the model's input and output files and uses a roubust variant of the Gauss-Marquardt-Levenberg method of nonlinear parameter estimation. Further, through adjustment of a number of control variables, a user is able to "tune" PEST's implementation of the method to suit the model for which parameters are sought. ADH and PEST work together to solve the current model problem.

i

 

1 2

i m

 

1

r i

2

( ;

i

,

i

, 

R w T i i

) 

T

i

ADH-PEST Output

Machine: AMD 3200+, CPU Time: 32 minutes Residual_L2_norm = f(w*)= 3.670591534e-07 Total number of linear iterations = 68178 Total number of nonlinear iterations = 2542 Parameter Name: shs=specific heat thk=thermal conductivity ems=bulk emissivity Material Property: 1= Undisturbed Silt, 2=Disturbed Soil, 3= Road Surface Solution w*= (shs1,thk1,shs2,thk2,shs3,thk3,ems3)= (3.74x10

-4 , 0.397, 1.03x10

-4 , 0.340, 1.56x10

-4 , 0.268, 0.938)

Hybrid Optimization Code

The hybrid optimization code is intended to solve large-scale problems of interest to ERDC. The code is based on the coupling of Simultaneous Perturbation Stochastic Approximation (SPSA) approach by Spall (a global derivative free optimization method) with two local and derivative dependent optimization methods: a globalized Newton-Krylov Interior-Point (NKIP) algorithm developed by Argaez, and the Global Levenberg-Marquardt (GLM) method introudced by Velazquez et. al. Finding a global solution in parameter estimation is difficult since models are usually nonlinear, non-smooth and subject to different sources of errors. SPSA performs random simultaneous perturbations of all model parameters to generate a descent direction at each iteration. Advantages of the SPSA algorithm are its simplicity, flexibility and low computational cost (at most 2 function evaluations at each iteration). A multi-start initial guess can be used with SPSA to help speed up the process of detecting the most promising search regions. Next, a sequence of surrogate models are built, and then the use of NKIP or GLM precedes until convergence to an optimal solution is reached.

Global Search Via SPSA

Explore Space

Enough Sampling?

Iterate and look for another prospective No Yes

Surrogate Model

Exploit sensitivity structure of the problem

Use of Derivatives STOP

Compute NLS residuals until convergence is reached

Nonlinear Least Squares (NLS) Solution via GLM/NKIP

Preliminary Numerical Results

We conducted an independent experimentation on a parameter estimation problem using sensor pressure data applying GLM. This analysis shows that even do we have 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. We also conducted the same experimentation using SPSA combined with NKIP (see poster presentation by Carlos Quintero), and we also obtained promising results.

Future Plans

We plan to couple the hybrid optimization algorithm with ADH to demonstrate the effectiveness of this scheme. We will compare the results obtained by the PEST and hybrid scheme.

References

1. A Hybrid Optimization Approach for Automated Parameter Estimation Problem, Miguel Argaez, Hector Klie, Carlos Quintero, Leticia Velazquez and Mary Wheeler, Technical Report 2007.

2. Projected Conjugate Gradient for Constrained Optimization, Miguel Argaez, Technical Report 2007.

3. PEST, Parameter Estimation and Uncertainty Analysis Software, 2006 4. Numerical Comparisons of path-following Strategies for a Primal-Dual Interior-Point Method for Nonlinear Programming}, Miguel Argaez, R. Tapia, and L. Velazquez, Journal of Optimization Theory and Applications, Vol. 114(2):255-272, 2002.

5. Selective Search for Global Optimization of Zero or Small Residual Least-Squares Problems: A Numerical Study, Leticia Velazquez, Richard Tapia, and Yin Zhang, Computational Optimization and Applications: An international Journal, Vol. 20(3):299-315, 2001.

6. User's manual for Groundwater/Surface Water Interaction using ADH, S. E. Howington, R. C. Berger, E. W. Jenkins, J. P. Hallberg, K. R. Kavanagh, and J. H. Schmidt, July 2003.

7. 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 8. J. C. Spall. Introduction to stochastic search and optimization: Estimation, simulation and control. John Wiley & Sons, Inc., New Jersey, 2003.

9. 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.

Acknowledgement

This work is being supported by the US Corps of Engineers Contract No. W912HZ-06-P-0402.

Contact Information

Mark Byers, Graduate Student The University of Texas at El Paso Department of Mathematical Sciences 500 W. University Avenue El Paso, Texas 79968-0514 Email: [email protected]

Phone: (915) 747-7005