February 25 th , 2014

1. POD Model Reduction: A Method for Reducing the Computational Burden of Solving Systems of Equations from Numerical Models Scott E. Boyce and William W.-G. Yeh February 25th, 2014 • Hydrology and Water Resources Program • Department of Civil and Environmental Engineering • University of California, Los Angeles • Contact: Boyce@engineer.com

2. Numerical Mathematical Models • Most hydrological systems are too complex to solve with a simple analytical solution. • Numerical approximations produce a system Ordinary Differential Equations (ODEs) or linear equations. • Three ways to improve speed of numerical approximations. • Wait for faster computers • Better Solvers • Model Reduction Techniques

3. Motivation For Model Reduction • Complex, highly-discretized models can have a substantial computational requirement to solve. • Speed of a single model run influences overall: • Management Decisions • Calibration Time (Parameter Estimation) • Projection Based Model Reduction Via Proper Orthogonal Decomposition (POD) • Project the system matrices onto a subspace that is easier to solve.

4. Visualizing Model Reduction • A gray scale image can be thought of as a matrix. • Each value in the matrix (pixel) contains a gray scale intensity. Think of each column as containing information about the image

5. All 1944 Terms

6. First 500

7. First 250

8. First 123

9. First 75

10. First 25

11. System of Equations • Linear systems of Equations or ODE’s are common when hydrological systems are approximated numerically. • The general forms of these two system of equations are:

12. Understanding

13. Model Reduction Projecting The System of Equations • Assume the following to be true: • Substitute into the system of ODEs • Apply the Galerkin Projection • Pre-multiply by PT

14. Making the Projection Matrix P • Collect solutions, called snapshots, from the model to construct P. • To be a proper projection basis matrix the set of snapshots have to be made orthonormal.

19. Proper Orthogonal Decomposition (POD) Singular Value Decomposition (SVD)

20. Confined Groundwater Flow Equation

21. Solving the Groundwater Equation • By applying finite differences or finite elements the spatial partial derivatives are transformed into a linear system of temporal ordinary differential equations

22. Synthetic model of theOristanoplain of Sardinia, Italy Modeled With SAT2D 57,888 Elements 29,197 Nodes 7 Hydraulic Conductivity Zones 6 Pumping Wells 1 Layer that is 100 m Thick Specific Storage is 10-5 m-1 5 day pump test simulation

23. Final Reduction Sizes • Original Size: 29,197 Equations → Simulation Time: ~33 s Collected snapshots at a constant pumping rate with different values of hydraulic conductivity. • Reduced Model: 127 Equations → Simulation Time: ~2.9 s

24. Reduce Model Accuracy For A Parameter Set

25. Conclusion • POD Model Reduction is a viable option for improving the speed of a linear system of equations or ODE’s • ~ 99% reduction in state dimension • ~ 90% reduction in solution time • While theoretically only applicable to linear systems, a nonlinear extensions require: • More snapshots to capture nonlinear behavior • Linearization of the nonlinear equations

26. Thank you for your time Questions?

27. Thank you for your time Questions?

28. Principle Component Analysis • PCA is similar to SVD, but instead analyzes the eigenvalues of a symmetric matrix. • Typically done on a covariance matrix. • Due to unnecessary work and numerical accuracy this is not recommend for model reduction.