arXiv subjects

Tommaso Taddei

Publications and source records attributed to Tommaso Taddei.

At least 19 recordsLinked to original sources

arXiv2026

Parametric vector flows for registration fields in bounded domains with applications to nonlinear interpolation of shock-dominated flows

We present a registration procedure for parametric model order reduction (MOR) in two- and three-dimensional bounded domains. In the MOR framework, registration methods exploit solution snapshots to identify a parametric coordinate transformation that improves the approximation of the solution set through linear subspaces. For each training parameter, optimization-based (or variational) registration methods minimize a target function that measures the alignment of the coherent structures of interest (e.g., shocks, shear layers, cracks) for different parameter values, over a family of bijections of the computational domain $\Omega$. We consider diffeomorphisms $\Phi$ that are vector flows of given velocity fields $v$ with vanishing normal component on $\partial \Omega$; we rely on a sensor to extract appropriate point clouds from the solution snapshots and we develop an expectation-maximization procedure to simultaneously solve the point cloud matching problem and to determine the velocity $v$ (and thus the bijection $\Phi$); finally, we combine our registration method with the nonlinear interpolation technique of [Iollo, Taddei, J. Comput. Phys., 2022] to perform accurate interpolations of fluid dynamic fields in the presence of shocks. Numerical results for a two-dimensional inviscid transonic flow past a NACA airfoil and a three-dimensional viscous transonic flow past an ONERA M6 wing illustrate the many elements of the methodology and demonstrate the effectiveness of nonlinear interpolation for shock-dominated fields.

physics.flu-dyn↗

arXiv2026

Mathematical aspects of registration methods in bounded domains

Registration methods in bounded domains have received significant attention in the model reduction literature, as a valuable tool for nonlinear approximation. The aim of this work is to provide a concise yet complete overview of relevant results for registration methods in $n$-dimensional domains, from the perspective of parametric model reduction. We present a thorough analysis of two classes of methods, vector flows and compositional maps: we discuss the enforcement of the bijectivity constraint and we comment on the approximation properties of the two methods, for Lipschitz $n$-dimensional domains.

math.NA↗

arXiv2025

High-order implicit Runge-Kutta time integrators for component-based model reduction of FSI problems

We propose a model order reduction framework for incompressible fluid-structure interaction (FSI) problems based on high-order implicit Runge-Kutta (IRK) methods. We consider separate reduced spaces for fluid velocity, fluid pressure and solid displacement; we enrich the velocity space with supremizer modes to ensure the inf-sup stability of the fluid subproblem; we consider bubble-port decomposition of fluid velocity and solid displacement to satisfy the kinematic conditions at the fluid structure interface. We resort to Galerkin projection to define the semi-discrete reduced-order model and we consider a Radau-IIA IRK method for time integration: the resulting algebraic system is solved using static condensation of the interface degrees of freedom. The reduced-order model preserves a semi-discrete energy balance inherited from the full-order model, and avoids the need for additional interface enrichment. Numerical experiments demonstrate that the proposed combination of high-order IRK schemes with bubble-port decoupling of velocity and displacement degrees of freedom yields stable and accurate reduced-order model for long-time integration of strongly-coupled parametric FSI problems.

math.NA↗

arXiv2024

Optimization-based model order reduction of fluid-structure interaction problems

We introduce optimization-based full-order and reduced-order formulations of fluid structure interaction problems. We study the flow of an incompressible Newtonian fluid which interacts with an elastic body: we consider an arbitrary Lagrangian Eulerian formulation of the fluid problem and a fully Lagrangian formulation of the solid problem; we rely on a finite element discretization of both fluid and solid equations. The distinctive feature of our approach is an implicit coupling of fluid and structural problems that relies on the solution to a constrained optimization problem with equality constraints. We discuss the application of projection-based model reduction to both fluid and solid subproblems: we rely on Galerkin projection for the solid equations and on least-square Petrov-Galerkin projection for the fluid equations. Numerical results for three model problems illustrate the many features of the formulation.

math.NA↗

arXiv2024

Point-set registration in bounded domains via the Fokker-Planck equation

We present a point set registration method in bounded domains based on the solution to the Fokker Planck equation. Our approach leverages (i) density estimation based on Gaussian mixture models; (ii) a stabilized finite element discretization of the Fokker Planck equation; (iii) a specialized method for the integration of the particles. We review relevant properties of the Fokker Planck equation that provide the foundations for the numerical method. We discuss two strategies for the integration of the particles and we propose a regularization technique to control the distance of the particles from the boundary of the domain. We perform extensive numerical experiments for two two-dimensional model problems to illustrate the many features of the method.

