A new approach for solving the multi-dimensional control optimization problems with first-order PDEs constraints
Anurag Jayswal, Tadeusz Antczak, Preeti Kardam
This paper aims to solve the multi-dimensional control optimization problem involving first-order PDEs constraints (MCOP). Firstly, we apply the modified objective function approach to (MCOP) and show that the solution sets of the original problem and its associated modified problem are equivalent. Further, we apply the penalty function method to transform the modified problem into an equivalent penalized problem. We also establish the relationship between a saddle-point of the modified problem and a minimizer of its penalized problem. We also give examples to verify the establish results.
Multimaterial topology optimization of a heated channel
The presentation will deal with a topology optimization problem in a heated channel. The problem, modelled using the incompressible Navier-Stokes equations coupled with the convection-diffusion equation, consists in minimizing pressure drop and maximizing heat transfer for some application in reduced mechanical air conditioning. The analysis focuses on the existence of a solution, the convergence of the numerical approximation using finite elements (including the convergence of the approximated optimum, using controls in BV), and the effect of using multiple materials to design the topology.
On Preconditioners for PDE-Constrained Optimization Problems with Higher-Order Discretization in the Time Variable
Santolo Leveque, John Pearson
Optimization problems with PDE constraints arise in numerous scientific applications. In this talk, we consider preconditioners for time-dependent PDE optimization, suitably discretized in time. The state-of-the-art is to apply a backward Euler method: this leads to desirable structures within the resulting matrix system, and effective preconditioners, but only a first-order accurate method. Here we present a second-order method, using Crank-Nicolson in time, and a new preconditioner for the more complex matrix. We show that this approach can obtain a more accurate solution in less CPU time.
Convergence analysis for approximations of optimal control problems subject to higher index differential-algebraic equations and mixed control-state constraints
Björn Martens, Matthias Gerdts
This paper establishes a convergence result for implicit Euler discretizations of optimal control problems with DAEs of higher index and mixed control-state constraints. The main difficulty of the analysis is caused by a structural discrepancy between the necessary conditions of the continuous problem and the necessary conditions of the discretized problems. This discrepancy does not allow one to compare the respective necessary conditions directly. We use an equivalent reformulation of the discretized problems to overcome this discrepancy and to prove first order convergence.