Adaptive machine learning-based surrogate modeling to accelerate PDE-constrained optimization in enhanced oil recovery

In this contribution, we develop an efficient surrogate modeling framework for simulation-based optimization of enhanced oil recovery, where we particularly focus on polymer flooding. The computational approach is based on an adaptive training procedure of a neural network that directly approximates an input-output map of the underlying PDE-constrained optimization problem. The training process thereby focuses on the construction of an accurate surrogate model solely related to the optimization path of an outer iterative optimization loop. True evaluations of the objective function are used to finally obtain certified results. Numerical experiments are given to evaluate the accuracy and efficiency of the approach for a heterogeneous five-spot benchmark problem.publishedVersio

Reduced basis method and error estimation for parametrized optimal control problems with control constraints

We propose a Reduced Basis method for the solution of parametrized optimal control problems with control constraints for which we extend the method proposed in Dedè, L. (SIAM J. Sci. Comput. 32:997, 2010) for the unconstrained problem. The case of a linear-quadratic optimal control problem is considered with the primal equation represented by a linear parabolic partial differential equation. The standard offline-online decomposition of the Reduced Basis method is employed with the Finite Element approximation as the "truth" one for the offline step. An error estimate is derived and an heuristic indicator is proposed to evaluate the Reduced Basis error on the optimal control problem at the online step; also, the indicator is used at the offline step in a Greedy algorithm to build the Reduced Basis space. We solve numerical tests in the two-dimensional case with applications to heat conduction and environmental optimal control problems. © 2011 Springer Science+Business Media, LLC

Reduced basis method for parametrized optimal control problems governed by PDEs

This master thesis aims at the development, analysis and computer implementation of effcient numerical methods for the solution of optimal control problems based on parametrized partial differential equations. Our goal isfto develop a new approach based on suitable model reduction paradigm --the reduced basis method (RB)-- for the rapid and reliable solution of control problems which may occur in several engineering contexts. In particular, we develop the methodology for parametrized quadratic optimization problem with either coercive elliptic equations or Stokes equations as constraints. Firstly, we recast the optimal control problem in the framework of mixed variational problems in order to take advantage of the already developed RB theory for Stokes-type problems. Then the usual ingredients of the RB methodology are provided: a Galerkin projection onto a low-dimensional space of basis functions properly selected by an adaptive procedure; an affine parametric dependence enabling to perform competitive Offine-Online splitting in the computational procedure; an efficient and rigorous a posteriori error estimation on the state, control and adjoint variables as well as on the cost functional. The reduction scheme is applied to several numerical tests conrming the theoretical results and demonstrating the efficiency of the proposed technique. Moreover an application to an (idealized) inverse problem in haemodynamics is discussed, showing the versatility and potentiality of the method in tackling parametrized optimal control problems that could arise in a a broad variety of application contexts

Reduced Models for Optimal Control, Shape Optimization and Inverse Problems in Haemodynamics

The objective of this thesis is to develop reduced models for the numerical solution of optimal control, shape optimization and inverse problems. In all these cases suitable functionals of state variables have to be minimized. State variables are solutions of a partial differential equation (PDE), representing a constraint for the minimization problem. The solution of these problems induce large computational costs due to the numerical discretization of PDEs and to iterative procedures usually required by numerical optimization (many-query context). In order to reduce the computational complexity, we take advantage of the reduced basis (RB) approximation for parametrized PDEs, once the state problem has been reformulated in parametrized form. This method enables a rapid and reliable approximation of parametrized PDEs by constructing low-dimensional, problem-specific approximation spaces. In case of PDEs defined over domains of variable shapes (e.g. in shape optimization problems) we need to introduce suitable, low-dimensional shape parametrization techniques in order to tackle the geometrical complexity. Free-Form Deformations and Radial-Basis Functions techniques have been analyzed and successfully applied with this aim. We analyze the reduced framework built by coupling these tools and apply it to the solution of optimal control and shape optimization problems. Robust optimization problems under uncertain conditions are also taken into consideration. Moreover, both deterministic and Bayesian frameworks are set in order to tackle inverse identification problems. As state equations, we consider steady viscous flow problems described by Stokes or Navier-Stokes equations, for which we provide a detailed analysis and construction of RB approximation and a posteriori error estimation. Several numerical test cases are also illustrated to show efficacy and reliability of RB approximations. We exploit this general reduced framework to solve some optimization and inverse problems arising in haemodynamics. More specifically, we focus on the optimal design of cardiovascular prostheses, such as bypass grafts, and on inverse identification of pathological conditions or flow/shape features in realistic parametrized geometries, such as carotid artery bifurcations