Iterative methods

Starting from a first estimation of the model parameters or from whatever appropriate model, the iterative method converges to the minimum of $L$ by modifying the current model according to the local properties of function $L$ . In the case of Newton-Raphson, damped least-square or gradient methods, the partial derivatives or the Jacobian matrix at the current model orientates the descent towards the solution (Tarantola (1987); Herrmann (1994); Nolet (1981),...). Calculating the partial derivatives allows a linearization of the problem and linear algebra is used to calculate a new estimate of the solution. The process is repeated through several iterations until finding an acceptable minimum. Downhill simplex (Press et al. (1992)) is an other iterative method that requires only function evaluations, not derivative. It is based on geometrical principles.

These kinds of methods are the exact opposite of the gridding method. The exploration of the parameter space is limited to the path followed during the successive iterations. They are mostly used for high dimensional parameter spaces for their ability to quickly converge to the solution. The number of function computations is very small compared to all other methods.

If there are more that one minimum or if the function $L$ has a complex shape with multiple secondary minima, those methods are likely to converge to one of them which is probably not the unique and the absolute minimum. The final solution highly depends upon the starting model. The non-uniqueness, a common phenomenon in inverse problems (Sambridge (2001)), can be apprehended only by a manual selection of "good" starting models. These methods are then inadequate when the nonlinearity becomes severe, and can produce optimistic resolution estimates, usually calculated around a single best data-fit model (Sambridge (2001)). Shapiro (1996) showed, that the solutions obtained from classical surface wave inversion schemes (damped least-square) are too restrictive and uncertainties are not correctly estimated.

From the starting model, the iterative process may lead the current model to whatever part of the parameter space, being in this case $\Re^n$ ^2.5. It depends upon the unknown shape of function $L$ . Indeed, $L$ is known for only a discrete number of points where the forward problem has been solved. In this framework, it is impossible to guarantee that the current model stays within a defined zone of the parameter space for all iterations. The limits of this zone are adjusted so that it encloses all potential solutions, given the prior knowledge we have about the model.