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This thesis concerns the solution of variational inequalities (VIs) with analytic center cutting plane methods (ACCPMs). A convex feasibility problem reformulation of the variational inequality is used; this reformulation applies to VIs defined with pseudo-monotone, single-valued mappings or with maximal monotone, multi-valued mappings. / Two cutting plane methods are presented: the first is based on linear cuts while the second uses quadratic cuts. The first method, ACCPM-VI (linear cuts), requires mapping evaluations but no Jacobian evaluations; in fact, no differentiability assumption is needed. The cuts are placed at approximate analytic centers that are tracked with infeasible primal-dual Newton steps. Linear equality constraints may be present in the definition of the VI's set of reference, and are treated explicitly. The set of reference is assumed to be polyhedral, or is convex and iteratively approximated by polyhedra. Alongside of the sequence of analytic centers, another sequence of points is generated, based on convex combinations of the analytic centers. This latter sequence is observed to converge to a solution much faster than the former sequence. / The second method, ACCPM-VI (quadratic cuts), has cuts based on both mapping evaluations and Jacobian evaluations. The use of such a richer information set allows cuts that guide more accurately the sequence of analytic centers towards a solution. Mappings are assumed to be strongly monotone. However, Jacobian approximations, relying only on mapping evaluations, are observed to work very well in practice, so that differentiability of the mappings may not be required. There are two versions of the ACCPM-VI (quadratic cuts), that differ in the way a new analytic center is reached after the introduction of a cut. One version uses a curvilinear search followed by dual Newton centering steps. The search entails a full eigenvector-eigenvalue decomposition of a dense matrix of the order of the number of variables. The other version uses two line searches, primal-dual Newton steps, but no eigenvector-eigenvalue decomposition. / The algorithms described in this thesis were implemented in the M ATLAB environment. Numerical tests were performed on a variety of problems, some new and some traditional applications of variational inequalities.

Two aspects of the theory of barotropic modons are examined in this thesis. First, sufficient neutral stability conditions are derived in the form of an integral constraint for westward and eastward-travelling modons. It is shown that eastward-travelling and westward-travelling modons are neutrally stable to perturbations in which the energy is contained mainly in spectral components with wavenumber magnitudes (|ƞ|) satisfying |ƞ|<κ and |ƞ|>κ, respectively, where κ is the modon wavenumber. These results imply that when κ/|ƞ|>1 the slope of the neutral stability curve proposed by McWilliams et al.(l98l) for eastward-travelling modons must begin to increase as κ/|ƞ| increases. The neutral stability condition is computed with mesoscale wavenumber eddy energy spectra representative of the atmosphere and ocean. Eastward-travelling atmospheric modons are neutrally stable to the observed seasonally- and annually-averaged atmospheric eddies. The neutral stability of westward-travelling atmospheric modons and oceanic modons cannot be inferred on the basis of the observed wavenumber eddy energy spectra for the atmosphere and ocean. Second, a leading order perturbation theory is developed to describe the propagation of barotropic modons in a slowly varying medium. Two problems are posed and solved. A perturbation solution is obtained describing the propagation of an eastward-travelling modon modulated by a weak bottom Ekman boundary layer. The results predict that the modon radius and translation speed decay exponentially and that the modon wavenumber increases exponentially, resulting in an exponential amplitude decay in the streamfunction and vorticity. These results agree with the numerical solution of McWilliams et al.(l98l). A leading order perturbation theory is also developed describing modon propagation over slowly varying topography. Nonlinear hyperbolic equations are derived to describe the evolution of the slowly varying modon radius, translation speed and wavenumber for arbitrary finite-amplitude topography. To leading order, the modon is unaffected by meridional gradients in topography. Analytical perturbation solutions for the modon radius, translation speed and wavenumber are obtained for small-amplitude topography. The perturbations take the form of westward and eastward-travelling transients and a stationary component proportional to the topography. The general solution is applied to ridge-like and escarpment-like topographic configurations. / Science, Faculty of / Mathematics, Department of / Graduate

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The HELMHOLTZ package, written in Maple V, is a collection of commands to support research in the variational calculus. These commands include the standard operators on differential forms, Euler-Lagrange operators, homotopy operators, Lie bracket, Lie derivatives, and the prolongation of a vector field. We give a brief introduction to the variational calculus. We describe each of the commands in the HELMHOLTZ package completely along with numerous examples of each. Applications of the package include verification of symmetry groups for differential equations, solving the inverse problem of the calculus of variations, computing generalized symmetries, and finding variational integrating factors. A complete listing of the Maple code for HELMHOLTZ is found in an appendix.

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