arXiv subjects

Paola F. Antonietti

Publications and source records attributed to Paola F. Antonietti.

At least 19 recordsLinked to original sources

arXiv2026

A novel mathematical and computational framework of amyloid-beta triggered seizure dynamics in Alzheimer's disease

The association of epileptic activity and Alzheimer's disease (AD) has been increasingly reported in both clinical and experimental studies, suggesting that amyloid-$β$ accumulation may directly affect neuronal excitability. Capturing these interactions requires a quantitative description that bridges the molecular alterations of AD with the fast electrophysiological dynamics of epilepsy. We introduce a novel mathematical model that extends the Barreto-Cressman ionic formulation by incorporating multiple mechanisms of calcium dysregulation induced by amyloid-$β$, including formation of $\mathrm{Ca}^{2+}$-permeable pores, overactivation of voltage-gated $\mathrm{Ca}^{2+}$ channels, and suppression of $\mathrm{Ca}^{2+}$-sensitive potassium currents. The resulting ionic model is coupled with the monodomain equation and discretized using a $p$-adaptive discontinuous Galerkin method on polytopal meshes, providing an effective balance between efficiency and accuracy in capturing the sharp spatiotemporal electrical wavefronts associated with epileptiform discharges. Numerical simulations performed on idealized and realistic brain geometries demonstrate that progressive amyloid-\textbeta{} accumulation leads to severe alterations in calcium homeostasis, increased neuronal hyperexcitability, and pathological seizure propagation. Specifically, high amyloid-$β$ concentrations produce secondary epileptogenic sources and spatially heterogeneous wavefronts, indicating that biochemical inhomogeneities play a critical role in shaping seizure dynamics. These results illustrate how multiscale modeling provides new mechanistic insights into the interplay between neurodegeneration and epilepsy in Alzheimer's disease.

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arXiv2026

A Neural-Network Approach for the Stability of Loosely-Coupled FSI Simulations in Hemodynamics

The development of accurate and efficient computational methods for fluid-structure interaction (FSI) problems remains an active research area due to its broad engineering and biomedical applications. Partitioned loosely coupled (LC) schemes are attractive for large-scale simulations because of their modularity and low computational cost. However, their applicability is limited by stability issues in high added-mass regimes, where fluid and structure densities are comparable, as in hemodynamics. In our recent work, we proposed an LC algorithm based on Dirichlet-Neumann interface coupling, the LC-DN-α scheme, which improves the stability of the coupling through a stabilization parameter α. Although theoretical stability has been demonstrated under suitable constraints on α, its optimal value depends on the physical and numerical characteristics of the FSI problem and is difficult to determine in practice. In this work, we propose a data-driven strategy to automatically estimate the optimal value of α. We develop a feed-forward neural network to predict α from the descriptors of the FSI problem. The model is trained on a dataset of baseline FSI problems in large added-mass regimes and evaluated on more realistic configurations to assess robustness and generalization. Numerical results show that the proposed model generalizes across the considered FSI scenarios and provides robust estimates of α under moderate variations in the problem setup, including changes in geometry and loading conditions, that do not completely alter the interaction mechanics represented in the training set. The methodology is intentionally developed and validated on simplified FSI problems, providing a proof of concept for data-driven strategies to improve the efficiency of partitioned FSI simulations and supporting their extension to more complex and realistic scenarios.

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arXiv2026

A hybrid reduced-order and high-fidelity discontinuous Galerkin Spectral Element framework for large-scale PMUT array simulations

