MUSCOP
swMATH ID:  6143 
Software Authors:  Potschka, Andreas 
Description:  A direct method for the numerical solution of optimization problems with timeperiodic PDE constraints.
In this thesis we develop a numerical method based on direct multiple shooting for optimal control problems (OCPs) constrained by timeperiodic partial differential equations (PDEs). The proposed method features asymptotically optimal scaleup of the numerical effort with the number of spatial discretization points. It consists of a linear iterative splitting approach (LISA) within a Newtontype iteration with globalization on the basis of natural level functions. We investigate the LISANewton method in the framework of Bock’s kappatheory and develop reliable aposteriori kappaestimators. Moreover we extend the inexact Newton method to an inexact sequential quadratic programming (SQP) method for inequality constrained problems and provide local convergence theory. In addition we develop a classical and a twogrid NewtonPicard preconditioner for LISA and prove grid independent convergence of the classical variant for a model problem. Based on numerical results we can claim that the twogrid version is even more efficient than the classical version for typical application problems. Moreover we develop a twogrid approximation for the Lagrangian Hessian which fits well in the twogrid NewtonPicard framework and yields a reduction of 68 For the solution of the occurring largescale quadratic programming problems (QPs) we develop a structure exploiting twostage approach. In the first stage we exploit the multiple shooting and NewtonPicard structure to reduce the largescale QP to an equivalent QP whose size is independent of the number of spatial discretization points. For the second stage we develop extensions for a parametric active set method (PASM) to achieve a reliable and efficient solver for the resulting, possibly nonconvex QP. Furthermore we construct three illustrative, counterintuitive toy examples which show that convergence of a oneshot onestep optimization method is neither necessary nor sufficient for the convergence of the forward problem method. For three regularization approaches to recover convergence our analysis shows that defacto loss of convergence cannot be avoided with these approaches. We have further implemented the proposed methods within a code called MUSCOP which features automatic derivative generation for the model functions and dynamic system solutions of first and second order, parallelization on the multiple shooting structure, and a hybrid language programming paradigm to minimize setup and solution time for new application problems. We demonstrate the applicability, reliability, and efficiency of MUSCOP and thus the proposed numerical methods and techniques on a sequence of PDE OCPs of growing difficulty ranging from linear academic problems, over highly nonlinear academic problems of mathematical biology to a highly nonlinear realworld chemical engineering problem in preparative chromatography: The simulated moving bed process. 
Homepage:  http://www.iwr.uniheidelberg.de/~Andreas.Potschka/software.html 
Keywords:  direct multiple shooting; optimal control; linear iterative splitting approach; Newtontype iteration; sequential quadratic programming; convergence; NewtonPicard preconditioner; numerical results; grid refinement; parametric active set method; automatic derivative generation; parallelization 
Related Software:  NewtonLib; ADOLC; LBFGSB; LBFGSB; LBFGS; UFL; FEniCS; DOLFIN; MUMPS; PETSc; Chebfun; deal.ii; DAESOLII; MINRES; CUTEst; MinRes; Trilinos; LiftOpt; SolvIND; NETLIB LP Test Set 
Cited in:  8 Documents 
Standard Articles
1 Publication describing the Software, including 1 Publication in zbMATH  Year 

A direct method for the numerical solution of optimization problems with timeperiodic PDE constraints. Zbl 1237.65062 Potschka, Andreas 
2001

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Cited by 7 Authors
7  Potschka, Andreas 
1  Bock, Hans Georg 
1  Fang, Liang 
1  Hante, Falk M. 
1  Meyers, Johan 
1  Mommer, Mario S. 
1  Vandewalle, Stefan G. 
Cited in 4 Serials
2  SIAM Journal on Numerical Analysis 
1  Journal of Computational Physics 
1  Numerical Algorithms 
1  Mathematical Programming. Series A. Series B 
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