Use of Machine Learning for Automated Convergence of Numerical Iterative Schemes

Convergence of a numerical solution scheme occurs when a sequence of increasingly refined iterative solutions approaches a value consistent with the modeled phenomenon. Approximations using iterative schemes need to satisfy convergence criteria, such as reaching a specific error tolerance or number of iterations. The schemes often bypass the criteria or prematurely converge because of oscillations that may be inherent to the solution. Using a Support Vector Machines (SVM) machine learning approach, an algorithm is designed to use the source data to train a model to predict convergence in the solution process and stop unnecessary iterations. The discretization of the Navier Stokes (NS) equations for a transient local hemodynamics case requires determining a pressure correction term from a Poisson-like equation at every time-step. The pressure correction solution must fully converge to avoid introducing a mass imbalance. Considering time, frequency, and time-frequency domain features of its residual’s behavior, the algorithm trains an SVM model to predict the convergence of the Poisson equation iterative solver so that the time-marching process can move forward efficiently and effectively. The fluid flow model integrates peripheral circulation using a lumped-parameter model (LPM) to capture the field pressures and flows across various circulatory compartments. Machine learning opens the doors to an intelligent approach for iterative solutions by replacing prescribed criteria with an algorithm that uses the data set itself to predict convergence

Fast simulations of patient-specific haemodynamics of coronary artery bypass grafts based on a POD-Galerkin method and a vascular shape parametrization

In this work a reduced-order computational framework for the study of haemodynamics in three-dimensional patient-specific configurations of coronary artery bypass grafts dealing with a wide range of scenarios is proposed. We combine several efficient algorithms to face at the same time both the geometrical complexity involved in the description of the vascular network and the huge computational cost entailed by time dependent patient-specific flow simulations. Medical imaging procedures allow to reconstruct patient-specific configurations from clinical data. A centerlines-based parametrization is proposed to efficiently handle geometrical variations. POD-Galerkin reduced-order models are employed to cut down large computational costs. This computational framework allows to characterize blood flows for different physical and geometrical variations relevant in the clinical practice, such as stenosis factors and anastomosis variations, in a rapid and reliable way. Several numerical results are discussed, highlighting the computational performance of the proposed framework, as well as its capability to carry out sensitivity analysis studies, so far out of reach. In particular, a reduced-order simulation takes only a few minutes to run, resulting in computational savings of 99% of CPU time with respect to the full-order discretization. Moreover, the error between full-order and reduced-order solutions is also studied, and it is numerically found to be less than 1% for reduced-order solutions obtained with just O(100) online degrees of freedom. (C) 2016 Elsevier Inc. All rights reserved

Novel Algorithms for Merging Computational Fluid Dynamics and 4D Flow MRI

Time-resolved three-dimensional spatial encoding combined with three-directional velocity-encoded phase contrast magnetic resonance imaging (termed as 4D flow MRI), can provide valuable information for diagnosis, treatment, and monitoring of vascular diseases. The accuracy of this technique, however, is limited by errors in flow estimation due to acquisition noise as well as systematic errors. Furthermore, available spatial resolution is limited to 1.5mm - 3mm and temporal resolution is limited to 30-40ms. This is often grossly inadequate to resolve flow details in small arteries, such as those in cerebral circulation. Recently, there have been efforts to address the limitations of the spatial and temporal resolution of MR flow imaging through the use of computational fluid dynamics (CFD). While CFD is capable of providing essentially unlimited spatial and temporal resolution, numerical results are very sensitive to errors in estimation of the flow boundary conditions. In this work, we present three novel techniques that combine CFD with 4D flow MRI measurements in order to address the resolution and noise issues. The first technique is a variant of the Kalman Filter state estimator called the Ensemble Kalman Filter (EnKF). In this technique, an ensemble of patient-specific CFD solutions are used to compute filter gains. These gains are then used in a predictor-corrector scheme to not only denoise the data but also increase its temporal and spatial resolution. The second technique is based on proper orthogonal decomposition and ridge regression (POD-rr). The POD method is typically used to generate reduced order models (ROMs) in closed control applications of large degree of freedom systems that result from discretization of governing partial differential equations (PDE). The POD-rr process results in a set of basis functions (vectors), that capture the local space of solutions of the PDE in question. In our application, the basis functions are generated from an ensemble of patient-specific CFD solutions whose boundary conditions are estimated from 4D flow MRI data. The CFD solution that should be most closely representing the actual flow is generated by projecting 4D flow MRI data onto the basis vectors followed by reconstruction in both MRI and CFD resolution. The rr algorithm was used for between resolution mapping. Despite the accuracy of using rr as the mapping step, due to manual adjustment of a coefficient in the algorithm we developed the third algorithm. In this step, the rr algorithm was substituded with a dynamic mode decomposition algorithm to preserve the robustness. These algorithms have been implemented and tested using a numerical model of the flow in a cerebral aneurysm. Solutions at time intervals corresponding to the 4D flow MRI temporal resolution were collected and downsampled to the spatial resolution of the imaging data. A simulated acquisition noise was then added in k-space. Finally, the simulated data affected by noise were used as an input to the merging algorithms. Rigorous comparison to state-of-the-art techniques were conducted to assess the accuracy and performance of the proposed method. The results provided denoised flow fields with less than 1\% overall error for different signal-to-noise ratios. At the end, a small cohort of three patients were corrected and the data were reconstructed using different methods, the wall shear stress (WSS) was calculated using different reconstructed data and the results were compared. As it has been shown in chapter 5, the calculated WSS using different methods results in mutual high and low shear stress regions, however, the exact value and patterns are significantly different