math.NA↗

arXiv2024

Compositional maps for registration in complex geometries

We develop and analyze a parametric registration procedure for manifolds associated with the solutions to parametric partial differential equations in two-dimensional domains. Given the domain $Ω\subset \mathbb{R}^2$ and the manifold $M=\{ u_μ : μ\in P\}$ associated with the parameter domain $P \subset \mathbb{R}^P$ and the parametric field $μ\mapsto u_μ \in L^2(Ω)$, our approach takes as input a set of snapshots from $M$ and returns a parameter-dependent mapping $Φ: Ω\times P \to Ω$, which tracks coherent features (e.g., shocks, shear layers) of the solution field and ultimately simplifies the task of model reduction. We consider mappings of the form $Φ=\texttt{N}(\mathbf{a})$ where $\texttt{N}:\mathbb{R}^M \to {\rm Lip}(Ω; \mathbb{R}^2)$ is a suitable linear or nonlinear operator; then, we state the registration problem as an unconstrained optimization statement for the coefficients $\mathbf{a}$. We identify minimal requirements for the operator $\texttt{N}$ to ensure the satisfaction of the bijectivity constraint; we propose a class of compositional maps that satisfy the desired requirements and enable non-trivial deformations over curved (non-straight) boundaries of $Ω$; we develop a thorough analysis of the proposed ansatz for polytopal domains and we discuss the approximation properties for general curved domains. We perform numerical experiments for a parametric inviscid transonic compressible flow past a cascade of turbine blades to illustrate the many features of the method.

math.NA↗

arXiv2024

Accelerated construction of projection-based reduced-order models via incremental approaches

We present an accelerated greedy strategy for training of projection-based reduced-order models for parametric steady and unsteady partial differential equations. Our approach exploits hierarchical approximate proper orthogonal decomposition to speed up the construction of the empirical test space for least-square Petrov-Galerkin formulations, a progressive construction of the empirical quadrature rule based on a warm start of the non-negative least-square algorithm, and a two-fidelity sampling strategy to reduce the number of expensive greedy iterations. We illustrate the performance of our method for two test cases: a two-dimensional compressible inviscid flow past a LS89 blade at moderate Mach number, and a three-dimensional nonlinear mechanics problem to predict the long-time structural response of the standard section of a nuclear containment building under external loading.

math.NA↗

arXiv2024

Projection-based model order reduction for prestressed concrete with an application to the standard section of a nuclear containment building

We propose a projection-based model order reduction procedure for the ageing of large prestressed concrete structures. Our work is motivated by applications in the nuclear industry, particularly in the simulation of containment buildings. Such numerical simulations involve a multi-modeling approach: a three-dimensional nonlinear thermo-hydro-visco-elastic rheological model is used for concrete; and prestressing cables are described by a one-dimensional linear thermo-elastic behavior. A kinematic linkage is performed in order to connect the concrete nodes and the steel nodes: coincident points in each material are assumed to have the same displacement. We develop an adaptive algorithm based on a Proper Orthogonal Decomposition (POD) in time and greedy in parameter to build a reduced order model (ROM). The nonlinearity of the operator entails that the computational cost of the ROM assembly scales with the size of the high-fidelity model. We develop an hyper-reduction strategy based on empirical quadrature to bypass this computational bottleneck: our approach relies on the construction of a reduced mesh to speed up online assembly costs of the ROM. We provide numerical results for a standard section of a double-walled containment building using a qualified and broadly-used industrial grade finite element solver for structural mechanics (code$\_$aster).

math.NA↗

arXiv2023

A non-overlapping optimization-based domain decomposition approach to component-based model reduction of incompressible flows

We present a component-based model order reduction procedure to efficiently and accurately solve parameterized incompressible flows governed by the Navier-Stokes equations. Our approach leverages a non-overlapping optimization-based domain decomposition technique to determine the control variable that minimizes jumps across the interfaces between sub-domains. To solve the resulting constrained optimization problem, we propose both Gauss-Newton and sequential quadratic programming methods, which effectively transform the constrained problem into an unconstrained formulation. Furthermore, we integrate model order reduction techniques into the optimization framework, to speed up computations. In particular, we incorporate localized training and adaptive enrichment to reduce the burden associated with the training of the local reduced-order models. Numerical results are presented to demonstrate the validity and effectiveness of the overall methodology.