Piezoelectric Micromachined Ultrasonic Transducers (PMUTs) are essential for next-generation ultrasonic sensing and imaging due to their bidirectional electromechanical behavior, compact design, and compatibility with low-voltage electronics. As PMUT arrays grow in size and complexity, efficiently modeling their coupled electromechanical-acoustic behavior becomes increasingly challenging. This work presents a novel computational framework that combines model order reduction with a Discontinuous Galerkin Spectral Element Method (DGSEM) paradigm to simulate large PMUT arrays. Each PMUT's mechanical behavior is represented using a reduced set of vibration modes, which are coupled to an acoustic domain model to describe the full array. To further improve efficiency, a secondary acoustic domain is connected via DG interfaces, enabling non-conforming mesh refinement, with variable approximation order, and accurate wave propagation. The framework is implemented in the SPectral Elements in Elastodynamics with Discontinuous Galerkin (SPEED) software, an open-source, parallelized platform leveraging domain decomposition, high-order polynomials, METIS graph partitioning, and MPI for scalable performance. The proposed methodology addresses key challenges in meshing, supporting high-fidelity simulations for both PMUT transmission and reception phases. Numerical results demonstrate the framework's accuracy, scalability, and efficiency for large PMUT array simulations.

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arXiv2026

A whole-brain model of amyloid beta accumulation and cerebral hypoperfusion in Alzheimer's disease

Accumulation of amyloid beta proteins is a defining feature of Alzheimer's disease, and is usually accompanied by cerebrovascular pathology. Evidence suggests that amyloid beta and cerebrovascular pathology are mutually reinforcing; in particular, amyloid beta suppresses perfusion by constricting capillaries, and hypoperfusion promotes the production of amyloid beta. Here, we propose a whole-brain model coupling amyloid beta and blood vessel through a hybrid model consisting of a reaction-diffusion system for the protein dynamics and porous-medium model of blood flow within and between vascular networks: arterial, capillary and venous. We discretize the resulting parabolic--elliptic system of PDEs by means of a high-order discontinuous Galerkin method in space and an implicit Euler scheme in time. Simulations in realistic brain geometries demonstrate the emergence of multistability, implying that a sufficiently large pathogenic protein seeds is necessary to trigger disease outbreak. Motivated by the "two-hit vascular hypothesis" of Alzheimer's disease that hypoperfusive vascular damage triggers amyloid beta pathology, we also demonstrate that localized hypoperfusion, in response to injury, can destabilize the healthy steady state and trigger brain-wide disease outbreak.

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arXiv2026

Discontinuous Galerkin discretization of coupled poroelasticity-elasticity problems

This work is concerned with the analysis of a space-time finite element discontinuous Galerkin method on polytopal meshes (XT-PolydG) for the numerical discretization of wave propagation in coupled poroelastic-elastic media. The mathematical model consists of the low-frequency Biot's equations in the poroelastic medium and the elastodynamics equation for the elastic one. To realize the coupling, suitable transmission conditions on the interface between the two domains are (weakly) embedded in the formulation. The proposed PolydG discretization in space is then coupled with a dG time integration scheme, resulting in a full space-time dG discretization. We present the stability analysis for both the continuous and the semidiscrete formulations, and we derive error estimates for the semidiscrete formulation in a suitable energy norm. The method is applied to a wide set of numerical test cases to verify the theoretical bounds. Examples of physical interest are also presented to investigate the capability of the proposed method in relevant geophysical scenarios.

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arXiv2026

The lymph 2.0 library: p-adaptive algorithms and parallel assembly strategies for polytopal DG methods

This work presents a new release of the lymph 2.0 library, an open-source MATLAB framework for high-order discontinuous Galerkin discretizations on general polytopal meshes. The lymph 2.0 version is extended to support discretizations with element-wise polynomial approximation degrees, which allows the design of $p$-adaptive strategies based on a posteriori error indicators. In addition, the library introduces a unified assembly framework that abstracts the construction of discrete operators from the underlying physical model, improving code modularity, parallelism, maintainability, and extensibility. Moreover, the proposed approach enables shared-memory parallelism through dedicated parallel tools. Several numerical examples demonstrate the effectiveness of the proposed developments in reducing the computational cost while preserving approximation accuracy.

