Singular value decomposition shows us that the calibrated parameter field (i.e. the MAP, or minimum a posteriori, solution to the inverse problem) is actually the projection of the "real" parameter field (which we will never know), onto a much smaller dimensional subspace. This is the so-called "solution subspace" of the inverse problem. Singular value decomposition is undertaken on the Jacobian matrix.

The dimensionality of the solution space is generally much smaller than that of the null space (its orthogonal complement). The dimensions of the solution space and null space add up to the total number of parameters.

The goal of model calibration is to find a set of parameters that has no null space component. This guarantees no bias in model predictions (because you can never be sure that you are in the right part of the null space).

It follows that if we can separate the solution space from the null space early in the calibration process, then we can work in this low-dimensional subspace from then on. By construction, the parameter set that calibrates the model lies in this subspace. In fact, to ease the numerical burden of model calibration, we can define new parameters that span this subspace. These become the only parameters that we need to adjust when we calibrate the model.

This is what happens when we use PEST's SVD-assist functionality.

First set up the PEST input dataset in the usual way. Include parameters, observations and Tikhonov regularisation in this file, just as you would for a normal PEST run.

Next, set the NOPTMAX variable to -1 or -2 in the "control data" section of the PEST control file. Then run PEST. PEST calculates a Jacobian matrix, stores it in a JCO file, and then shuts down. (Parallelise these model runs to make calculation of this matrix faster if you like.)

Now you must build a new PEST control file which features only super-parameters. This is done automatically using the SVDAPREP program supplied with PEST.

Then run PEST. Model runs can be parallelised in the usual way, this saving even more time. Furthermore, when adjusting super-parameters, the first iteration of the SVD-assisted inversion process does not require calculation of a Jacobian matrix; PEST calculates the super parameter Jacobian matrix from the base parameter Jacobian matrix. So, after its initial model run, PEST can go straight into the Marquardt-lambda-testing phase of the first iteration. For subsequent iterations of the inversion process, a Jacobian matrix must be calculated - but only for super parameters. This requires only as many model runs as the dimensionality of the solution space.

Note that PESTPP-GLM does not require construction of a separate PEST control file when it estimates super-parameters.

How many super-parameters to use?

The SUPCALC utility (supplied with PEST) can tell you the minimum number of super-parameters to use.

However, it is a good idea to use as many super-parameters as your computing resources allow. So if, because of inverse problem nonlinearity, the border between the solution and null spaces shifts as parameter values change, you can still reduce model-to-measurement fit with a parameter set that is virtually null space free.

Best parameters

As a PEST inversion process progresses, the best parameters achieved so far can always be found in a parameter value file. This has an extension of ".par"; its filename base is the same as that of the PEST control file.

Things are a little different when undertaking SVD-assisted inversion. Optimised values of super-parameters are always available in the parameter value (i.e. PAR) file. However we cannot parameterise a model directly with these. What we need to know are optimised values of base parameters, for these are the parameters that inform the model. These are available in a file that has an extension of ".bpa" (for "best parameter values") and whose filename base is the same as that of the base PEST control file. When the inversion process is finished, you can use the PARREP utility to place these optimised parameter values into a new PEST control file.