Monday, April 12, 2010: 11:25 a.m.
Tabor Auditorium (Westin Tabor Center, Denver)
Johan Valstar
,
Subsurface and Groundwater Systems, Deltares, Utrecht, Netherlands
Jarno Verkaik
,
Subsurface and Groundwater Systems, Deltares, Utrecht, Netherlands
A new inverse method is introduced for groundwater flow modeling. It is based on a combination of the representer method and a preconditioned conjugate gradient solver. The method has been applied in oceanography and meteorology and has been addressed as the iterative representer method. Unfortunately, the standard representer method in inverse groundwater flow modeling is already an iterative algorithm, so we renamed the method to avoid confusion. The standard representer method requires representer values for each measurement. Each representer calculation takes two groundwater flow model evaluations, which makes the method computationally demanding for inverse problems with a large number of measurements. The representer values however are only used to solve a single matrix equation, after which the solution vector of this equation is the only information that is actually needed for the remainder of the analysis.
The preconditioned conjugate gradient is an efficient method to solve a matrix equations (Ax=b) if the product between the matrix A and an arbitrary vector can be calculated and a preconditioning matrix that resembles matrix A is used. It turns out that the required matrix vector multiplication with an arbitrary vector can be performed using the representer setup
A reasonable preconditioning matrix can be obtained using a Monte Carlo analysis, as it is known that matrix A equals the linearized covariance of the measurement predictions. For large inverse problems with many measurements (> 5000) the full preconditioning takes a considerable amount of computation time and memory storage and an alternative was developed by approximating the preconditioning matrix with a fraction of its (most important) eigenvalues and the corresponding eigenvectors.
Preliminary results show that the number of groundwater flow model evaluations is reduced by more than 98 % for a large inverse model with over 5000 measurements.