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arXiv2026

Optimized high-order IMEX-RK schemes for degenerate diffusion-reaction problems with application to travelling waves phenomena

We study a class of IMplicit-EXplicit Runge--Kutta (IMEX-RK) schemes for the numerical approximation of reaction and diffusion-reaction problems arising in a variety of biological and physical applications. Such models may admit travelling wave solutions, with the Fisher--Kolmogorov equation representing a prototypical example. Motivated by this feature, the proposed time integration schemes are designed to accurately capture sharp propagating fronts. We also investigate a less standard use of IMEX-RK methods that circumvents a splitting of reaction terms into linear and nonlinear components, while still requiring the solution of linear systems at each stage. This semi-implicit formulation, referred to as SI-IMEX-RK, enables a targeted treatment of stiffness by isolating its relevant contributions. The time discretization is coupled with a high-order polygonal discontinuous Galerkin method for space discretization, resulting in a flexible and robust framework for the treatment of multiscale dynamics in complex geometries. A comprehensive validation strategy is presented to assess the accuracy and stability properties of the proposed schemes across a hierarchy of increasingly challenging test problems.

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arXiv2026

A Scalable Deflated Conjugate Gradient Solver for the Time-Dependent Pseudo-Stress Stokes Problem

We propose a novel iterative solution framework for the unsteady Stokes equations in the pseudo-stress formulation. When solving this class of problems by using implicit time-integration schemes, standard solvers suffer from deteriorating convergence properties for small time steps, independently of the chosen space discretisation method. This is due to the singular modes of the dev-dev operator. For this reason, we introduce a computational framework obtained by combining a deflated Conjugate Gradient method with a W-cycle multigrid scheme that employs a Restricted Additive Schwarz smoother. The key point is to choose the deflation subspace so that the inner system to be solved within a deflated Conjugate Gradient scheme corresponds to a Laplace problem defined on the singular modes of the original dev-dev operator. This results to be independent of the spatial discretisation method and allows one to use efficient multigrid iterative solvers. Numerical experiments show that the proposed strategy significantly accelerates the Conjugate Gradient convergence and provides stable performance with respect to the time step, confirming its robustness for solving linear systems in the pseudo-stress framework.

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arXiv2026

Splitting strategies for the fully-coupled nonlinear thermo-hydro-mechanical problem

We propose novel semi-decoupled and fully-decoupled iterative algorithms for efficiently solving the fully-coupled nonlinear four-field thermo-poroelastic model discretized in space by discontinuous Galerkin method on polytopal grids. We present the model problem, its four-field formulation, and the arbitrary-order weighted symmetric interior penalty scheme exploited for its spatial discretization. Such a scheme is robust with respect to strong heterogeneities in the model coefficients. Then, we present the two solution strategies and prove that under suitable conditions both schemes are convergent. A wide set of numerical simulations is presented to assess the convergence and robustness properties of the proposed method. Moreover, we test the scheme with literature and physically sound test cases for proof-of-concept applications in the geophysical context.

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arXiv2026

Polytopal Discontinuous Galerkin Discretizations of Coupled Non-Newtonian Stokes-Darcy Systems

We propose and analyze a polytopal discontinuous Galerkin method for the numerical approximation of a coupled non-Newtonian Stokes-Darcy system modeling the interaction between a non-Newtonian free-flow fluid and a non-Newtonian flow through a porous medium. Due to its geometric flexibility and arbitrary-order accuracy, the proposed discretization scheme is well-suited to configurations with complex geometries. We provide a complete a-priori analysis that considers shear-dependent and velocity-dependent non-Newtonian viscosity models for the free-flow and porous media regions, respectively. The well-posedness, stability, and error bounds of the method are established in the framework of generalized inf-sup theory. Error estimates are confirmed by numerical results.

