Reduced order methods for inverse problem in CFD

The computational approximation of solutions to inverse problem in fluid-dynamic is often a formidable task. A general inverse problem can be seen as Partial Differential Equations (PDE) constrained optimization problem. Most of these problems for which the solution of direct problem depends on a single or multiple parameters. Historically, parametric optimal control problems are a powerful and elegant mathematical framework to fill the gap between observed data and model equations to make numerical schemes more reliable and accurate for prevision, especially in Finite Element approximations. In optimal control problems, although widely exploited by various branches of scientific research, it is necessary to minimize a functional cost dependent on the state variables of the system, through the control of one or more variables that influence the solution of the state problem. These problems are characterized by considerable computational complexity, due to the numerical discretization of the PDEs and the iterative procedures required for numerical optimization. All the more reason, in the event that various physical and/or geometric reasons were present, a method capable of reducing the dimension of the optimization problem could prove to be a useful tool to save computational costs due to simulation. In this Master thesis, we analyze the inverse problem like PDEs-constrained optimal control problems from a theoretical point of view, to derive the optimality system to be solved numerically in a optimize-than-discretize approach. In this work the reduced bases method is introduced as a Galerkin projection in a space of reduced dimensions, generated by base functions appropriately chosen through the Proper Orthogonal Decomposition (POD) algorithm. We construct the implementation for inverse problem test case in which we want to learn the viscosity constant in a lid-driven steady cavity flow from observed data. We then analyze the numerical resolution at Full-Order level through the Conjugated Gradient Method. At reduced order level we adopt a optimize-then-reduce procedure, where we first apply the iterative method to the (continuous) system of optimality conditions, then we discretize the various steps of the algorithm at reduced order level. At reduced-order level we develop an algorithm that emulates the steps of Full-order solution which we call Reduced Conjugate Gradient Method, to avoid the saddle-point structure of reduced optimal system. We can conclude that applying to our test case we verify and underline its potential. In fact, the basics method allows you to quickly and accurately solve this type of parameterized problem.