arXiv subjects

Annalisa Quaini

Publications and source records attributed to Annalisa Quaini.

At least 19 recordsLinked to original sources

arXiv2026

Nonlinear filtering stabilizations for the quasi-geostrophic equations

Numerical simulations of ocean flows typically require fine computational meshes to resolve the Munk scale, leading to high computational costs. Filtering-based large eddy simulation (LES) provides a way to relax the mesh size requirement by modeling the effects of the unresolved scales. For the implementation of this strategy, we propose a three-step algorithm called Evolve-Filter-Relax (EFR) that requires (i) the solution of a QGE problem, (ii) a nonlinear Helmholtz filter for the potential vorticity field leveraging an indicator function, and (iii) a final relaxation step. We show that the EFR algorithm can be interpreted as a splitting scheme for a perturbed QGE problem with additional dissipation and provide a practical choice for the relaxation parameter. For comparison, we also investigate a nonlinear Bardina regularization of the QGE. Numerical results on a classical benchmark show that both the EFR approach and the nonlinear Bardina regularization significantly improve the accuracy and stability of coarse mesh simulations with no LES model. Additionally, the EFR method with a deconvolution-based indicator function delivered the best balance between accuracy, stability, and computational efficiency in a test case involving a more realistic geometry (Mediterranean Sea).

math.NA↗

arXiv2026

A finite element framework for simulating residential burglary in realistic urban geometries

We consider a partial differential equation (PDE) model to predict residential burglary derived from a probabilistic agent-based model through a mean-field limit operation. The PDE model is a nonlinear, coupled system of two equations in two variables (attractiveness of residential sites and density of criminals), similar to the Keller-Segel model for aggregation based on chemotaxis. Unlike previous works, which applied periodic boundary conditions, we enforce boundary conditions that arise naturally from the variational formulation of the PDE problem, i.e., the starting point for the application of a finite element method. These conditions specify the value of the normal derivatives of the system variables at the boundary. For the numerical solution of the PDE problem discretized in time and space, we propose a scheme that decouples the computation of the attractiveness from the computation of the criminal density at each time step, resulting in the solution of two linear algebraic systems per iteration. Through numerous numerical tests, we demonstrate the robustness and computational efficiency of this approach. Leveraging the flexibility allowed by the finite element method, we show results for spatially heterogeneous model parameters and a realistic geometry (city of Chicago). The paper includes a discussion of future perspectives to build multiscale, 'multi-physics' models that can become a tool for the community. The robust and efficient code developed for this paper, which is shared open-source, is intended as the solid base for this broader research program.

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arXiv2026

An isotropic recovery-based error estimator algorithm for mesh adaptation in a finite volume environment with application to atmospheric flows

We develop an Isotropic Recovery-based Error Estimator (IREE) to drive mesh adaptation within a finite volume framework. Recovery-based error estimators are widely used in practice thanks to their simplicity, relying solely on the available discrete solution and a suitable post-processing step. While recovery-based estimators are well established in the finite element framework, their application to finite volume methods, despite the widespread use of the latter in industrial and commercial codes, remains largely unexplored and motivates the present study. We assess the performance of the proposed IREE-driven mesh adaptation procedure through well-known benchmarks for dry atmospheric flows modeling and compare it against a widely-used plain mesh adaptation algorithm. Both qualitative and quantitative results show that the IREE algorithm is more effective at suppressing numerical instabilities compared with the plain approach and, for most of the simulation time, provides a superior accuracy, though at a moderately higher computational cost. In addition, both the adaptive approaches offer substantial computational savings (between 35% and 94%) relative to simulations based on fixed uniformly fine meshes. These findings demonstrate that IREE-based mesh adaptation is a promising and effective strategy for atmospheric flow simulations and has the potential to substantially reduce the computational effort of generating reanalysis-quality data.

