arXiv subjects

Maria Strazzullo

Publications and source records attributed to Maria Strazzullo.

At least 19 recordsLinked to original sources

arXiv2026

Numerical Analysis of the Virtual Element Approximation for the Smagorinsky Turbulence Model

In this paper, we consider the Smagorinsky model for the Navier-Stokes equations within a virtual element framework. Under the standard assumption of small data, we prove the existence and uniqueness of a solution. Assuming more regularity to the solutions, we derive the known convergence rates $h$ for the a priori error estimates of the Smagorinsky model in two dimensional domains. We additionally prove that divergence-free virtual discretizations provide improved convergence orders, with weaker regularity assumptions than in the finite element literature. We conclude the paper with numerical results that corroborate the theory.

math.NA↗

arXiv2026

Graph-Instructed Neural Networks for parametric problems with varying boundary conditions

This work addresses the accurate and efficient simulation of physical phenomena governed by parametric Partial Differential Equations (PDEs) characterized by varying boundary conditions, where parametric instances modify not only the physics of the problem but also the imposition of boundary constraints on the computational domain. In such scenarios, classical Galerkin projection-based reduced order techniques encounter a fundamental bottleneck. Parametric boundaries typically necessitate a re-formulation of the discrete problem for each new configuration, and often, these approaches are unsuitable for real-time applications. To overcome these limitations, we propose a novel methodology based on Graph-Instructed Neural Networks (GINNs). The GINN framework effectively learns the mapping between the parametric description of the computational domain and the corresponding PDE solution. Our results demonstrate that the proposed GINN-based models, can efficiently represent highly complex parametric PDEs, serving as a robust and scalable asset for several applied-oriented settings when compared with fully connected architectures.

math.NA↗

arXiv2026

Reinforcement Learning-Based Filters for Convection-Dominated Flows: Reference-Free and Reference-Guided Training

We propose a reinforcement learning (RL) framework for the dynamic selection of the filter parameter in Evolve-Filter (EF) regularization strategies for incompressible turbulent flows. Instead of prescribing the filter radius heuristically, the RL agent learns to adaptively control the filtering intensity in time, balancing numerical stability and physical accuracy. The methodology is assessed on two benchmark problems with fundamentally different dynamics: flow past a cylinder and decaying homogeneous turbulence. Both reference-guided and reference-free reward formulations are investigated. In the reference-guided setting, the agent is trained using direct numerical simulation (DNS) data over a limited time window and then evaluated in extrapolation. In the reference-free setting, the reward relies exclusively on physics-based quantities, without access to reference solutions, i.e., completely eliminating the computational costs related to DNS simulations. The results show that the proposed RL-EF strategies prevent numerical blow-up while avoiding the excessive dissipation typical of standard EF approaches based on a fixed Kolmogorov length scale. The learned policies accurately reproduce the relevant flow dynamics across scales, preserving the correct balance between large-scale structures and small-scale dissipation. Notably, the reference-free reward achieves performance comparable to the reference-guided approach, demonstrating that stable and spectrally consistent filtering strategies can be learned even without DNS data, drastically reducing the computational costs of the training phase. The proposed framework provides a robust and flexible alternative to manually tuned regularization parameters, enabling adaptive, physically consistent control of filtering in turbulent flow simulations.

math.NA↗

arXiv2025

Machine Learning enhanced parametric Reynolds-averaged Navier-Stokes equations at the full- and reduced-order levels

In this contribution, we focus on the Reynolds-averaged Navier-Stokes (RANS) models and their exploitation to build reliable reduced-order models to further accelerate predictions for real-time applications and many-query scenarios. Specifically, we investigate how machine learning can be employed to enhance the predictive capabilities of the model, both at the full-order model (FOM) and reduced-order model (ROM) levels. We explore a novel integration of these two areas. We generate the FOM snapshots, essential for ROM construction, using a data-driven RANS model: the $\nu_t$-Vector Basis Neural Network. This is the first time that this machine learning procedure covers a large parametric variation, and we propose tailored training strategies to increase the accuracy of the FOM model. At the ROM level, we compare the results obtained by standard proper orthogonal decomposition (POD) in an intrusive Galerkin setting (PODG) and POD neural network approach (PODNN). The numerical validation is conducted on a classic turbulent square-duct flow benchmark, with the bulk Reynolds number as the sole varying parameter. Our investigation reveals that the PODG method proves to be unstable and inaccurate for turbulent flow prediction, while PODNN demonstrates superior performance in terms of accuracy and computational efficiency.