math.NA↗

arXiv2023

Model order reduction by convex displacement interpolation

We present a nonlinear interpolation technique for parametric fields that exploits optimal transportation of coherent structures of the solution to achieve accurate performance. The approach generalizes the nonlinear interpolation procedure introduced in [Iollo, Taddei, J. Comput. Phys., 2022] to multi-dimensional parameter domains and to datasets of several snapshots. Given a library of high-fidelity simulations, we rely on a scalar testing function and on a point set registration method to identify coherent structures of the solution field in the form of sorted point clouds. Given a new parameter value, we exploit a regression method to predict the new point cloud; then, we resort to a boundary-aware registration technique to define bijective mappings that deform the new point cloud into the point clouds of the neighboring elements of the dataset, while preserving the boundary of the domain; finally, we define the estimate as a weighted combination of modes obtained by composing the neighboring snapshots with the previously-built mappings. We present several numerical examples for compressible and incompressible, viscous and inviscid flows to demonstrate the accuracy of the method. Furthermore, we employ the nonlinear interpolation procedure to augment the dataset of simulations for linear-subspace projection-based model reduction: our data augmentation procedure is designed to reduce offline costs -- which are dominated by snapshot generation -- of model reduction techniques for nonlinear advection-dominated problems.

math.NA↗

arXiv2023

Registration-based model reduction of parameterized PDEs with spatio-parameter adaptivity

We propose an automated nonlinear model reduction and mesh adaptation framework for rapid and reliable solution of parameterized advection-dominated problems, with emphasis on compressible flows. The key features of our approach are threefold: (i) a metric-based mesh adaptation technique to generate an accurate mesh for a range of parameters, (ii) a general (i.e., independent of the underlying equations) registration procedure for the computation of a mapping $Φ$ that tracks moving features of the solution field, and (iii) an hyper-reduced least-square Petrov-Galerkin reduced-order model for the rapid and reliable estimation of the mapped solution. We discuss a general paradigm -- which mimics the refinement loop considered in mesh adaptation -- to simultaneously construct the high-fidelity and the reduced-order approximations, and we discuss actionable strategies to accelerate the offline phase. We present extensive numerical investigations for a quasi-1D nozzle problem and for a two-dimensional inviscid flow past a Gaussian bump to display the many features of the methodology and to assess the performance for problems with discontinuous solutions.

math.NA↗

arXiv2022

A projection-based reduced-order model for parametric quasi-static nonlinear mechanics using an open-source industrial code

We propose a projection-based model order reduction procedure for a general class of parametric quasi-static problems in nonlinear mechanics with internal variables. The methodology is integrated in the industrial finite element code code aster. Model order reduction aims to lower the computational cost of engineering studies that involve the simulation to a costly high-fidelity differential model for many different parameters, which correspond, for example to material properties or initial and boundary conditions. We develop an adaptive algorithm based on a POD-Greedy strategy, and we develop an hyper-reduction strategy based on an element-wise empirical quadrature in order to speed up the assembly costs of the reduced-order model by building an appropriate reduced mesh. We introduce a cost-efficient error indicator which relies on the reconstruction of the stress field by a Gappy-POD strategy. We present numerical results for a three-dimensional elastoplastic system in order to illustrate and validate the methodology.

math.NA↗

arXiv2022

A one-shot overlapping Schwarz method for component-based model reduction: application to nonlinear elasticity

We propose a component-based (CB) parametric model order reduction (pMOR) formulation for parameterized nonlinear elliptic partial differential equations (PDEs) based on overlapping subdomains. Our approach reads as a constrained optimization statement that penalizes the jump at the components' interfaces subject to the approximate satisfaction of the PDE in each local subdomain. Furthermore, the approach relies on the decomposition of the local states into a port component -- associated with the solution on interior boundaries -- and a bubble component that vanishes at ports: since the bubble components are uniquely determined by the solution value at the corresponding port, we can recast the constrained optimization statement into an unconstrained statement, which reads as a nonlinear least-squares problem and can be solved using the Gauss-Newton method. We present thorough numerical investigations for a two-dimensional neo-Hookean nonlinear mechanics problem to validate our method; we further discuss the well-posedness of the mathematical formulation and the \emph{a priori} error analysis for linear coercive problems.