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arXiv2026

Structure-preserving local discontinuous Galerkin discretization of conformational conversion systems

We investigate a two-state conformational conversion system and introduce a novel structure-preserving numerical scheme that couples a local discontinuous Galerkin space discretization with the backward Euler time-integration method. The model is first reformulated in terms of auxiliary variables involving suitable nonlinear transformations, which allow us to enforce positivity and boundedness at the numerical level. Then, we prove a discrete entropy-stability inequality, which we use to show the existence of discrete solutions, as well as to establish the convergence of the scheme by means of some discrete compactness arguments. As a by-product of the theoretical analysis, we also prove the existence of global weak solutions satisfying the system's physical bounds. Numerical results validate the theoretical results and assess the capabilities of the proposed method in practice.

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arXiv2026

High-fidelity and Network-based Spatio-temporal Mathematical Models of Alzheimer's Disease Progression and their Validation Against PET-SUVR Imaging Data

Alzheimer's disease is the most common neurodegenerative disorder. Its pathological development is connected with the misfolding and accumulation of two toxic proteins: amyloid-beta and tau proteins. Mathematical models provide a valuable quantitative tool for monitoring disease progression. In this work, we proposed and compare a novel framework where the spatio-temporal dynamics of amyloid-beta and tau proteins is modeled based on employing either three-dimensional patient-specific geometries or through reduced network-based models defined on the brain connectome. More specifically, a high-fidelity biophysical model is proposed on three-dimensional brain geometries reconstructed from magnetic resonance imaging, whereas a network-based reduced formulation is defined on the brain connectome. For both approaches, a suitable numerical discretisation is proposed. A sensitivity analysis is presented to quantify the influence of model parameters on protein concentration patterns as well as compare the quality of the predictions. For both approaches, the results are validated against PET-SUVR clinical data using 18FAZD4694 for amyloid-beta and 18FMK6240 for tau protein. The results indicate that the three-dimensional model provides the most accurate and biologically consistent description of the disease progression, but remains computationally demanding. On the other hand, the reduced graph-based model is cheaper, but it is not always able to achieve reliable results.

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arXiv2026

A Multilevel Monte Carlo Virtual Element Method for Uncertainty Quantification of Elliptic Partial Differential Equations

We introduce a Monte Carlo Virtual Element estimator based on Virtual Element discretizations for stochastic elliptic partial differential equations with random diffusion coefficients. We prove estimates for the statistical approximation error for both the solution and suitable linear quantities of interest. A Multilevel Monte Carlo Virtual Element method is also developed and analyzed to mitigate the computational cost of the plain Monte Carlo strategy. The proposed approach exploits the flexibility of the Virtual Element method on general polytopal meshes and employs sequences of coarser spaces constructed via mesh agglomeration, providing a practical realization of the multilevel hierarchy even in complex geometries. This strategy substantially reduces the number of samples required on the finest level to achieve a prescribed accuracy. We prove convergence of the multilevel method and analyze its computational complexity, showing that it yields significant cost reductions compared to standard Monte Carlo methods for a prescribed accuracy. Extensive numerical experiments support the theoretical results and demonstrate the efficiency of the proposed method.

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arXiv2026

Mathematical and numerical modeling of coupled oxygen dynamics and neuronal electrophysiology

Modeling and simulating how oxygen supply shapes neuronal excitability is crucial for advancing the understanding of brain function in pathological scenarios, such as ischemia. This condition is caused by a reduced blood supply, leading to the deprivation of oxygen and other metabolites; this energy deficit impairs ionic pumps and causes cellular swelling. In this work, this phenomenon is modeled through a volumetric variation law that links cell swelling to local oxygen concentration and the percentage of blood flow reduction. The swelling law links volume changes to local oxygen and the degree of blood-flow depression, providing a simple mechanistic pathway from hypoxia to tortuosity-driven transport impairment. The interplay between oxygen supply and excitability in brain tissue is described by coupling the monodomain model with specific neuronal ionic and metabolic models that characterize ion and metabolite concentration dynamics. The numerical approximation of this coupled multiscale problem is particularly challenging, owing to the presence of sharp and fast-propagating wavefronts and complex geometrical domains. To address these challenges, suitable space- and time-adaptive schemes are employed to capture the action potential dynamics accurately. This multiscale model is discretized in space with the high-order p-adaptive discontinuous Galerkin method on polygonal and polyhedral grids (PolyDG) and integrated in time with a Crank-Nicolson scheme. We numerically investigate different pathological scenarios on a two-dimensional idealized square domain and on a realistic geometry, both discretized with a polygonal grid, analyzing how subclinical and severe ischemia can affect brain electrophysiology and metabolic concentrations.