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arXiv2026

Crime hotspot dynamics in residential burglary models with police response

We develop and analyze mathematical models for residential burglary that incorporates police deployment through a delayed feedback mechanism. Motivated by empirical observations from publicly available crime and policing data, we extend a well-known agent-based model by introducing a dynamic police response driven by crime information that becomes available only after a finite delay. Taking the mean-field limit, we derive a coupled continuum system consisting of three partial differential equations and one ordinary differential equation describing the interactions among criminal density, environmental attractiveness, delayed crime signal, and police deployment. Linear stability analysis of homogeneous steady states reveals that response delays can destabilize otherwise stable equilibria through Hopf bifurcations. As a result, the model predicts sustained temporal oscillations and dynamically evolving crime hotspots. Numerical simulations of both the agent-based and continuum models confirm the theoretical analysis and uncover rich spatio-temporal behaviors, including moving, splitting, and merging hotspots. Through a parametric study, we investigate the roles of police density, crime information delay, and neighborhood effects in controlling stability, hotspot size, and oscillatory behavior. Our results indicate that timely access to crime data plays a more important role than police density in stabilizing crime levels.

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arXiv2026

Digital Twins in Coronary Artery Disease: A Mathematical Roadmap

The combination of data and models, enhanced by AI methodologies, leads to the paradigm called Digital Twins. This concept is expected to bring unprecedented support to personalized medicine. The combination of mathematical and numerical models with diagnostic devices that provide patient-specific knowledge in a bidirectional framework can be a formidable decision support for clinicians. In this paper, we consider some mathematical aspects of constructing a Digital Twin to prevent and treat Coronary Artery Disease. The keywords for the bidirectional communication between twins in our system are (i) Data Assimilation and (ii) Probabilistic Graphic Models. In particular, a quantity of paramount interest in the evaluation and prognosis of Coronary Artery Disease is the Wall Shear Stress, i.e., the tangential component of normal stress on the arterial wall. By considering steps for the personalization and the synthesis of Wall Shear Stress estimation, we propose a mathematical roadmap for constructing a Digital Twin system that could help prevent infarcts, one of the most lethal diseases in the world.

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arXiv2026

Enhancing the accuracy of under-resolved numerical simulations of atmospheric flows with super resolution

Super-resolution (SR) techniques based on deep learning have recently emerged as a promising approach to enhance the spatial resolution of computational fluid dynamics simulations while containing computational cost. In this paper, we investigate several SR architectures to improve coarse-grid simulations of mesoscale atmospheric flows, with training data generated from simulations of the weakly compressible Euler equations. We compare a baseline convolutional neural network (CNN), an attention-enhanced CNN, a multi-scale CNN designed to capture flow structures across different spatial scales, and a diffusion-based SR model. The methods are evaluated on two standard atmospheric benchmarks: the rising thermal bubble and the density current. Results show that the baseline CNN can accurately reconstruct simpler flow features, while more complex flows require multi-scale architectures. Overall, SR based on the multi-scale CNN provides the best balance of accuracy, robustness, and computational efficiency, outperforming even a state-of-the-art diffusion-based approach. We also analyze the sensitivity of the models to the size of the training dataset, highlighting limitations and trade-offs of the proposed SR strategies.

physics.flu-dyn↗

arXiv2026

StabOp: A Data-Driven Stabilization Operator for Reduced Order Modeling

Spatial filters have played a central role in large eddy simulation and, more recently, in reduced order model (ROM) stabilization for convection-dominated flows. Nevertheless, important open questions remain: in under-resolved regimes, which filter is most suitable for a given stabilization or closure model? Moreover, once a filter is selected, how should its parameters, such as the filter radius, be determined? Addressing these questions is essential for the reliable design and performance of filter-based stabilization strategies. To answer these questions, we propose a novel strategy that differs fundamentally from current filter-based approaches: we replace traditional spatial filters with a data-driven stabilization operator (StabOp) that yields accurate results for a given resolution, quantity of interest, and stabilization strategy. Although the new StabOp can be used for both classical discretizations and ROMs, and for different types of filter-based stabilization or closure, for clarity we focus on ROMs and the Leray stabilization. To build the new StabOp, we postulate its model form as a linear, quadratic, or nonlinear mapping, and then solve a PDE-constrained optimization problem to minimize a given loss function. Using the resulting StabOp in the Leray ROM (L-ROM) yields a new stabilized ROM, StabOp-L-ROM. To assess the StabOp-L-ROM, we compare it with the L-ROM and the standard ROM in numerical simulations of four flows: 2D flow past a cylinder at Re=500, lid-driven cavity at Re=10000, 3D flow past a hemisphere at Re=2200, and minimal channel flow at Re=5000. Our numerical results show that the StabOp-L-ROM can be orders of magnitude more accurate than the classical L-ROM tuned with an optimal filter radius in the predictive regime. Furthermore, while the new StabOp smooths the input flow fields, its smoothing mechanism differs from that of classical spatial filters.

