5. Conclusion
A traditional model reduction technique for groundwater flow has been combined with an interpolation scheme to further reduce nonlinear components. The result is a reduced model of an uncon- fined flow equation that can be solved entirely in the reduced dimension with no dependence on the original, full model complexity. This additional approximation allows for faster calculations of nonlinear operations at each time step while sacrificing a tractably small amount of accuracy. As simulation models get more complex, with finer discretization, larger domains, and more nonlinear processes, faster calculations become more important. The combined model reduction approach with POD and DEIM greatly improves a modeler’s ability to obtain solutions quickly. The results from the two test problems show a two to three orders of dimension reduction. A key advantage of the POD-DEIM model is that nonlinear operations are carried in the reduced space. The faster overall simulation times are critical when embedding within or linking the model to any form of optimization (e.g., parameter estimation, experimental design, resource allocation) or extensive uncertainty analysis (e.g., Monte Carlo). While more and more optimization algorithms are taking advantage of parallel computing power, long simulation runtimes still inhibit the attainment of optimal solutions in reasonable amounts of time. Therefore, reduced models such as those developed with POD-DEIM can be used within parallel architectures to facilitate searching very large feasible regions— regions with dimensions so large that they would otherwise be impossible to explore.
