A Reduced Computation Method for Choosing the Regularization Parameter for Tikhonov Problems

This paper presents a method for choosing the regularization parameter (α) appearing in Tikhonov regularization based on the L-curve. The point of intersection between the L-curve and a straight line of arbitrary slope tangent to the L-curve is used as the criterion for identifying the corner of the L-curve and the corresponding value of α. This condition is shown to result in a new scalar algebraic equation for α in terms of the components of the SVD of the problem. A root of this equation yields the optimal α. The formulation of the problem in this way contrasts existing methods which require multiple solutions of the original regularization problem. Since computing the SVD remains costly for large problems, we then use the new method as the basis for rational approximation through the use of only a limited number of the singular values and vectors. Simulation results show that the proposed suboptimal method can provide a reasonable value of regularization parameter even when very few singular values and vectors are used in its definition.