arXiv subjects

Luca Franco Pavarino

Publications and source records attributed to Luca Franco Pavarino.

10 recordsLinked to original sources

arXiv2026

Second-Order IMEX Time-Stepping Methods for Efficient Bidomain Multi-Electrode Array Simulations of Cardiac Stem Cell Monolayers

Multi-electrode Arrays (MEAs) enable tissue-level electrophysiological studies of human induced pluripotent stem cell-derived cardiomyocytes (hiPSC-CMs) by recording extracellular field potentials. Recent computational models have improved MEA simulations by coupling detailed electrode descriptions with the Bidomain framework, a parabolic-elliptic system of nonlinear PDEs coupled with a stiff ionic model. The standard numerical strategy relies on operator splitting techniques that decouple the PDE and ODE components. In most implementations, the ionic subsystem is treated explicitly while the diffusive operator is handled implicitly, resulting in a first-order implicit-explicit (IMEX) time discretization. Although computationally convenient, this approach limits temporal accuracy and may reduce efficiency in large-scale simulations. In this work, we investigate higher-order IMEX Runge-Kutta schemes within a Strang-based operator splitting, specifically tailored for the MEA model of hiPSC-CMs monolayers. First- and second-order schemes are compared in terms of computational cost and global error against a high-fidelity reference solution obtained with a very small time step. For a fixed computational cost, the second-order schemes achieve errors that are 2-3 orders of magnitude smaller than those of the first-order method. These results demonstrate that higher-order IMEX integration significantly improves the accuracy-to-cost ratio of MEA simulations, providing a practical and reliable approach for large-scale electrophysiological studies.

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arXiv2026

A Neural Latent Dynamics Approach for Solving Inverse Problems in Cardiac Electrophysiology

Solving inverse problems in cardiac electrophysiology consists in the recovery of physiological parameters from surface electrocardiogram (ECG) measurements, a task which is often computationally unfeasible due to the severe ill-posedness and the prohibitive computational complexity of PDE-constrained optimization. In this work, we introduce a data-driven framework leveraging Latent Dynamics Networks (LDNets) to construct efficient surrogate models of the forward operator. By mapping low-dimensional parameters, representing ectopic activation sites or ischemic region descriptors, to the ECG signals via latent dynamics governed by neural ordinary differential equations, our approach circumvents the computational burden of evaluating high-fidelity cardiac models during iterative parameter estimation. The surrogate is trained offline on high-fidelity data, enabling rapid and robust inversion. We validate the proposed framework through rigorous numerical experiments with synthetic data across both 2d and 3d geometries. Results show that the LDNet-based surrogate achieves precise reconstruction of cardiac parameters while drastically reducing computational overhead, thereby enabling near real-time clinical applications.

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arXiv2025

Learning High-dimensional Ionic Model Dynamics Using Fourier Neural Operators

Ionic models, described by systems of stiff ordinary differential equations, are fundamental tools for simulating the complex dynamics of excitable cells in both Computational Neuroscience and Cardiology. Approximating these models using Artificial Neural Networks poses significant challenges due to their inherent stiffness, multiscale nonlinearities, and the wide range of dynamical behaviors they exhibit, including multiple equilibrium points, limit cycles, and intricate interactions. While in previous studies the dynamics of the transmembrane potential has been predicted in low dimensionality settings, in the present study we extend these results by investigating whether Fourier Neural Operators can effectively learn the evolution of all the state variables within these dynamical systems in higher dimensions. We demonstrate the effectiveness of this approach by accurately learning the dynamics of three well-established ionic models with increasing dimensionality: the two-variable FitzHugh-Nagumo model, the four-variable Hodgkin-Huxley model, and the forty-one-variable O'Hara-Rudy model. To ensure the selection of near-optimal configurations for the Fourier Neural Operator, we conducted automatic hyperparameter tuning under two scenarios: an unconstrained setting, where the number of trainable parameters is not limited, and a constrained case with a fixed number of trainable parameters. Both constrained and unconstrained architectures achieve comparable results in terms of accuracy across all the models considered. However, the unconstrained architecture required approximately half the number of training epochs to achieve similar error levels, as evidenced by the loss function values recorded during training. These results underline the capabilities of Fourier Neural Operators to accurately capture complex multiscale dynamics, even in high-dimensional dynamical systems.

