Giovanna Guidoboni

Lucia Carichino, Giovanna Guidoboni, Marcela Szopos

The goal of this work is to develop a novel splitting approach for the numerical solution of multiscale problems involving the coupling between Stokes equations and ODE systems, as often encountered in blood flow modeling applications. The proposed algorithm is based on a semi-discretization in time based on operator splitting, whose design is guided by the rationale of ensuring that the physical energy balance is maintained at the discrete level. As a result, unconditional stability with respect to the time step choice is ensured by the implicit treatment of interface conditions within the Stokes substeps, whereas the coupling between Stokes and ODE substeps is enforced via appropriate initial conditions for each substep. Notably, unconditional stability is attained without the need of subiterating between Stokes and ODE substeps. Stability and convergence properties of the proposed algorithm are tested on three specific examples for which analytical solutions are derived.

Riccardo Sacco, Aurelio Giancarlo Mauri, Giovanna Guidoboni

This work focuses on a class of elliptic boundary value problems with diffusive, advective and reactive terms, motivated by the study of three-dimensional heterogeneous physical systems composed of two or more media separated by a selective interface. We propose a novel approach for the numerical approximation of such heterogeneous systems combining, for the first time: (1) a dual mixed hybrid (DMH) finite element method (FEM) based on the lowest order Raviart-Thomas space (RT0); (2) a Three-Field (3F) formulation; and (3) a Streamline Upwind/Petrov-Galerkin (SUPG) stabilization method. Using the abstract theory for generalized saddle-point problems and their approximation, we show that the weak formulation of the proposed method and its numerical counterpart are both uniquely solvable and that the resulting finite element scheme enjoys optimal convergence properties with respect to the discretization parameter. In addition, an efficient implementation of the proposed formulation is presented. The implementation is based on a systematic use of static condensation which reduces the method to a nonconforming finite element approach on a grid made by three-dimensional simplices. Extensive computational tests demonstrate the theoretical conclusions and indicate that the proposed DMH-RT0 FEM scheme is accurate and stable even in the presence of marked interface jump discontinuities in the solution and its associated normal flux. Results also show that in the case of strongly dominating advective terms, the proposed method with the SUPG stabilization is capable of resolving accurately steep boundary and/or interior layers without introducing spurious unphysical oscillations or excessive smearing of the solution front.

Ahmet Demirkaya, Tales Imbiriba, Kyle Lockwood et al.

Modeling biological dynamical systems is challenging due to the interdependence of different system components, some of which are not fully understood. To fill existing gaps in our ability to mechanistically model physiological systems, we propose to combine neural networks with physics-based models. Specifically, we demonstrate how we can approximate missing ordinary differential equations (ODEs) coupled with known ODEs using Bayesian filtering techniques to train the model parameters and simultaneously estimate dynamic state variables. As a study case we leverage a well-understood model for blood circulation in the human retina and replace one of its core ODEs with a neural network approximation, representing the case where we have incomplete knowledge of the physiological state dynamics. Results demonstrate that state dynamics corresponding to the missing ODEs can be approximated well using a neural network trained using a recursive Bayesian filtering approach in a fashion coupled with the known state dynamic differential equations. This demonstrates that dynamics and impact of missing state variables can be captured through joint state estimation and model parameter estimation within a recursive Bayesian state estimation (RBSE) framework. Results also indicate that this RBSE approach to training the NN parameters yields better outcomes (measurement/state estimation accuracy) than training the neural network with backpropagation through time in the same setting.