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arXiv2025

Phase-separated lipid vesicles: continuum modeling, simulation, and validation

The paper presents a complete research cycle comprising continuum-based modeling, computational framework development, and validation setup to predict phase separation and surface hydrodynamics in lipid bilayer membranes. We starting with an overview of the key physical characteristics of lipid bilayers, including their composition, mechanical properties, and thermodynamics, and then discuss continuum models of multi-component bilayers. The most complex model is a Navier--Stokes--Cahn--Hilliard (NSCH) type system, describing the coupling of incompressible surface fluid dynamics with phase-field dynamics on arbitrarily curved geometries. It is discretized using trace finite element methods, which offer geometric flexibility and stability in representing surface PDEs. Numerical studies are conducted to examine physical features such as coarsening rates and interfacial dynamics. The computational results obtained from the NSCH model are compared against experimental data for membrane compositions with distinct phase behaviors, demonstrating that including both phase-field models and surface hydrodynamics is essential to accurately reproduce domain evolution observed in epi-fluorescence microscopy. Lastly, we extend the model to incorporate external forces that enable the simulation of vesicles containing cationic lipids, used to enhance membrane fusion.

math.NA↗

arXiv2025

Randomized Proper Orthogonal Decomposition for data-driven reduced order modeling of a two-layer quasi-geostrophic ocean model

The two-layer quasi-geostrophic equations (2QGE) serve as a simplified model for simulating wind-driven, stratified ocean flows. However, their numerical simulation remains computationally expensive due to the need for high-resolution meshes to capture a wide range of turbulent scales. This becomes especially problematic when several simulations need to be run because of, e.g., uncertainty in the parameter settings. To address this challenge, we propose a data-driven reduced order model (ROM) for the 2QGE that leverages randomized proper orthogonal decomposition (rPOD) and long short-term memory (LSTM) networks. To efficiently generate the snapshot data required for model construction, we apply a nonlinear filtering stabilization technique that allows for the use of larger mesh sizes compared to a direct numerical simulations (DNS). Thanks to the use of rPOD to extract the dominant modes from the snapshot matrices, we achieve up to 700 times speedup over the use of deterministic POD. LSTM networks are trained with the modal coefficients associated with the snapshots to enable the prediction of the time- and parameter-dependent modal coefficients during the online phase, which is hundreds of thousands of time faster than a DNS. We assess the accuracy and efficiency of our rPOD-LSTM ROM through an extension of a well-known benchmark called double-gyre wind forcing test. The dimension of the parameter space in this test is increased from two to four.

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arXiv2025

Combining Extended Convolutional Autoencoders and Reservoir Computing for Accurate Reduced-Order Predictions of Atmospheric Flows

Forecasting atmospheric flows with traditional discretization methods, also called full order methods (e.g., finite element methods or finite volume methods), is computationally expensive. We propose to reduce the computational cost with a Reduced Order Model (ROM) that combines Extended Convolutional Autoencoders (E-CAE) and Reservoir Computing (RC). Thanks to an extended network depth, the E-CAE encodes the high-resolution data coming from the full order method into a compact latent representation and can decode it back into high-resolution with 75% lower reconstruction error than standard CAEs. The compressed data are fed to an RC network, which predicts their evolution. The advantage of RC networks is a reduced computational cost in the training phase compared to conventional predictive models. We assess our data-driven ROM through well-known 2D and 3D benchmarks for atmospheric flows. We show that our ROM accurately reconstructs and predicts the future system dynamics with errors below 6% in 2D and 8% in 3D, while significantly reducing the computational cost of a full-order simulation. Compared to other ROMs available in the literature, such as Dynamic Mode Decomposition and Proper Orthogonal Decomposition with Interpolation, our ROM is as efficient but more accurate. Thus, it is a promising alternative to high-dimensional atmospheric simulations.

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arXiv2024

Data driven learning to enhance a kinetic model of distressed crowd dynamics

The mathematical modeling of crowds is complicated by the fact that crowds possess the behavioral ability to develop and adapt moving strategies in response to the context. For example, in emergency situations, people tend to alter their walking strategy in response to fear. To be able to simulate these situations, we consider a kinetic model of crowd dynamics that features the level of stress as a parameter and propose to estimate this key parameter by solving an inverse crowd dynamics problem. This paper states the mathematical problem and presents a method for its numerical solution. We show some preliminary results based on a synthetic data set, i.e., test cases where the exact stress level is known and the crowd density data are generated numerically by solving a forward crowd dynamics problem.