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arXiv2026

Conforming and discontinuous discretizations of non-isothermal Darcy-Forchheimer flows

We present and analyze in a unified setting two schemes for the numerical discretization of a Darcy-Forchheimer fluid flow model coupled with an advection-diffusion equation modeling the temperature distribution in the fluid. The first approach is based on fully discontinuous Galerkin discretization spaces. In contrast, in the second approach, the velocity is approximated in the Raviart-Thomas space, and the pressure and temperature are still piecewise discontinuous. A fixed-point linearization strategy, naturally inducing an iterative splitting solution, is proposed for treating the nonlinearities of the problem. We present a unified stability analysis and prove the convergence of the iterative algorithm under mild requirements on the problem data. A wide set of two- and three-dimensional simulations is presented to assess the error decay and demonstrate the practical performance of the proposed approaches in physically sound test cases.

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arXiv2026

Numerical Verification of PolyDG Algebraic Solvers for the Pseudo-Stress Stokes Problem

This work focuses on the development of efficient solvers for the pseudo-stress formulation of the unsteady Stokes problem, discretised by means of a discontinuous Galerkin method on polytopal grids (PolyDG). The introduction of the pseudo-stress variable is motivated by the growing interest in non-Newtonian flow models and coupled interface problems, where the stress field plays a fundamental role in the physical description. The space-time discretisation of the problem is obtained by combining the PolyDG approach in space with the implicit Euler method for time integration. The resulting linear system, characterised by a symmetric, positive, definite matrix, exhibits deteriorating convergence with standard solvers as the time step decreases. To address this issue, we investigate two tailored strategies: deflated Conjugate Gradient, which mitigates the effect of the most problematic eigenmodes, and collective Block-Jacobi, which exploits the block structure of the system matrix. Numerical experiments show that both approaches yield iteration counts effectively independent of $Δt$, ensuring robust performance with respect to the time step. Future work will focus on extending this robustness to the spatial discretisation parameter $h$ by integrating multigrid strategies with the time-robust solvers developed in this study.

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arXiv2026

The Reduced Basis Multigrid scheme for the Virtual Element Method

We present a non-nested W-cycle multigrid scheme for the lowest order Virtual Element Method on polygonal meshes. To avoid the implicit definition of the Virtual Element space, which poses several issues in the computation of intergrid operators that underpin multigrid methods, the proposed scheme uses a fully-conforming auxiliary space constructed by cheaply computing the virtual basis functions via the reduced basis method.

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arXiv2026

Virtual Element methods for non-Newtonian shear-thickening fluid flow problems

In this work, we present a comprehensive theoretical analysis for Virtual Element discretizations of incompressible non-Newtonian flows governed by the Carreau-Yasuda constitutive law, in the shear-thickening regime (r > 2) including both degenerate (delta = 0) and non-degenerate (delta > 0) cases. The proposed Virtual Element method features two distinguishing advantages: the construction of an exactly divergence-free discrete velocity field and compatibility with general polygonal meshes. The analysis presented in this work extends a previous work, where only shear-thinning behavior (1 < r < 2) was considered. Indeed, the theoretical analysis of the shear-thickening setting requires several novel analytical tools, including: an inf-sup stability analysis of the discrete velocity-pressure coupling in non-Hilbertian norms, a stabilization term specifically designed to address the nonlinear structure as the exponent r > 2; and the introduction of a suitable discrete norm tailored to the underlying nonlinear constitutive relation. Numerical results demonstrate the practical performance of the proposed formulation.

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