Download we include it here.

Advamced Regularisation and SVD-Assist
8-2
calibration dataset.
The first part of this chapter continues the discussion of the previous chapter, showing how
more sophistication can be added to the Tikhonov regularisation methodologies discussed
therein. The second part of this chapter introduces singular value decomposition as a
parameter estimation methodology. Following this, PEST’s unique “SVD-assist”
methodology is discussed, this being a method that combines the two main types of
regularisation techniques into a single, extremely powerful methodology, that not only results
in enhanced numerical stability of the inversion process, but can also result in enormous
efficiency gains, thus allowing the use of regularised inversion in conjunction with complex
models whose run times are large.
This chapter finishes by describing some memory conservation functionality that has been
introduced to PEST in order to allow it to better accommodate the estimation of hundreds, or
even thousands, of parameters, using thousands, or even tens of thousands of regularisation
constraints. It also describes an alternative solver for the “normal equation” through which
the parameter upgrade vector is calculated. This is a conjugate gradient solver. It must be
said, however, that at the time of writing, this solver has not lived up to expectations.
However it has been retained within PEST in the hope that future research will lead to a
method of preconditioning that will greatly increase its speed in highly parameterised
modelling contexts.
8.2 Multiple Regularisation Groups
8.2.1 Definition of Multiple Groups
In the previous chapter it was asserted that an observation or prior information item must be
assigned to the observation group “regul” if it is to be used in the regularisation process, and
if the residual associated with this item is thus to contribute to the regularisation objective
function rather than to the measurement objective function. In fact, PEST can accommodate
the existence of multiple regularisation groups rather than the single group “regul”. Any
observation group whose name begins with the letters “regul” is considered to be a
“regularisation group”. Thus if PEST is run in regularisation mode, members of the
observation groups “regul1”, “regular”, etc will be considered to be regularisation
observations or regularisation prior information equations.
Multiple regularisation groups can be of use where different types of information are
employed in the regularisation process. As is explained in the previous chapter, in
implementing the regularised inversion process, PEST calculates a regularisation weight
factor by which the regularisation objective function is multiplied before being combined
with the measurement objective function to form the total objective function. However unless
PEST is instructed otherwise (see below) the relative weighting assigned to different
components of the regularisation objective function must be decided by the user. This can be
difficult to do if the contribution of each regularisation group to the total regularisation
objective function is unknown. By allowing different types of regularisation observations or
prior information to be placed into groups of different name, PEST is able to print the
contribution made by these groups to the objective function. This, in turn, allows the user to
ensure that one such group does not dominate the regularisation objective function, and is not