Second-order approximation of the pseudoinverse for operator deconvolutions and families of ill-posed problems

We identify a more accurate means of deconvolving compact nondegenerate operators than that provided by independent application of standard regularization techniques to individual members of the implied continuum of first kind integral equations. For the problem defined by h(x, t) = integral(Y) f(x, y)g(y, t)dY, where it is required to find g(y, t) given noise-corrupted versions of square-integrable f(x, y) and h(x, t), it is shown that greater accuracy results from optimizing solutions for each member of the sequence of integral equations derived from individual components of the singular value expansion of h(x, t), than from optimizing solutions to each of the equations determined by different fixed values of t. This result has particular implications for treatment of the bioelectric/biomagnetic source imaging problems, where the characteristic need to perform such a deconvolution has been previously handled by producing an individually regularized solution to the above equation for each t in a collection of times spanning an interval of interest, or by inadequate augmentation of the dimensionality of the regularization parameter. A numerical example illustrates the potential advantages of the method.