TY - JOUR
T1 - A weighted POD-reduction approach for parametrized PDE-constrained optimal control problems with random inputs and applications to environmental sciences
AU - Carere, Giuseppe
AU - Strazzullo, Maria
AU - Ballarin, Francesco
AU - Rozza, Gianluigi
AU - Stevenson, Rob
PY - 2021
Y1 - 2021
N2 - Reduced basis approximations of Optimal Control Problems (OCPs) governed by steady partial differential equations (PDEs) with random parametric inputs are analyzed and constructed. Such approximations are based on a Reduced Order Model, which in this work is constructed using the method of weighted Proper Orthogonal Decomposition. This Reduced Order Model then is used to efficiently compute the reduced basis approximation for any outcome of the random parameter. We demonstrate that such OCPs are well-posed by applying the adjoint approach, which also works in the presence of admissibility constraints and in the case of non linear-quadratic OCPs, and thus is more general than the conventional Lagrangian approach. We also show that a step in the construction of these Reduced Order Models, known as the aggregation step, is not fundamental and can in principle be skipped for noncoercive problems, leading to a cheaper online phase. Numerical applications in three scenarios from environmental science are considered, in which the governing PDE is steady and the control is distributed. Various parameter distributions are taken, and several implementations of the weighted Proper Orthogonal Decomposition are compared by choosing different quadrature rules.
AB - Reduced basis approximations of Optimal Control Problems (OCPs) governed by steady partial differential equations (PDEs) with random parametric inputs are analyzed and constructed. Such approximations are based on a Reduced Order Model, which in this work is constructed using the method of weighted Proper Orthogonal Decomposition. This Reduced Order Model then is used to efficiently compute the reduced basis approximation for any outcome of the random parameter. We demonstrate that such OCPs are well-posed by applying the adjoint approach, which also works in the presence of admissibility constraints and in the case of non linear-quadratic OCPs, and thus is more general than the conventional Lagrangian approach. We also show that a step in the construction of these Reduced Order Models, known as the aggregation step, is not fundamental and can in principle be skipped for noncoercive problems, leading to a cheaper online phase. Numerical applications in three scenarios from environmental science are considered, in which the governing PDE is steady and the control is distributed. Various parameter distributions are taken, and several implementations of the weighted Proper Orthogonal Decomposition are compared by choosing different quadrature rules.
KW - Environmental applications
KW - Optimal control problems
KW - Partial differential equations
KW - Reduced order models
KW - Uncertainty quantification
KW - Environmental applications
KW - Optimal control problems
KW - Partial differential equations
KW - Reduced order models
KW - Uncertainty quantification
UR - http://hdl.handle.net/10807/193348
U2 - 10.1016/j.camwa.2021.10.020
DO - 10.1016/j.camwa.2021.10.020
M3 - Article
SN - 0898-1221
VL - 102
SP - 261
EP - 276
JO - COMPUTERS & MATHEMATICS WITH APPLICATIONS
JF - COMPUTERS & MATHEMATICS WITH APPLICATIONS
ER -