Approximating the singular value expansion of a compact operator

The singular values and singular vectors of a compact operator T can be estimated by discretizing T (in a variety of ways) and then computing the singular value decomposition of a suitably scaled Galerkin matrix. In general, the singular values and singular vectors converge at the same rate, which is governed by the error (in the operator norm) in approximating T by the discretized operator. However, when the discretization is accomplished by projection (variational approximation), the computed singular values converge at an increased rate; the typical case is that the errors in the singular values are asymptotically equal to the square of the errors in the singular vectors (this statement must be modified if the approximations to the left and right singular vectors converge at different rates). Moreover, in the case of variational approximation, the error in the singular vectors can be compared with the optimal approximation error, with the two being asymptotically equal in the typical case