physics.flu-dyn↗

arXiv2025

General Order Virtual Element Approximation for the Smagorinsky turbulence model

In this paper, we investigate a Smagorinsky model in a virtual element framework to simulate convection-dominated Navier-Stokes equations. We conduct a two-dimensional numerical investigation to assess the performance of the general order virtual element approximation in this context. First, we examine numerically the convergence of the method with respect to the meshsize to certify the novel virtual element numerical discretization, which includes, for the first time, a discretization of the Smagorinsky term. Moreover, we present a numerical study of a lid-driven cavity for different Reynolds numbers (up to 10000) and meshes (uniform, anisotropic, and isotropic with hanging nodes). The results highlight the main advantage of using the virtual elements method in this context: the isotropic refinement with hanging nodes enhances the accuracy of the solution compared to the anisotropic mesh, uses fewer degrees of freedom with respect to the uniform mesh, and yields the most stable behavior in terms of convergence of the Newton solver.

math.NA↗

arXiv2025

Deflation-based certified greedy algorithm and adaptivity for bifurcating nonlinear PDEs

This work deals with tailored reduced order models for bifurcating nonlinear parametric partial differential equations, where multiple coexisting solutions arise for a given parametric instance. Approaches based on proper orthogonal decomposition have been widely investigated in the literature, but they usually rely on some \emph{a-priori} knowledge about the bifurcating model and lack any error estimation. On the other hand, standard certified reduced basis techniques fail to represent correctly the branching behavior, since the error estimator is no longer reliable. The main goal of the contribution is to overcome these limitations by introducing two novel algorithms: (i) the adaptive-greedy, detecting the bifurcation point starting from scarce information over the parametric space, and (ii) the deflated-greedy, certifying multiple coexisting branches simultaneously. The former approach takes advantage of the features of the reduced manifold to detect the bifurcation, while the latter exploits the deflation and continuation methods to discover the bifurcating solutions and enrich the reduced space. We test the two strategies for the Coanda effect held by the Navier-Stokes equations in a sudden-expansion channel. The accuracy of the approach and the error certification are compared with vanilla-greedy and proper orthogonal decomposition.

math.NA↗

arXiv2025

Dynamical Low-Rank Approximation Strategies for Nonlinear Feedback Control Problems

This paper addresses the stabilization of dynamical systems in the infinite horizon optimal control setting using nonlinear feedback control based on State-Dependent Riccati Equations (SDREs). While effective, the practical implementation of such feedback strategies is often constrained by the high dimensionality of state spaces and the computational challenges associated with solving SDREs, particularly in parametric scenarios. To mitigate these limitations, we introduce the Dynamical Low-Rank Approximation (DLRA) methodology, which provides an efficient and accurate framework for addressing high-dimensional feedback control problems. DLRA dynamically constructs a compact, low-dimensional representation that evolves with the problem, enabling the simultaneous resolution of multiple parametric instances in real-time. We propose two novel algorithms to enhance numerical performances: the cascade-Newton-Kleinman method and Riccati-based DLRA (R-DLRA). The cascade-Newton-Kleinman method accelerates convergence by leveraging Riccati solutions from the nearby parameter or time instance, supported by a theoretical sensitivity analysis. R-DLRA integrates Riccati information into the DLRA basis construction to improve the quality of the solution. These approaches are validated through nonlinear one-dimensional and two-dimensional test cases showing transport-like behavior, demonstrating that R-DLRA outperforms standard DLRA and Proper Orthogonal Decomposition-based model order reduction in both speed and accuracy, offering a superior alternative to Full Order Model solutions.

math.NA↗

arXiv2025

Data-driven Optimization for the Evolve-Filter-Relax regularization of convection-dominated flows

