Overview

So far, we have diseussed various algorithms for computing a regularized solution. Every algorithm has its advantages and disadvantages in terms of implementation issues, filter properties. etc. However, no regularization method is complete without a method for choosing the regularization parmeter. either the continuous parameter ? or the discrete parameter k. and in this chapter we discuss several parameter-choice methods and their implementations. Surveys of parameter-choice methods are rare in the literature; we are currently aware of [171] and [333].

Whenever possible, we discuss these methods in terms of a continuous regularization parameter ?, but we emphasize that similar results hold when ? is replaced by a discrete regularization parameter k. We will also restrict our discussion to the standard-form case whenever the extensions to the general case are obvious, and we use the notation x _? = x _{I n, ?}.

Most of the parameter-choice methods are based on residual norms and, in the case of the L-curve, also on the solution's "size" ?, typically its seminorm. When solving general-form problems via a standard-form transformation, it is therefore important to recall the relations

cf. (2.39) and (2.45). These relations ensure that application of a norm-based parameter-choice rule to the original problem with A and b, or to the standard form problem with A and b, yields exactly the same regularization parameter.

For certain problems it may be advantageous to use more information about the residual...