Gianluigi Rozza

• PhD Mathematics EPFL, MS Aerospace Eng. PoliMI
• Professor (Full) at International School for Advanced Studies

It is well known that in the computational fluid dynamics simulations related to the cardiovascular system the enforcement of outflow boundary conditions is a crucial point. In fact, they highly affect the computed flow and a wrong setup could lead to unphysical results. In this chapter we discuss the main features of two different ways for the est...

Numerical simulations of turbulent flows are well known to pose extreme computational challenges because of the huge number of dynamical degrees of freedom required to correctly describe the complex multiscale statistical correlations of the velocity. On the other hand, kinetic mesoscale approaches based on the Boltzmann equation, have the potentia...

This article provides a reduced-order modelling framework for turbulent compressible flows discretized by the use of finite volume approaches. The basic idea behind this work is the construction of a reduced-order model capable of providing closely accurate solutions with respect to the high fidelity flow fields. Full-order solutions are often obta...

This work investigates model reduction techniques for nonlinear parameterized and time-dependent PDEs, specifically focusing on bifurcating phenomena in Computational Fluid Dynamics (CFD). We develop interpretable and non-intrusive Reduced Order Models (ROMs) capable of capturing dynamics associated with bifurcations by identifying a minimal set of...

Autoregressive and recurrent networks have achieved remarkable progress across various fields, from weather forecasting to molecular generation and Large Language Models. Despite their strong predictive capabilities, these models lack a rigorous framework for addressing uncertainty, which is key in scientific applications such as PDE solving, molec...

This paper deals with the development of a Reduced-Order Model (ROM) to investigate haemodynamics in cardiovascular applications. It employs the use of Proper Orthogonal Decomposition (POD) for the computation of the basis functions and the Galerkin projection for the computation of the reduced coefficients. The main novelty of this work lies in th...

This work presents an overview of several nonlinear reduction strategies for data compression from various research fields, and a comparison of their performance when applied to problems characterized by diffusion and/or advection terms. We aim to create a common framework by unifying the notation referring to a common two-stage pipeline. At the sa...

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...

Traditional linear approximation methods, such as proper orthogonal decomposition and the reduced basis method, are ineffective for transport-dominated problems due to the slow decay of the Kolmogorov $n$-width. This results in reduced-order models that are both inefficient and inaccurate. In this work, we present an approach for the model reductio...

Patient-specific modeling of cardiovascular flows with high-fidelity is challenging due to its dependence on accurately estimated velocity boundary profiles, which are essential for precise simulations and directly influence wall shear stress calculations - key in predicting cardiovascular diseases like atherosclerosis. This data, often derived fro...

In this work, we focus on the early design phase of cruise ship hulls, where the designers are tasked with ensuring the structural resilience of the ship against extreme waves while reducing steel usage and respecting safety and manufacturing constraints. The ship's geometry is already finalized and the designer can choose the thickness of the prim...

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 thes...

In this paper, we introduce a data-driven filter to analyze the relationship between Implicit Large-Eddy Simulations (ILES) and Direct Numerical Simulations (DNS) in the context of the Spectral Difference method. The proposed filter is constructed from a linear combination of sharp-modal filters where the weights are given by a convolutional neural...

This Special Issue comprises the second collection of papers submitted by both the Editorial Board Members (EBMs) of the journal Mathematical and Computational Applications (MCA) and the outstanding scholars working in the core research fields of MCA [...]

This paper presents a projection-based reduced order modelling (ROM) framework for unsteady parametrized optimal control problems (OCP$_{(\mu)}$s) arising from cardiovascular (CV) applications. In real-life scenarios, accurately defining outflow boundary conditions in patient-specific models poses significant challenges due to complex vascular morp...

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...

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 insta...

The complexity of the cardiovascular system needs to be accurately reproduced in order to promptly acknowledge health conditions; to this aim, advanced multifidelity and multiphysics numerical models are crucial. On one side, Full Order Models (FOMs) deliver accurate hemodynamic assessments, but their high computational demands hinder their real-ti...

