Iterative procedures for space shuttle main engine performance models

Performance models of the Space Shuttle Main Engine (SSME) contain iterative strategies for determining approximate solutions to nonlinear equations reflecting fundamental mass, energy, and pressure balances within engine flow systems. Both univariate and multivariate Newton-Raphson algorithms are employed in the current version of the engine Test Information Program (TIP). Computational efficiency and reliability of these procedures is examined. A modified trust region form of the multivariate Newton-Raphson method is implemented and shown to be superior for off nominal engine performance predictions. A heuristic form of Broyden's Rank One method is also tested and favorable results based on this algorithm are presented

On solving trust-region and other regularised subproblems in optimization

The solution of trust-region and regularisation subproblems which arise in unconstrained optimization is considered. Building on the pioneering work of Gay, Mor´e and Sorensen, methods which obtain the solution of a sequence of parametrized linear systems by factorization are used. Enhancements using high-order polynomial approximation and inverse iteration ensure that the resulting method is both globally and asymptotically at least superlinearly convergent in all cases, including in the notorious hard case. Numerical experiments validate the effectiveness of our approach. The resulting software is available as packages TRS and RQS as part of the GALAHAD optimization library, and is especially designed for large-scale problems

Randomized Riemannian Preconditioning for Orthogonality Constrained Problems

Optimization problems with (generalized) orthogonality constraints are prevalent across science and engineering. For example, in computational science they arise in the symmetric (generalized) eigenvalue problem, in nonlinear eigenvalue problems, and in electronic structures computations, to name a few problems. In statistics and machine learning, they arise, for example, in canonical correlation analysis and in linear discriminant analysis. In this article, we consider using randomized preconditioning in the context of optimization problems with generalized orthogonality constraints. Our proposed algorithms are based on Riemannian optimization on the generalized Stiefel manifold equipped with a non-standard preconditioned geometry, which necessitates development of the geometric components necessary for developing algorithms based on this approach. Furthermore, we perform asymptotic convergence analysis of the preconditioned algorithms which help to characterize the quality of a given preconditioner using second-order information. Finally, for the problems of canonical correlation analysis and linear discriminant analysis, we develop randomized preconditioners along with corresponding bounds on the relevant condition number