Direct Multiple Shooting for Parabolic PDE Constrained Optimization

2015 ◽  
pp. 159-181 ◽  
Author(s):  
Andreas Potschka
Keyword(s):  
Constrained Optimization ◽  
Multiple Shooting ◽  
Pde Constrained Optimization ◽  
Parabolic Pde

Diagonalization-based parallel-in-time algorithms for parabolic PDE-constrained optimization problems

2020 ◽  
Vol 26 ◽  
pp. 88
Author(s):  
Shu-Lin Wu ◽  
Tao Zhou
Keyword(s):  
Constrained Optimization ◽  
Optimization Problems ◽  
Main Idea ◽  
Euler Method ◽  
Computational Time ◽  
Constrained Optimization Problems ◽  
Pde Constrained Optimization ◽  
Kkt System ◽  
Backward Euler ◽  
Parabolic Pde

Solving parabolic PDE-constrained optimization problems requires to take into account the discrete time points all-at-once, which means that the computation procedure is often time-consuming. It is thus desirable to design robust and analyzable parallel-in-time (PinT) algorithms to handle this kind of coupled PDE systems with opposite evolution directions. To this end, for two representative model problems which are, respectively, the time-periodic PDEs and the initial-value PDEs, we propose a diagonalization-based approach that can reduce dramatically the computational time. The main idea lies in carefully handling the associated time discretization matrices that are denoted by Bper and Bini for the two problems. For the first problem, we diagonalize Bper directly and this results in a direct PinT algorithm (i.e., non-iterative). For the second problem, the main idea is to design a suitable approximation B̂per of Bini, which naturally results in a preconditioner of the discrete KKT system. This preconditioner can be used in a PinT pattern, and for both the Backward-Euler method and the trapezoidal rule the clustering of the eigenvalues and singular values of the preconditioned matrix is justified. Compared to existing preconditioners that are designed by approximating the Schur complement of the discrete KKT system, we show that the new preconditioner leads to much faster convergence for certain Krylov subspace solvers, e.g., the GMRES and BiCGStab methods. Numerical results are presented to illustrate the advantages of the proposed PinT algorithm.

scholarly journals On quantitative stability in infinite-dimensional optimization under uncertainty

2021 ◽  
Author(s):  
M. Hoffhues ◽  
W. Römisch ◽  
T. M. Surowiec
Keyword(s):  
Stochastic Optimization ◽  
Constrained Optimization ◽  
Optimization Problems ◽  
Optimization Under Uncertainty ◽  
Optimal Solutions ◽  
Probability Metrics ◽  
Pde Constrained Optimization ◽  
Quantitative Stability ◽  
Infinite Dimensional ◽  
Optimal Value

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.

scholarly journals A Low-Rank in Time Approach to PDE-Constrained Optimization

2015 ◽  
Vol 37 (1) ◽  
pp. B1-B29 ◽  
Author(s):  
Martin Stoll ◽  
Tobias Breiten
Keyword(s):  
Constrained Optimization ◽  
Low Rank ◽  
Pde Constrained Optimization

scholarly journals Large Scale Non-Linear Programming for PDE Constrained Optimization

2002 ◽  
Author(s):  
BART G VAN BLOEMEN WAANDERS ◽  
ROSCOE A BARTLETT ◽  
KEVIN R LONG ◽  
PAUL T BOGGS ◽  
ANDREW G SALINGER
Keyword(s):  
Linear Programming ◽  
Constrained Optimization ◽  
Large Scale ◽  
Pde Constrained Optimization ◽  
Non Linear Programming ◽  
Non Linear
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