Diego Renner

Jacques Y. Xing, Boyang Xia, Diego Renner et al.

We extend earlier international efforts to optimise hexahedral-based spectral element methods on GPUs and vectorised CPUs to mixed element meshes additionally involving prismatic, pyramidic, and tetrahedral shapes using tensorial expansions. We demonstrate that common finite element operators (such as the mass and Helmholtz matrices) benefit from alternative implementation strategies depending on the element shape, choice of polynomial order, and system architecture in order to achieve optimal performance. In addition, we introduce a new approach/interpretation to efficiently evaluate more complex operations involving inner products with the derivative of the expansions as part of the integrand such as the stiffness matrix. This approach seeks to maximise operations using the collocation properties of the nodal tensorial expansion associated with classical quadrature rules. Our GPU performance tests demonstrate that the throughput of the Helmholtz operator on tetrahedral elements is at most 2.5 times slower than on hexahedral elements, despite tetrahedra having a factor of six greater floating-point operations.

Diego Renner, Georgios Kissas

One of the goals of personalized medicine is to tailor diagnostics to individual patients. Diagnostics are performed in practice by measuring quantities, called biomarkers, that indicate the existence and progress of a disease. In common cardiovascular diseases, such as hypertension, biomarkers that are closely related to the clinical representation of a patient can be predicted using computational models. Personalizing computational models translates to considering patient-specific flow conditions, for example, the compliance of blood vessels that cannot be a priori known and quantities such as the patient geometry that can be measured using imaging. Therefore, a patient is identified by a set of measurable and nonmeasurable parameters needed to well-define a computational model; else, the computational model is not personalized, meaning it is prone to large prediction errors. Therefore, to personalize a computational model, sufficient information needs to be extracted from the data. The current methods by which this is done are either inefficient, due to relying on slow-converging optimization methods, or hard to interpret, due to using `black box` deep-learning algorithms. We propose a personalized diagnostic procedure based on a differentiable 0D-1D Navier-Stokes reduced order model solver and fast parameter inference methods that take advantage of gradients through the solver. By providing a faster method for performing parameter inference and sensitivity analysis through differentiability while maintaining the interpretability of well-understood mathematical models and numerical methods, the best of both worlds is combined. The performance of the proposed solver is validated against a well-established process on different geometries, and different parameter inference processes are successfully performed.