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arXiv2025

Learning cardiac activation and repolarization times with operator learning

Solving partial or ordinary differential equation models in cardiac electrophysiology is a computationally demanding task, particularly when high-resolution meshes are required to capture the complex dynamics of the heart. Moreover, in clinical applications, it is essential to employ computational tools that provide only relevant information, ensuring clarity and ease of interpretation. In this work, we exploit two recently proposed operator learning approaches, namely Fourier Neural Operators (FNO) and Kernel Operator Learning (KOL), to learn the operator mapping the applied stimulus in the physical domain into the activation and repolarization time distributions. These data-driven methods are evaluated on synthetic 2D and 3D domains, as well as on a physiologically realistic left ventricle geometry. Notably, while the learned map between the applied current and activation time has its modelling counterpart in the Eikonal model, no equivalent partial differential equation (PDE) model is known for the map between the applied current and repolarization time. Our results demonstrate that both FNO and KOL approaches are robust to hyperparameter choices and computationally efficient compared to traditional PDE-based Monodomain models. These findings highlight the potential use of these surrogate operators to accelerate cardiac simulations and facilitate their clinical integration.

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arXiv2025

Block BDDC/FETI-DP Preconditioners for Three-Field mixed finite element Discretizations of Biot's consolidation model

In this paper, we construct and analyze a block dual-primal preconditioner for Biot's consolidation model approximated by three-field mixed finite elements based on a displacement, pressure, and total pressure formulation. The domain is decomposed into nonoverlapping subdomains, and the continuity of the displacement component across the subdomain interface is enforced by introducing a Lagrange multiplier. After eliminating all displacement variables and the independent subdomain interior components of pressure and total pressure, the problem is reduced to a symmetric positive definite linear system for the subdomain interface pressure, total pressure, and the Lagrange multiplier. This reduced system is solved by a preconditioned conjugate gradient method, with a block dual-primal preconditioner using a Balancing Domain Decomposition by Constraints (BDDC) preconditioner for both the interface total pressure block and the interface pressure blocks, as well as a Finite Element Tearing and Interconnecting-Dual Primal (FETI-DP) preconditioner for the Lagrange multiplier block. By analyzing the conditioning of the preconditioned subsystem associated with the interface pressure and total pressure components, we obtain a condition number bound of the preconditioned system, which is scalable in the number of subdomains, poly-logarithmic in the ratio of subdomain and mesh sizes, and robust with respect to the parameters of the model. Extensive numerical experiments confirm the theoretical result of the proposed algorithm.

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arXiv2025

Parameter Robust Isogeometric Methods for a Four-Field Formulation of Biot's Consolidation Model

In this paper, a novel isogeometric method for Biot's consolidation model is constructed and analyzed, using a four-field formulation where the unknown variables are the solid displacement, solid pressure, fluid flux, and fluid pressure. Mixed isogeometric spaces based on B-splines basis functions are employed in the space discretization, allowing a smooth representation of the problem geometry and solution fields. The main result of the paper is the proof of optimal error estimates that are robust with respect to material parameters for all solution fields, particularly in the case of nearly incompressible materials. The analysis does not require a uniformly positive storage coefficient. The results of numerical experiments in two and three dimensions confirm the theoretical error estimates and high-order convergence rates attained by the proposed isogeometric Biot discretization and assess its performance with respect to the mesh size, spline polynomial degree, spline regularity, and material parameters.