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arXiv2024

Optimal Transport-Based Displacement Interpolation with Data Augmentation for Reduced Order Modeling of Nonlinear Dynamical Systems

We present a novel reduced-order Model (ROM) that leverages optimal transport (OT) theory and displacement interpolation to enhance the representation of nonlinear dynamics in complex systems. While traditional ROM techniques face challenges in this scenario, especially when data (i.e., observational snapshots) is limited, our method addresses these issues by introducing a data augmentation strategy based on OT principles. The proposed framework generates interpolated solutions tracing geodesic paths in the space of probability distributions, enriching the training dataset for the ROM. A key feature of our approach is its ability to provide a continuous representation of the solution's dynamics by exploiting a virtual-to-real time mapping. This enables the reconstruction of solutions at finer temporal scales than those provided by the original data. To further improve prediction accuracy, we employ Gaussian Process Regression to learn the residual and correct the representation between the interpolated snapshots and the physical solution. We demonstrate the effectiveness of our methodology with atmospheric mesoscale benchmarks characterized by highly nonlinear, advection-dominated dynamics. Our results show improved accuracy and efficiency in predicting complex system behaviors, indicating the potential of this approach for a wide range of applications in computational physics and engineering.

math.NA↗

arXiv2024

A Robin-Robin splitting method for the Stokes-Biot fluid-poroelastic structure interaction model

We develop and analyze a splitting method for fluid-poroelastic structure interaction. The fluid is described using the Stokes equations and the poroelastic structure is described using the Biot equations. The transmission conditions on the interface are mass conservation, balance of stresses, and the Beavers-Joseph-Saffman condition. The splitting method involves single and decoupled Stokes and Biot solves at each time step. The subdomain problems use Robin boundary conditions on the interface, which are obtained from the transmission conditions. The Robin data is represented by an auxiliary interface variable. We prove that the method is unconditionally stable and establish that the time discretization error is $\mathcal{O}(\sqrt{T}Δt)$, where $T$ is the final time and $Δt$ is the time step. We further study the iterative version of the algorithm, which involves an iteration between the Stokes and Biot sub-problems at each time step. We prove that the iteration converges to a monolithic scheme with a Robin Lagrange multiplier used to impose the continuity of the velocity. Numerical experiments are presented to illustrate the theoretical results.

math.NA↗

arXiv2024

Data-driven reduced order modeling of a two-layer quasi-geostrophic ocean model

The two-layer quasi-geostrophic equations (2QGE) is a simplified model that describes the dynamics of a stratified, wind-driven ocean in terms of potential vorticity and stream function. Its numerical simulation is plagued by a high computational cost due to the size of the typical computational domain and the need for high resolution to capture the full spectrum of turbulent scales. In this paper, we present a data-driven reduced order model (ROM) for the 2QGE that drastically reduces the computational time to predict ocean dynamics, especially when there are variable physical parameters. The main building blocks of our ROM are: i) proper orthogonal decomposition (POD) and ii) long short-term memory (LSTM) recurrent neural networks. Snapshots data are collected from a high-resolution simulation for part of the time interval of interest and for given parameter values in the case of variable parameters. POD is applied to each field variable to extract the dominant modes and a LSTM model is trained on the modal coefficients associated with the snapshots for each variable. Then, the trained LSTM models predict the modal coefficients for the remaining part of the time interval of interest and for a new parameter value. To illustrate the predictive performance of our POD-LSTM ROM and the corresponding time savings, we consider an extension of the so-called double-gyre wind forcing test. We show that the POD-LSTM ROM is accurate in predicting both time-averaged fields and time-dependent quantities (modal coefficients, enstrophy, and kinetic energy), even when retaining only 10-20\% of the singular value energy of the system. The computational speed up for the prediction is about up to 1E+07 compared to a finite volume based full order method.

math.NA↗

arXiv2024

Bridging Large Eddy Simulation and Reduced Order Modeling of Convection-Dominated Flows through Spatial Filtering: Review and Perspectives

