Summary Hierarchical model reduction is intended to solve efficiently partial differential equations in domains with a geometrically dominant direction. In many engineering applications, these problems are often reduced to 1‐dimensional differential systems. This guarantees computational efficiency yet dumps local accuracy as nonaxial dynamics are dropped. Hierarchical model reduction recovers the secondary components of the dynamics of interest with a combination of different discretization techniques, following up a natural separation of variables. The dominant direction is generally solved by the finite element method or isogeometric analysis to guarantee flexibility, while the transverse components are solved by spectral methods, to guarantee a small number of degrees of freedom. By judiciously selecting the number of transverse modes, the method has been proven to improve significantly the accuracy of purely 1‐dimensional solvers, with great computational efficiency. A Cartesian framework has been used so far both in slab domains and cylindrical pipes (including arteries) mapped to Cartesian reference domains. In this paper, we investigate the alternative use of a polar coordinates system for the transverse dynamics in circular or elliptical pipes. This seems a natural choice for applications like computational hemodynamics. In spite of this, the selection of a basis function set for the transverse dynamics is troublesome. As pointed out in the literature—even for simple elliptical problems—there is no “best” basis available. In this paper, we perform an extensive investigation of hierarchical model reduction in polar coordinates to discuss different possible choices for the transverse basis, pointing out pros and cons of the polar coordinate system.

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Abstract. Overlapping domain decomposition is a technique for solving complex problems described by Partial Differential Equations in a parallel framework. The performance of this approach strongly depends on the size and the position of the overlap since on the one hand more overlap increases the computational costs for the single subdomain but on the other hand it accelerates the iterative procedure of the global decomposition solving. In this paper we test the overlapping domain decomposition method on the finite element discretization of a diffusion reaction problem in both idealized and real 3D geometries. Results confirm that the detection of the optimal overlapping in real cases is not trivial but has the potential to significantly reduce the computational costs of the entire solution process.

High Performance Computing (HPC) is a mainstream mode of exploration and analysis in different fields, not only technical but also social and life sciences. A well established HPC domain is medicine, and cardiovascular sciences in particular. The adoption of CFD as a tool for diagnosis, prognosis, and treatment planning in the clinical routine is however still an open challenge. This computational tool, required by Computer Aided Clinical Trials and Surgical Planning, calls for significant computational resources to face both large volume of patients and diverse timelines ranging from election to emergency scenarios. Traditional local clusters may be not adequate to deliver the computational needs. Alternative solutions like grids and on-demand cloud resources need to be seriously considered. This paper proposes methodologies and protocols to identify the optimal choice of computing platforms for hemodynamics computations that will be increasingly needed in the future and the optimal scheduling of the tasks across the selected resources. We focus on hemodynamics in patient-specific settings and present extensive results on different platforms. We propose a way to measure and estimate performance and running time under realis

This work introduces the network uncertainty quantification (NetUQ) method for performing uncertainty propagation in systems composed of interconnected components. The method assumes the existence of a collection of components, each of which is characterized by exogenous-input random variables (e.g., material properties), endogenous-input random variables (e.g., boundary conditions defined by another component), output random variables (e.g., quantities of interest), and a local uncertainty-propagation operator (e.g., provided by stochastic collocation) that computes output random variables from input random variables. The method assembles the full-system network by connecting components, which is achieved simply by associating endogenous-input random variables for each component with output random variables from other components; no other inter-component compatibility conditions are required. The network uncertainty-propagation problem is: Compute output random variables for all components given all exogenous-input random variables. To solve this problem, the method applies classical relaxation methods (i.e., Jacobi and Gauss-Seidel iteration with Anderson acceleration), which require only blackbox evaluations of component uncertainty-propagation operators. Compared with other available methods, this approach is applicable to any network topology (e.g., no restriction to feed-forward or two-component networks), promotes component independence by enabling components to employ tailored uncertaintypropagation operators, supports general functional representations of random variables, and requires no offline preprocessing stage. Also, because the method propagates functional representations of random variables throughout the network (and not, e.g., probability density functions), the joint distribution of any set of random variables throughout the network can be estimated a posteriori in a straightforward manner. We perform supporting convergence and error analysis and execute numerical experiments that demonstrate the weak-and strong-scaling performance of the method.