Numerical stabilization techniques are often employed in under-resolved simulations of convection-dominated flows to improve accuracy and mitigate spurious oscillations. Specifically, the evolve--filter--relax (EFR) algorithm is a framework which consists in evolving the solution, applying a filtering step to remove high-frequency noise, and relaxing through a convex combination of filtered and original solutions. The stability and accuracy of the EFR solution strongly depend on two parameters, the filter radius $\delta$ and the relaxation parameter $\chi$. Standard choices for these parameters are usually fixed in time, and related to the full order model setting, i.e., the grid size for $\delta$ and the time step for $\chi$. The key novelties with respect to the standard EFR approach are: (i) time-dependent parameters $\delta(t)$ and $\chi(t)$, and (ii) data-driven adaptive optimization of the parameters in time, considering a fully-resolved simulation as reference. In particular, we propose three different classes of optimized-EFR (Opt-EFR) strategies, aiming to optimize one or both parameters. The new Opt-EFR strategies are tested in the under-resolved simulation of a turbulent flow past a cylinder at $Re=1000$. The Opt-EFR proved to be more accurate than standard approaches by up to 99$\%$, while maintaining a similar computational time. In particular, the key new finding of our analysis is that such accuracy can be obtained only if the optimized objective function includes: (i) a global metric (as the kinetic energy), and (ii) spatial gradients' information.

math.NA↗

arXiv2024

Variational Multiscale Evolve and Filter Strategies for Convection-Dominated Flows

The evolve-filter (EF) model is a filter-based numerical stabilization for under-resolved convection-dominated flows. EF is a simple, modular, and effective strategy for both full-order models (FOMs) and reduced-order models (ROMs). It is well-known, however, that when the filter radius is too large, EF can be overdiffusive and yield inaccurate results. To alleviate this, EF is usually supplemented with a relaxation step. The relaxation parameter, however, is very sensitive with respect to the model parameters. In this paper, we propose a novel strategy to alleviate the EF overdiffusivity for a large filter radius. Specifically, we leverage the variational multiscale (VMS) framework to separate the large resolved scales from the small resolved scales in the evolved velocity, and we use the filtered small scales to correct the large scales. Furthermore, in the new VMS-EF strategy, we use two different ways to decompose the evolved velocity: the VMS Evolve-Filter-Filter-Correct (VMS-EFFC) and the VMS Evolve-Postprocess-Filter-Correct (VMS-EPFC) algorithms. The new VMS-based algorithms yield significantly more accurate results than the standard EF in both the FOM and the ROM simulations of a flow past a cylinder at Reynolds number Re = 1000.

math.NA↗

arXiv2024

Stabilization-Free General Order Virtual Element Methods for Neumann Boundary Optimal Control Problems in Saddle Point Formulation

In this work, we explore the application of Stabilization-Free Virtual Element Methods for Neumann boundary Optimal Control Problems in saddle point formulation. The method is proposed for arbitrary polynomial order of accuracy and general polygonal meshes. Our contribution includes a rigorous a priori error estimate that holds for general polynomial order. On the numerical side, we present (i) an initial convergence test that reflects our theoretical findings, (ii) a second test analyzing the role of the stabilization term in the Virtual Element Method (VEM) formulation and its influence on the approximation error, and (iii) a third test case based on a more application-oriented experiment. The stabilization-free approach is proposed as an alternative strategy to circumvent issues related to the choice of the stabilization parameter in standard VEM formulations.

math.NA↗

arXiv2024

A Streamline upwind Petrov-Galerkin Reduced Order Method for Advection-Dominated Partial Differential Equations under Optimal Control