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arXiv2025

Convergence analysis of BDDC preconditioners for composite DG discretizations of the cardiac cell-by-cell model

A Balancing Domain Decomposition by Constraints (BDDC) preconditioner is constructed and analyzed for the solution of composite Discontinuous Galerkin discretizations of reaction-diffusion systems of ordinary and partial differential equations arising in cardiac cell-by-cell models. Unlike classical Bidomain and Monodomain cardiac models, which rely on homogenized descriptions of cardiac tissue at the macroscopic level, the cell-by-cell models enable the representation of individual cardiac cells, cell aggregates, damaged tissues, and nonuniform distributions of ion channels on the cell membrane. The resulting discrete cell-by-cell models exhibit discontinuous global solutions across the cell boundaries. Therefore, the proposed BDDC preconditioner employs appropriate dual and primal spaces with additional constraints to transfer information between cells (subdomains) without affecting the overall discontinuity of the global solution. A scalable convergence rate bound is proved for the resulting BDDC cell-by-cell preconditioned operator, while numerical tests validate this bound and investigate its dependence on the discretization parameters.

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arXiv2024

GDSW preconditioners for composite Discontinuous Galerkin discretizations of multicompartment reaction-diffusion problems

The aim of the present work is to design, analyze theoretically, and test numerically, a generalized Dryja-Smith-Widlund (GDSW) preconditioner for composite Discontinuous Galerkin discretizations of multicompartment parabolic reaction-diffusion equations, where the solution can exhibit natural discontinuities across the domain. We prove that the resulting preconditioned operator for the solution of the discrete system arising at each time step converges with a scalable and quasi-optimal upper bound for the condition number. The GDSW preconditioner is then applied to the EMI (Extracellular - Membrane - Intracellular) reaction-diffusion system, recently proposed to model microscopically the spatiotemporal evolution of cardiac bioelectrical potentials. Numerical tests validate the scalability and quasi-optimality of the EMI-GDSW preconditioner, and investigate its robustness with respect to the time step size as well as jumps in the diffusion coefficients.

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arXiv2022

A numerical study of the spectral properties of Isogeometric collocation matrices for acoustic wave problems

This paper focuses on the spectral properties of the mass and stiffness matrices for acoustic wave problems discretized with Isogeometric analysis (IGA) collocation methods in space and Newmark methods in time. Extensive numerical results are reported for the eigenvalues and condition numbers of the acoustic mass and stiffness matrices in the reference square domain with Dirichlet, Neumann and absorbing boundary conditions, varying the polynomial degree $p$, mesh size $h$, regularity $k$, of the IGA discretization and the time step $Δt$ and parameter $β$ of the Newmark method. Results on the sparsity of the matrices and the eigenvalue distribution with respect to the degrees of freedom d.o.f. and the number of nonzero entries nz are also reported. The results are comparable with the available spectral estimates for IGA Galerkin matrices associated to the Poisson problem with Dirichlet boundary conditions, and in some cases the IGA collocation results are better than the corresponding IGA Galerkin estimates.

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arXiv2022

Parallel Newton-Krylov-BDDC and FETI-DP deluxe solvers for implicit time discretizations of the cardiac Bidomain equations

Two novel parallel Newton-Krylov Balancing Domain Decomposition by Constraints (BDDC) and Dual-Primal Finite Element Tearing and Interconnecting (FETI-DP) solvers are here constructed, analyzed and tested numerically for implicit time discretizations of the three-dimensional Bidomain system of equations. This model represents the most advanced mathematical description of the cardiac bioelectrical activity and it consists of a degenerate system of two non-linear reaction-diffusion partial differential equations (PDEs), coupled with a stiff system of ordinary differential equations (ODEs). A finite element discretization in space and a segregated implicit discretization in time, based on decoupling the PDEs from the ODEs, yields at each time step the solution of a non-linear algebraic system. The Jacobian linear system at each Newton iteration is solved by a Krylov method, accelerated by BDDC or FETI-DP preconditioners, both augmented with the recently introduced {\em deluxe} scaling of the dual variables. A polylogarithmic convergence rate bound is proven for the resulting parallel Bidomain solvers. Extensive numerical experiments on linux clusters up to two thousands processors confirm the theoretical estimates, showing that the proposed parallel solvers are scalable and quasi-optimal.

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