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Inexact Newton with Iterated Sensitivities for Optimization of Partial Differential Equation
Abstract
Solving optimal control problems subject to partial differential equations (PDEs) is a challenging but important task, for instance, optimal heating or cooling. The problem arising from the discretization can easily have 106 decision variables and therefore it is important to provide methods that can solve such problems efficiently. Thus, the interaction between the optimization method and the numerical simulation is crucial for such problems. The Inexact Newton with Iterated Sensitivities (INIS) method solves a particular class of nonlinear programming (NLP) problems, which arise from optimal control formulations where a subset of the variables are implicitly defined by nonlinear equality constraints. The system of nonlinear equality constraints is called the forward problem. In contrast to other inexact Newton-type optimization methods, the INIS method preserves the local convergence properties and the asymptotic contraction rate of the inexact Newton-type method with the same Jacobian approximation applied to the forward problem. The INIS method is especially suited for problems where the number of states is significantly larger than the number of controls. This is the case for PDE constrained optimal control problems with boundary controls, which we regard in this thesis. Moreover, we consider a forward problem that arises from the discretization of a PDE defined on a 2-dimensional domain. An efficient method for solving linear systems that result from the discretization of PDEs is the multi-grid (MG) method. This thesis shows that the MG method can be combined with the INIS method resulting in the INIS-MG algorithm. The algorithm is applied to a test problem and compared to ipopt, a software package for large-scale nonlinear optimization. Furthermore, the theoretical properties of the INIS method are verified for the PDE constrained case.
The numerical experiments show that even a MATLAB implementation of the INIS- MG algorithm can outperform state of the art NLP solvers like ipopt.
feed/2024/04/26/inexact_newton_with_iterated_sensitivities_for_optimization_of_partial_differential_equation.txt ยท Last modified: 2024/04/26 13:28 by Horea Caramizaru
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