In this paper we will consider distributed Linear-Quadratic Optimal Control Problems dealing with Advection-Diffusion PDEs for high values of the Péclet number. In this situation, computational instabilities occur, both for steady and unsteady cases. A Streamline Upwind Petrov-Galerkin technique is used in the optimality system to overcome these unpleasant effects. We will apply a finite element method discretization in an optimize-then-discretize approach. Concerning the parabolic case, a stabilized space-time framework will be considered and stabilization will also occur in both bilinear forms involving time derivatives. Then we will build Reduced Order Models on this discretization procedure and two possible settings can be analyzed: whether or not stabilization is needed in the online phase, too. In order to build the reduced bases for state, control, and adjoint variables we will consider a Proper Orthogonal Decomposition algorithm in a partitioned approach. It is the first time that Reduced Order Models are applied to stabilized parabolic problems in this setting. The discussion is supported by computational experiments, where relative errors between the FEM and ROM solutions are studied together with the respective computational times.

math.NA↗

arXiv2023

New Feedback Control and Adaptive Evolve-Filter-Relax Regularization for the Navier-Stokes Equations in the Convection-Dominated Regime

We propose, analyze, and investigate numerically a novel feedback control strategy for high Reynolds number flows. For both the continuous and the discrete (finite element) settings, we prove that the new strategy yields accurate results for high Reynolds numbers that were not covered by current results. We also show that the new feedback control yields more accurate results than the current control approaches in marginally-resolved numerical simulations of a two-dimensional flow past a circular cylinder at Reynolds numbers $Re=1000$. We note, however, that for realistic control parameters, the stabilizing effect of the new feedback control strategy is not sufficient in the convection-dominated regime. Our second contribution is the development of an adaptive evolve-filter-relax (aEFR) regularization that stabilizes marginally-resolved simulations in the convection-dominated regime and increases the accuracy of the new feedback control in realistic parameter settings. For the finite element setting, we prove that the novel feedback control equipped with the new aEFR method yields accurate results for high Reynolds numbers. Furthermore, our numerical investigation shows that the new strategy yields accurate results for reduced order models that dramatically decrease the size of the feedback control problem.

math.NA↗

arXiv2023

POD-based reduced order methods for optimal control problems governed by parametric partial differential equation with varying boundary control

In this work we propose tailored model order reduction for varying boundary optimal control problems governed by parametric partial differential equations. With varying boundary control, we mean that a specific parameter changes where the boundary control acts on the system. This peculiar formulation might benefit from model order reduction. Indeed, fast and reliable simulations of this model can be of utmost usefulness in many applied fields, such as geophysics and energy engineering. However, varying boundary control features very complicated and diversified parametric behaviour for the state and adjoint variables. The state solution, for example, changing the boundary control parameter, might feature transport phenomena. Moreover, the problem loses its affine structure. It is well known that classical model order reduction techniques fail in this setting, both in accuracy and in efficiency. Thus, we propose reduced approaches inspired by the ones used when dealing with wave-like phenomena. Indeed, we compare standard proper orthogonal decomposition with two tailored strategies: geometric recasting and local proper orthogonal decomposition. Geometric recasting solves the optimization system in a reference domain simplifying the problem at hand avoiding hyper-reduction, while local proper orthogonal decomposition builds local bases to increase the accuracy of the reduced solution in very general settings (where geometric recasting is unfeasible). We compare the various approaches on two different numerical experiments based on geometries of increasing complexity.

math.NA↗

arXiv2023

An extended physics informed neural network for preliminary analysis of parametric optimal control problems

In this work we propose an extension of physics informed supervised learning strategies to parametric partial differential equations. Indeed, even if the latter are indisputably useful in many applications, they can be computationally expensive most of all in a real-time and many-query setting. Thus, our main goal is to provide a physics informed learning paradigm to simulate parametrized phenomena in a small amount of time. The physics information will be exploited in many ways, in the loss function (standard physics informed neural networks), as an augmented input (extra feature employment) and as a guideline to build an effective structure for the neural network (physics informed architecture). These three aspects, combined together, will lead to a faster training phase and to a more accurate parametric prediction. The methodology has been tested for several equations and also in an optimal control framework.

cs.LG↗

arXiv2023

Stabilized Weighted Reduced Order Methods for Parametrized Advection-Dominated Optimal Control Problems governed by Partial Differential Equations with Random Inputs