In this work, we present the modelling and numerical simulation of a molten glass fluid flow in a furnace melting basin. We first derive a model for a molten glass fluid flow and present numerical simulations based on the finite element method (FEM). We further discuss and validate the results obtained from the simulations by comparing them with ex...

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 th...

Incorporating probabilistic terms in mathematical models is crucial for capturing and quantifying uncertainties of real-world systems. However, stochastic models typically require large computational resources to produce meaningful statistics. For such reason, the development of reduction techniques becomes essential for enabling efficient and scal...

The Sparse Identification of Nonlinear Dynamics (SINDy) framework is a robust method for identifying governing equations, successfully applied to ordinary, partial, and stochastic differential equations. In this work we extend SINDy to identify delay differential equations by using an augmented library that includes delayed samples and Bayesian opt...

Advection-dominated problems are commonly noticed in nature, engineering systems, and a wide range of industrial processes. For these problems, linear approximation methods (proper orthogonal decomposition and reduced basis method) are not suitable, as the Kolmogorov $n$-width decay is slow, leading to inefficient and inaccurate reduced order model...

In this manuscript, we combine non-intrusive reduced-order models (ROMs) with space-dependent aggregation techniques to build a mixed-ROM , able to accurately capture the flow dynamics in different physical settings. The flow prediction obtained using the mixed formulation is derived from a convex combination of the predictions of several previousl...

Using Domain Decomposition (DD) algorithm on non--overlapping domains, we compare couplings of different discretisation models, such as Finite Element (FEM) and Reduced Order (ROM) models for separate subcomponents. In particular, we consider an optimisation-based DD model where the coupling on the interface is performed using a control variable re...

In this work, we address parametric non–stationary fluid dynamics problems within a model order reduction setting based on domain decomposition. Starting from the optimisation–based domain decomposition approach, we derive an optimal control problem, for which we present a convergence analysis in the case of non–stationary incompressible Navier–Sto...

This work aims to introduce a heuristic timestep-adaptive algorithm for Computational Fluid Dynamics (CFD) and Fluid-Structure Interaction (FSI) problems where the flow is dominated by the pressure. In such scenarios, many time-adaptive algorithms based on the interplay of implicit and explicit time schemes fail to capture the fast transient dynami...

In this manuscript we propose and analyze weighted reduced order methods for stochastic Stokes and Navier-Stokes problems depending on random input data (such as forcing terms, physical or geometrical coefficients, boundary conditions). We will compare weighted methods such as weighted greedy and weighted POD with non-weighted ones in case of stoch...

The purpose of this work is to investigate the inf-sup stability of reduced basis (RB) method applied to parametric Stokes problem. While performing the Galerkin projection on the reduced space, the inf-sup approximation stability has always been a challenge for the RB community, even if the construction of reduced basis is done using a stable high...

This chapter focuses on the combination of reduced order models and data-driven techniques applied to the study of turbulent flows in order to improve the pressure and velocity accuracy of standard reduced order methods. We focus on reduced order models constructed by means of Proper Orthogonal Decomposition with Galerkin approach, enhanced with tw...

In this paper, we discuss reduced order modelling approaches to bifurcating systems arising from continuum mechanics benchmarks. The investigation of the beam’s deflection is a relevant topic of investigation with fundamental implications on their design for structural analysis and health. When the beams are exposed to external forces, their equili...

This study introduces a first step for constructing a hybrid reduced-order models (ROMs) for segregated fluid-structure interaction in an Arbitrary Lagrangian-Eulerian (ALE) approach at a high Reynolds number using the Finite Volume Method (FVM). The ROM is driven by proper orthogonal decomposition (POD) with hybrid techniques that combines the cla...

In this paper, we propose an equation-based parametric Reduced Order Model (ROM), whose accuracy is improved with data-driven terms added into the reduced equations. These additions have the aim of reintroducing contributions that in standard ROMs are not taken into account. In particular, in this work we consider two types of contributions: the tu...

