Regularization and linear boundary value problems

Abstract

The method of regularization is used in a general setting to obtain least squares solutions of thelinear equation Lx = y, permitting applications to linear boundary value problems of ordinary and partial differential equations. In regularization the functional

is minimized over the domain , where L is assumed to be aclosed densely defined linear operator from a Hilbert space X into a Hilbert space Y, α is a nonzero parameter, and T is a linearoperator from X into a Hilbert space Z with . Under suitable conditions on L and T, it is shown that G_α achieves a minimum at a unique point x_α in , and by using the alternative method the x_α are shown to converge to a least squares solution of Lx = y with rate of convergence of order α² . The regularization method is also recast as a least squares process. Finally, the important special case Z = X and T = I is examined in detail, andthe method is applied to the numerical solution of some model boundary value problems.