Abstract. Issues of indefinite preconditioning of reduced Newton systems arising in optimization with interior point methods are addressed in this paper. Constraint preconditioners have shown much promise in this context. However, there are situations in which an unfavorable sparsity pattern of Jacobian matrix may adversely affect the preconditioner and make its inverse representation unacceptably dense hence too expensive to be used in practice. A remedy to such situations is proposed in this paper. An approximate constraint preconditioner is considered in which sparse approximation of the Jacobian is used instead of the complete matrix. Spectral analysis of the preconditioned matrix is performed and bounds on its non-unit eigenvalues are provided. Preliminary computational results are encouraging.

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SUMMARYIn this paper, the characteristic-based split (CBS) algorithm is presented to solve water motion problem due to tides in the Venice lagoon. The basic equations are the well-known vertically integrated shallowwater equations expressed in terms of total water depth and horizontal velocities. The approximation of any additional scalar variables, such as temperature, concentration or pollutant dispersion from the appropriate advection-diffusion governing equations is also outlined. Numerical results illustrate the performance of the proposed algorithm on both advection-diffusion and shallow-water problems.

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