Reduced Order Models (ROMs) have gained a great attention by the scientific community in the last years thanks to their capabilities of significantly reducing the computational cost of the numerical simulations, which is a crucial objective in applications like real time control and shape optimization. This contribution aims to provide a brief over...

In this chapter, we consider a scalar advection-diffusion PDE modeling the Graetz flow problem in a two-dimensional and geometrically parametrized domain. The aim is to obtain efficient evaluations of the thermal field and the output of interest, defined as the integral of the temperature over the outflow boundary. The solution is approximated usin...

In this chapter, we consider a vector elliptic PDE modeling a parametric linear elasticity problem in a two-dimensional heterogeneous domain. The aim is to obtain efficient evaluations of the vector displacement field and the output of interest, defined as the integrated horizontal displacement over the loaded boundary. The solution is approximated...

In this chapter, we introduce the readers to the main notions regarding the Reduced Basis approximation based on Finite Element method for parametrized Partial Differential Equations. We recall definitions and techniques useful to understand and analyze the application of Reduced Order Modeling to the worked out problems discussed in the chapters o...

In this chapter, we consider a scalar elliptic PDE modeling a parametric steady heat conduction problem in a two-dimensional domain. The aim is to obtain efficient evaluations of the thermal field and the output of interest, defined as the average temperature over the boundary. The solution is approximated using reduced order modeling techniques ba...

In this chapter, we consider a scalar elliptic PDE modeling a steady-state heat conduction problem in a two-dimensional and geometrically parametrized holed domain. The aim is to obtain efficient evaluations of the thermal field and the output of interest, defined as the average conduction temperature distribution at the inner walls. The solution i...

A data-driven reduced order model (ROM) based on a proper orthogonal decomposition-radial basis function (POD-RBF) approach is adopted in this paper for the analysis of blood flow dynamics in a patient-specific case of atrial fibrillation (AF). The full order model (FOM) is represented by incompressible Navier–Stokes equations, discretized with a f...

Parameter space reduction has been proved to be a crucial tool to speed-up the execution of many numerical tasks such as optimization, inverse problems, sensitivity analysis, and surrogate models’ design, especially when in presence of high-dimensional parametrized systems. In this work we propose a new method called local active subspaces (LAS), w...

The proper orthogonal decomposition (POD) has been applied on a full-scale horizontal-axis wind turbine (HAWT) to shed light on the wake characteristics behind the wind turbine. In reality, the blade tip experiences high deflections even at the rated conditions which definitely alter the wake flow field, and in the case of a wind farm, may complica...

This chapter provides an extended overview about Reduced Order Models (ROMs), with a focus on their features in terms of efficiency and accuracy. In particular, the aim is to browse the more common ROM frameworks, considering both intrusive and data-driven approaches. We present the validation of such techniques against several test cases. The firs...

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 un...

The article presents the application of inductive graph machine learning surrogate models for accurate and efficient prediction of 3D flow for industrial geometries, explicitly focusing here on external aerodynamics for a motorsport case. The final aim is to build a surrogate model that can provide quick predictions, bypassing in this way the unfea...

In this work, we present GAROM, a new approach for reduced order modeling (ROM) based on generative adversarial networks (GANs). GANs attempt to learn to generate data with the same statistics of the underlying distribution of a dataset, using two neural networks, namely discriminator and generator. While widely applied in many areas of deep learni...

In this article, we propose a shape optimization pipeline for propeller blades, applied to naval applications. The geometrical features of a blade are exploited to parametrize it, allowing to obtain deformed blades by perturbating their parameters. The optimization is performed using a genetic algorithm that exploits the computational speed‐up of r...

Parallel to the need for new technologies and renewable energy resources to address sustainability, the emerging field of Artificial Intelligence (AI) has experienced continuous high-speed growth in the application of its capabilities of modelling, managing, processing, and making sense of data in the entire areas related to the production and mana...