Reduced order models (ROMs) have achieved a lot of success in reducing the computational cost of traditional numerical methods across many disciplines. For convection-dominated (e.g., turbulent) flows, however, standard ROMs generally yield inaccurate results, usually affected by spurious oscillations. Thus, ROMs are usually equipped with numerical stabilization or closure models to account for the effect of the discarded modes. The literature on ROM closures and stabilizations is large and growing fast. In this paper, we focus on one particular type of ROM closures and stabilizations that are inspired by Large Eddy Simulation (LES). These ROMs, which we call LES-ROMs, are extremely easy to implement, very efficient, and accurate. Carefully tuned LES-ROMs can accurately capture the average physical quantities of interest in challenging convection-dominated flows in many applications. LES-ROM are constructed by leveraging spatial filtering, i.e., the same principle used to build classical LES models. This ensures a modeling consistency between LES-ROMs and the approaches that generated the data used to train them. It also ``bridges'' two distinct research fields (LES and ROMs), disconnected until now. This paper is a review of LES-ROMs. It starts with a description of a versatile LES strategy called evolve-filter-relax (EFR) that has been successfully used as a full order method. We then show how the EFR strategy, and spatial filtering in general, can be leveraged to construct LES-ROMs. Several applications of LES-ROMs are presented. Finally, we draw conclusions and outline several research directions and open questions in the LES-ROM development. While we do not claim this review to be comprehensive, we certainly hope it serves as a brief and friendly introduction to this exciting research area, which has a lot of potential in practical numerical simulation of convection-dominated flows.

physics.flu-dyn↗

arXiv2024

Computational study of numerical flux schemes for mesoscale atmospheric flows in a Finite Volume framework

We develop, and implement in a Finite Volume environment, a density-based approach for the Euler equations written in conservative form using density, momentum, and total energy as variables. Under simplifying assumptions, these equations are used to describe non-hydrostatic atmospheric flow. The well-balancing of the approach is ensured by a local hydrostatic reconstruction updated in runtime during the simulation to keep the numerical error under control. To approximate the solution of the Riemann problem, we consider four methods: Roe-Pike, HLLC, AUSM+-up and HLLC-AUSM. We assess our density-based approach and compare the accuracy of these four approximated Riemann solvers using two two classical benchmarks, namely the smooth rising thermal bubble and the density current.

math.NA↗

arXiv2024

Stabilized POD Reduced Order Models for convection-dominated incompressible flows

We present a comparative computational study of two stabilized Reduced Order Models (ROMs) for the simulation of convection-dominated incompressible flow (Reynolds number of the order of a few thousands). Representative solutions in the parameter space, which includes either time only or time and Reynolds number, are computed with a Finite Volume method and used to generate a reduced basis via Proper Orthogonal Decomposition (POD). Galerkin projection of the Navier-Stokes equations onto the reduced space is used to compute the ROM solution. To ensure computational efficiency, the number of POD modes is truncated and ROM solution accuracy is recovered through two stabilization methods: i) adding a global constant artificial viscosity to the reduced dimensional model, and ii) adding a different value of artificial viscosity for the different POD modes. We test the stabilized ROMs for fluid flow in an idealized medical device consisting of a conical convergent, a narrow throat, and a sudden expansion. Both stabilization methods significantly improve the ROM solution accuracy over a standard (non-stabilized) POD-Galerkin model.

physics.flu-dyn↗

arXiv2024

Linear and nonlinear filtering for a two-layer quasi-geostrophic ocean model

Although the two-layer quasi-geostrophic equations (2QGE) are a simplified model for the dynamics of a stratified, wind-driven ocean, their numerical simulation is still plagued by the need for high resolution to capture the full spectrum of turbulent scales. Since such high resolution would lead to unreasonable computational times, it is typical to resort to coarse low-resolution meshes combined with the so-called eddy viscosity parameterization to account for the diffusion mechanisms that are not captured due to mesh under-resolution. We propose to enable the use of further coarsened meshes by adding a (linear or nonlinear) differential low-pass to the 2QGE, without changing the eddy viscosity coefficient. While the linear filter introduces constant (additional) artificial viscosity everywhere in the domain, the nonlinear filter relies on an indicator function to determine where and how much artificial viscosity is needed. Through several numerical results for a double-gyre wind forcing benchmark, we show that with the nonlinear filter we obtain accurate results with very coarse meshes, thereby drastically reducing the computational time (speed up ranging from 30 to 300).

math.NA↗