Minimum Principles for Ill-Posed Problems

Ill-posed problems Ax = h are discussed in which A is Hermitian,and postive definite; a bound ║Bx║ ≤ β is prescribed. A minimum principle is given for an approximate solution x^. Comparisons are made with the least-squares solutions of K. Miller, A. Tikhonov, et al. Applications are made to deconvolution, the backward heat equation, and the inversion of ill-conditioned matrices. If A and B are positive-definite, commuting matrices, the approximation x^ is shown to be about as accurate as the least-squares solution and to be more quickly and accurately computable.