The simulation of atmospheric flows by means of traditional discretization methods remains computationally intensive, hindering the achievement of high forecasting accuracy in short time frames. In this paper, we apply three reduced order models that have successfully reduced the computational time for different applications in computational fluid...

This work addresses the development of a physics-informed neural network (PINN) with a loss term derived from a discretized time-dependent reduced-order system. In this work, first, the governing equations are discretized using a finite difference scheme (whereas, any other discretization technique can be adopted), then projected on a reduced or la...

In this chapter, we consider a vector elliptic PDE modeling a linear elasticity problem for bridge designing in a two-dimensional and geometrically parametrized domain. The aim is to obtain efficient evaluations of the vector displacement field and the output of interest, defined as the integrated vertical displacement over the loaded boundary. The...

In this chapter, we consider a scalar advection-diffusion PDE modeling a stabilized advection dominated flow problem in a two-dimensional domain. The aim is to obtain efficient evaluations of the unknown field for high Péclet number and the output of interest, defined as the average temperature over the boundary. The solution is approximated using...

In this chapter, we consider a parabolic PDE modeling the unsteady conduction problem in a two-dimensional domain. The aim is to obtain efficient evaluations of the time evolution of the thermal field and the output of interest, defined as the average temperature over the boundary. The solution is approximated using reduced order modeling technique...

In this chapter, we consider a vector steady nonlinear PDE modeling the bifurcating Coanda effect for an incompressible flow in a two-dimensional channel. The aim is to obtain efficient evaluations of the velocity and pressure fields and the output of interest, defined as the integral of vertical velocity, which is related to the symmetry breaking...

In this chapter, we consider a scalar elliptic PDE modeling the steady heat conduction problem with a non-affine Gaussian flux in a two-dimensional domain. The aim is to obtain efficient evaluations of the thermal field and the output of interest, defined as the integral of the source term. The solution is approximated using reduced order modeling...

In this chapter, we consider a vector time-dependent nonlinear PDE modeling the unsteady Navier-Stokes system for an incompressible flow in a two-dimensional domain with a cylindrical obstacle. The aim is to obtain efficient evaluations of the velocity and pressure fields and the output of interest, defined as the average of the solution over the w...

In this chapter, we consider a linear-quadratic optimal control problem with a vector elliptic PDE constraint in a two-dimensional and geometrically parametrized domain. The aim is to obtain efficient evaluations of the optimal temperature distribution, the adjoint, the optimal control field and the output of interest, defined as the cost functiona...

In this chapter, we consider a vector nonlinear parabolic PDE modeling the unsteady FitzHugh–Nagumo system in a one-dimensional domain. The aim is to obtain efficient evaluations of the unknown fields describing the voltage and the recovery voltage. The solution is approximated using reduced order modeling techniques based on the POD-Galerkin metho...

In this chapter, we consider a vector elliptic PDE modeling a contact problem in linear elasticity with friction in a two-dimensional and geometrically parametrized domain. The aim is to obtain efficient evaluations of the vector displacement field and the output of interest, defined as the integrated vertical displacement over the loaded boundary....

In this chapter, we consider a vector elliptic stochastic PDE modeling a steady heat conduction in a two-dimensional heterogeneous domain based on random input data. The aim is to obtain efficient evaluation of the statistics for the thermal field and the output of interest, defined as the average temperature over the whole domain. The solution is...

In this chapter, we consider a scalar nonlinear elliptic PDE as a benchmark for complex material problems in a two-dimensional domain. The aim is to obtain efficient evaluations of the unknown field and the output of interest, defined as its average over the whole domain. The solution is approximated using reduced order modeling techniques based on...

In this chapter, we consider a vector steady nonlinear PDE modeling the Navier–Stokes system for an incompressible flow in a two-dimensional domain. The aim is to obtain efficient evaluations of the velocity and pressure fields and the output of interest, defined as the average of the solution over the whole domain. The solution is approximated usi...

