| In Part I of the thesis, we investigate the volume-matching problem, which is the problem of finding a nonnegative surface which minimizes the Dirichlet integral subjected to the volume-matching constraints. An existence and uniqueness result is obtained for a general one-sided interpolation problem which includes the volume-matching problem as a special case. A variational characterization is obtained via the generalized Kuhn Tucker theorem. Making use of regularity results in variational inequality, we can also obtain a characterization by local behavior and boundary conditions. Convergence of the piecewise linear finite element approximation is proved together with an order of magnitude estimate of the approximation error. The discretized problem is a large scale quadratic programming problem which is too large to be solved by standard quadratic programming packages. An algorithm is presented which exploits the sparsity and structure of the matrices involved. The algorithm makes use of ideas from the gradient projection method, and the iterative methods for solving large linear systems.;In part II of the thesis a criterion is suggested for the smoothing parameter of density estimates. The bootstrap method can be used to approximate the expected Kullback-Leibler information between the estimated and the true density. The criterion proposes choosing as the smoothing parameter the value which minimizes this approximated expected information distance. The empirically successful modified likelihood is seen to be related to the jackknife version of the bootstrap approximation to the expected information. |
