This paper presents a projection-based reduced order modelling (ROM) framework for unsteady parametrized optimal control problems (OCP$_{(\mu)}$s) arising from cardiovascular (CV) applications. In real-life scenarios, accurately defining outflow boundary conditions in patient-specific models poses significant challenges due to complex vascular morphologies, physiological conditions, and high computational demands. These challenges make it difficult to compute realistic and reliable CV hemodynamics by incorporating clinical data such as 4D magnetic resonance imaging. To address these challenges, we focus on controlling the outflow boundary conditions to optimize CV flow dynamics and minimize the discrepancy between target and computed flow velocity profiles. The fluid flow is governed by unsteady Navier--Stokes equations with physical parametric dependence, i.e. the Reynolds number. Numerical solutions of OCP$_{(\mu)}$s require substantial computational resources, highlighting the need for robust and efficient ROMs to perform real-time and many-query simulations. Here, we aim at investigating the performance of a projection-based reduction technique that relies on the offline-online paradigm, enabling significant computational cost savings. The Galerkin finite element method is used to compute the high-fidelity solutions in the offline phase. We implemented a nested-proper orthogonal decomposition (nested-POD) for fast simulation of OCP$_{(\mu)}$s that encompasses two stages: temporal compression for reducing dimensionality in time, followed by parametric-space compression on the precomputed POD modes. We tested the efficacy of the methodology on vascular models, namely an idealized bifurcation geometry and a patient-specific coronary artery bypass graft, incorporating stress control at the outflow boundary, observing consistent speed-up with respect to high-fidelity strategies.

翻译：本文提出了一种基于投影的降阶建模（ROM）框架，用于处理心血管（CV）应用中产生的非稳态参数化最优控制问题（OCP$_{(\mu)}$s）。在实际场景中，由于复杂的血管形态、生理条件和高计算需求，在患者特异性模型中精确定义流出边界条件面临重大挑战。这些挑战使得结合4D磁共振成像等临床数据计算真实可靠的心血管血流动力学变得困难。为解决这些挑战，我们专注于控制流出边界条件以优化心血管流动动力学，并最小化目标与计算流速剖面之间的差异。流体流动由具有物理参数依赖性（即雷诺数）的非稳态Navier--Stokes方程控制。OCP$_{(\mu)}$s的数值求解需要大量计算资源，这凸显了对稳健高效ROM以执行实时和多查询模拟的需求。本文旨在研究一种基于投影的降阶技术的性能，该技术依赖于离线-在线范式，能够显著节省计算成本。在离线阶段采用Galerkin有限元方法计算高保真解。我们实现了嵌套本征正交分解（nested-POD）用于OCP$_{(\mu)}$s的快速模拟，该过程包含两个阶段：首先进行时间压缩以降低时间维度，随后对预计算的POD模态进行参数空间压缩。我们在血管模型（即理想化分叉几何结构和患者特异性冠状动脉旁路移植血管）上测试了该方法的有效性，结合流出边界处的应力控制，观察到相对于高保真策略具有一致的加速效果。