Reduced order methods for parametrized non-linear and time dependent optimal flow control problems, towards applications in biomedical and environmental sciences Maria Strazzullo ♯ , Zakia Zainib ♯ , Francesco Ballarin ♯ , and Gianluigi Rozza ♯ ♯ mathlab, Mathematics Area, International School for Advanced Studies (SISSA), Via Bonomea 265, I-34136 Trieste, Italy Abstract We introduce reduced order methods as an efficient strategy to solve parametrized non-linear and time dependent optimal flow control problems governed by partial differential equations. Indeed, the optimal control problems require a huge computational effort in order to be solved, most of all in physical and/or geometrical parametrized settings. Reduced order methods are a reliable and suitable approach, increasingly gaining popularity, to achieve rapid and accurate optimal solutions in several fields, such as in biomedical and environmental sciences. In this work, we employ a POD-Galerkin reduction approach over a parametrized optimality system, derived from the Karush- Kuhn-Tucker conditions. The methodology presented is tested on two boundary control problems, governed respectively by (i) time dependent Stokes equations and (ii) steady non-linear Navier-Stokes equations. 1 Introduction Parametrized optimal flow control problems (OFCP( μ)s) constrained to parametrized partial differen- tial equations (PDE( μ)s) are a very versatile mathematical model which arises in several applications, see e.g. [7, 9, 4]. These problems are computationally expensive and challenging even in a sim- pler non-parametrized context. The computational cost becomes unfeasible when these problems involve time dependency [1, 14] or non-linearity [5, 6, 4], in addition to physical and/or geometrical parametrized settings that describe several configurations and phenomena. A suitable strategy to lower this expensive computational effort is to employ reduced order methods (ROMs) in the context of OCP( μ)s, which recast them in a cheap, yet reliable, low dimensional framework [8, 12]. We ex- ploit these techniques in order to solve boundary OCP( μ)s on a bifurcation geometry [13] which can be considered as (i) a riverbed in environmental sciences and as (ii) a bypass graft for cardiovascular applications. In the first research field, reduced parametrized optimal control framework (see e.g. [10, 11]) can be of utmost importance. It perfectly fits in forecasting and data assimilated models and it could be exploited in order to prevent possibly dangerous natural situations [15]. The presented test case is governed by time dependent Stokes equations, which are an essential tool in marine sciences in order to reliably simulate evolving natural phenomena. Furthermore, discrepancies between computational modelling in cardiovascular mechanics and reality usually ought to high computational cost and lack of optimal quantification of boundary conditions, especially the outflow boundary conditions. In this work, we present application of the aforementioned numerical framework combining OFCP( µ) and reduced order methods in the bifurcation geometry. The aim is to quantify the outflow conditions automatically while matching known physiological data for different parameter-dependent scenarios [2, 17]. In this test case, Navier-Stokes equations will model the fluid flow. The work is outlined as follows: in section 2, the problem formulation and the methodology are summarized. Section 3 shows the numerical results for the two boundary OCP( μ)s, based on [9, 13]. Conclusions follow in section 4. 1 arXiv:1912.07886v2 [math.NA] 25 May 2020