In this work, we analyze Parametrized Advection-Dominated distributed Optimal Control Problems with random inputs in a Reduced Order Model (ROM) context. All the simulations are initially based on a finite element method (FEM) discretization; moreover, a space-time approach is considered when dealing with unsteady cases. To overcome numerical instabilities that can occur in the optimality system for high values of the P\'eclet number, we consider a Streamline Upwind Petrov-Galerkin technique applied in an optimize-then-discretize approach. We combine this method with the ROM framework in order to consider two possibilities of stabilization: Offline-Only stabilization and Offline-Online stabilization. Moreover we consider random parameters and we use a weighted Proper Orthogonal Decomposition algorithm in a partitioned approach to deal with the issue of uncertainty quantification. Several quadrature techniques are used to derive weighted ROMs: tensor rules, isotropic sparse grids, Monte-Carlo and quasi Monte-Carlo methods. We compare all the approaches analyzing relative errors between the FEM and ROM solutions and the computational efficiency based on the speedup-index.

math.NA↗

arXiv2022

Driving bifurcating parametrized nonlinear PDEs by optimal control strategies: application to Navier-Stokes equations with model order reduction

This work deals with optimal control problems as a strategy to drive bifurcating solution of nonlinear parametrized partial differential equations towards a desired branch. Indeed, for these governing equations, multiple solution configurations can arise from the same parametric instance. We thus aim at describing how optimal control allows to change the solution profile and the stability of state solution branches. First of all, a general framework for nonlinear optimal control problem is presented in order to reconstruct each branch of optimal solutions, discussing in detail the stability properties of the obtained controlled solutions. Then, we apply the proposed framework to several optimal control problems governed by bifurcating Navier-Stokes equations in a sudden-expansion channel, describing the qualitative and quantitative effect of the control over a pitchfork bifurcation, and commenting in detail the stability eigenvalue analysis of the controlled state. Finally, we propose reduced order modeling as a tool to efficiently and reliably solve parametric stability analysis of such optimal control systems, which can be challenging to perform with standard discretization techniques such as Finite Element Method.

math.NA↗

arXiv2022

A data-driven partitioned approach for the resolution of time-dependent optimal control problems with dynamic mode decomposition

This work recasts time-dependent optimal control problems governed by partial differential equations in a Dynamic Mode Decomposition with control framework. Indeed, since the numerical solution of such problems requires a lot of computational effort, we rely on this specific data-driven technique, using both solution and desired state measurements to extract the underlying system dynamics. Thus, after the Dynamic Mode Decomposition operators construction, we reconstruct and perform future predictions for all the variables of interest at a lower computational cost with respect to the standard space-time discretized models. We test the methodology in terms of relative reconstruction and prediction errors on a boundary control for a Graetz flow and on a distributed control with Stokes constraints.

math.OC↗

arXiv2022

Consistency of the Full and Reduced Order Models for Evolve-Filter-Relax Regularization of Convection-Dominated, Marginally-Resolved Flows

Numerical stabilization is often used to eliminate (alleviate) the spurious oscillations generally produced by full order models (FOMs) in under-resolved or marginally-resolved simulations of convection-dominated flows. In this paper we investigate the role of numerical stabilization in reduced order models (ROMs) of convection-dominated, marginally-resolved flows. Specifically, we investigate the FOM-ROM consistency, i.e., whether the numerical stabilization is beneficial both at the FOM and the ROM level. As a numerical stabilization strategy, we focus on the evolve-filter-relax (EFR) regularization algorithm, which centers around spatial filtering. To investigate the FOM-ROM consistency, we consider two ROM strategies: (i) the EFR-ROM, in which the EFR stabilization is used at the FOM level, but not at the ROM level; and (ii) the EFR-EFRROM, in which the EFR stabilization is used both at the FOM and at the ROM level. We compare the EFR-ROM with the EFR-EFRROM in the numerical simulation of a 2D flow past a circular cylinder in the convection-dominated, marginally-resolved regime. We also perform model reduction with respect to both time and Reynolds number. Our numerical investigation shows that the EFR-EFRROM is more accurate than the EFR-ROM, which suggests that FOM-ROM consistency is beneficial in convection-dominated, marginally-resolved flows.

physics.flu-dyn↗