Mixed-Integer Nonlinear PDE-Constrained Optimization for Multi-Modal Chromatography

Author(s):  
Dominik H. Cebulla ◽  
Christian Kirches ◽  
Andreas Potschka
Author(s):  
Stefan Güttel ◽  
John W Pearson

Abstract We devise a method for nonlinear time-dependent partial-differential-equation-constrained optimization problems that uses a spectral-in-time representation of the residual, combined with a Newton–Krylov method to drive the residual to zero. We also propose a preconditioner to accelerate this scheme. Numerical results indicate that this method can achieve fast and accurate solution of nonlinear problems for a range of mesh sizes and problem parameters. The numbers of outer Newton and inner Krylov iterations required to reach the attainable accuracy of a spatial discretization are robust with respect to the number of collocation points in time and also do not change substantially when other problem parameters are varied.


2012 ◽  
Vol 20 (3) ◽  
pp. 293-310 ◽  
Author(s):  
Kevin Long ◽  
Paul T. Boggs ◽  
Bart G. van Bloemen Waanders

Sundance is a package in the Trilinos suite designed to provide high-level components for the development of high-performance PDE simulators with built-in capabilities for PDE-constrained optimization. We review the implications of PDE-constrained optimization on simulator design requirements, then survey the architecture of the Sundance problem specification components. These components allow immediate extension of a forward simulator for use in an optimization context. We show examples of the use of these components to develop full-space and reduced-space codes for linear and nonlinear PDE-constrained inverse problems.


Author(s):  
M. Hoffhues ◽  
W. Römisch ◽  
T. M. Surowiec

AbstractThe vast majority of stochastic optimization problems require the approximation of the underlying probability measure, e.g., by sampling or using observations. It is therefore crucial to understand the dependence of the optimal value and optimal solutions on these approximations as the sample size increases or more data becomes available. Due to the weak convergence properties of sequences of probability measures, there is no guarantee that these quantities will exhibit favorable asymptotic properties. We consider a class of infinite-dimensional stochastic optimization problems inspired by recent work on PDE-constrained optimization as well as functional data analysis. For this class of problems, we provide both qualitative and quantitative stability results on the optimal value and optimal solutions. In both cases, we make use of the method of probability metrics. The optimal values are shown to be Lipschitz continuous with respect to a minimal information metric and consequently, under further regularity assumptions, with respect to certain Fortet-Mourier and Wasserstein metrics. We prove that even in the most favorable setting, the solutions are at best Hölder continuous with respect to changes in the underlying measure. The theoretical results are tested in the context of Monte Carlo approximation for a numerical example involving PDE-constrained optimization under uncertainty.


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