In this chapter, we consider a scalar elliptic PDE modeling the steady-state heat transfer problem through a two-dimensional geometrically parametrized fin. The aim is to obtain efficient evaluations of the thermal field and the output of interest, defined as a measure of its dissipative capabilities. The solution is approximated using reduced orde...

In this poster, we present a computational approach tailored for biomedical applications, leveraging parametrized optimal control problems (OCP_μ) and reduction techniques. Our method involves solving unsteady Navier-Stokes (NS) equations using a Galerkin finite element approach, with specified initial and/or boundary conditions, while minimizing a...

We introduce GEA (Geophysical and Environmental Applications), a new open-source atmosphere and ocean modeling framework within the finite volume C++ library OpenFOAM®. Here, we present the development of a non-hydrostatic atmospheric model consisting of a pressure-based solver for the Euler equations written in conservative form using density, mom...

The aim of this work is to present a model reduction technique in the framework of optimal control problems for partial differential equations. We combine two approaches used for reducing the computational cost of the mathematical numerical models: domain–decomposition (DD) methods and reduced–order modelling (ROM). In particular, we consider an op...

To shed light on the effect of the icing phenomenon on the vertical-axis wind turbine (VAWT) wake characteristics, we present a high-fidelity computational fluid dynamics simulation of the flow field of H-Darrieus turbine under the icing conditions. To address continuous geometry alteration due to the icing and predefined motion of the VAWT, a pseu...

Multi‐fidelity models are of great importance due to their capability of fusing information coming from different numerical simulations, surrogates, and sensors. We focus on the approximation of high‐dimensional scalar functions with low intrinsic dimensionality. By introducing a low dimensional bias we can fight the curse of dimensionality affecti...

Reduced order models (ROMs) are widely used in scientific computing to tackle high-dimensional systems. However, traditional ROM methods may only partially capture the intrinsic geometric characteristics of the data. These characteristics encompass the underlying structure, relationships, and essential features crucial for accurate modeling. To ov...

Reduced order models (ROMs) are widely used in scientific computing to tackle high-dimensional systems. However, traditional ROM methods may only partially capture the intrinsic geometric characteristics of the data. These characteristics encompass the underlying structure, relationships, and essential features crucial for accurate modeling. To ov...

In the context of simulation‐based methods, multiple challenges arise, two of which are considered in this work. As a first challenge, problems including time‐dependent phenomena with complex domain deformations, potentially even with changes in the domain topology, need to be tackled appropriately. The second challenge arises when computational re...

A slow decaying Kolmogorov n-width of the solution manifold of a parametric partial differential equation precludes the realization of efficient linear projection-based reduced-order models. This is due to the high dimensionality of the reduced space needed to approximate with sufficient accuracy the solution manifold. To solve this problem, neural...

Real-world applications of computational fluid dynamics often involve the evaluation of quantities of interest for several distinct geometries that define the computational domain or are embedded inside it. For example, design optimization studies require the realization of response surfaces from the parameters that determine the geometrical deform...

Friedrichs' systems (FS) are symmetric positive linear systems of first-order partial differential equations (PDEs), which provide a unified framework for describing various elliptic, parabolic and hyperbolic semi-linear PDEs such as the linearized Euler equations of gas dynamics, the equations of compressible linear elasticity and the Dirac-Klein-...

In this work, we address parametric non-stationary fluid dynamics problems within a model order reduction setting based on domain decomposition. Starting from the domain decomposition approach, we derive an optimal control problem, for which we present the convergence analysis. The snapshots for the high-fidelity model are obtained with the Finite...

In this work, we present the modelling and numerical simulation of a molten glass fluid flow in a furnace melting basin. We first derive a model for a molten glass fluid flow and present numerical simulations based on the Finite Element Method (FEM). We further discuss and validate the results obtained from the simulations by comparing them with ex...

In the present work, we introduce a novel approach to enhance the precision of reduced order models by exploiting a multi-fidelity perspective and DeepONets. Reduced models provide a real-time numerical approximation by simplifying the original model. The error introduced by the such operation is usually neglected and sacrificed in order to reach a...