math.NA↗

arXiv2022

An optimization-based registration approach to geometry reduction

We develop and assess an optimization-based approach to parametric geometry reduction. Given a family of parametric domains, we aim to determine a parametric diffeomorphism $Φ$ that maps a fixed reference domain $Ω$ into each element of the family, for different values of the parameter; the ultimate goal of our study is to determine an effective tool for parametric projection-based model order reduction of partial differential equations in parametric geometries. For practical problems in engineering, explicit parameterizations of the geometry are likely unavailable: for this reason, our approach takes as inputs a reference mesh of $Ω$ and a point cloud $\{y_i^{\rm raw}\}_{i=1}^Q$ that belongs to the boundary of the target domain $V$ and returns a bijection $Φ$ that approximately maps $Ω$ in $V$. We propose a two-step procedure: given the point clouds $\{x_j\}_{j=1}^N\subset \partial Ω$ and $\{y_i^{\rm raw}\}_{i=1}^Q \subset \partial V$, we first resort to a point-set registration algorithm to determine the displacements $\{ v_j \}_{j=1}^N$ such that the deformed point cloud $\{y_j:= x_j+v_j \}_{j=1}^N$ approximates $\partial V$; then, we solve a nonlinear non-convex optimization problem to build a mapping $Φ$ that is bijective from $Ω$ in $\mathbb{R}^d$ and (approximately) satisfies $Φ(x_j) = y_j$ for $j=1,\ldots,N$.We present a rigorous mathematical analysis to justify our approach; we further present thorough numerical experiments to show the effectiveness of the proposed method.

math.NA↗

arXiv2022

Mapping of coherent structures in parameterized flows by learning optimal transportation with Gaussian models

We present a general (i.e., independent of the underlying model) interpolation technique based on optimal transportation of Gaussian models for parametric advection-dominated problems. The approach relies on a scalar testing function to identify the coherent structure we wish to track; a maximum likelihood estimator to identify a Gaussian model of the coherent structure; and a nonlinear interpolation strategy that relies on optimal transportation maps between Gaussian distributions. We show that well-known self-similar solutions can be recast in the frame of optimal transportation by appropriate rescaling; we further present several numerical examples to motivate our proposal and to assess strengths and limitations; finally, we discuss an extension to deal with more complex problems.

math.NA↗

arXiv2022

Localized model reduction for nonlinear elliptic partial differential equations: localized training, partition of unity, and adaptive enrichment

We propose a component-based (CB) parametric model order reduction (pMOR) formulation for parameterized {nonlinear} elliptic partial differential equations (PDEs). CB-pMOR is designed to deal with large-scale problems for which full-order solves are not affordable in a reasonable time frame or parameters' variations induce topology changes that prevent the application of monolithic pMOR techniques. We rely on the partition-of-unity method (PUM) to devise global approximation spaces from local reduced spaces, and on Galerkin projection to compute the global state estimate. We propose a randomized data compression algorithm based on oversampling for the construction of the components' reduced spaces: the approach exploits random boundary conditions of controlled smoothness on the oversampling boundary. We further propose an adaptive residual-based enrichment algorithm that exploits global reduced-order solves on representative systems to update the local reduced spaces. We prove exponential convergence of the enrichment procedure for linear coercive problems; we further present numerical results for a two-dimensional nonlinear diffusion problem to illustrate the many features of our proposal and demonstrate its effectiveness.

math.NA↗

arXiv2021

A projection-based model reduction method for nonlinear mechanics with internal variables: application to thermo-hydro-mechanical systems

We propose a projection-based monolithic model order reduction (MOR) procedure for a class of problems in nonlinear mechanics with internal variables. The work is is motivated by applications to thermo-hydro-mechanical (THM) systems for radioactive waste disposal. THM equations model the behaviour of temperature, pore water pressure and solid displacement in the neighborhood of geological repositories, which contain radioactive waste and are responsible for a significant thermal flux towards the Earth's surface. We develop an adaptive sampling strategy based on the POD-Greedy method, and we develop an element-wise empirical quadrature hyper-reduction procedure to reduce assembling costs. We present numerical results for a two-dimensional THM system to illustrate and validate the proposed methodology.

math.NA↗

arXiv2021

Registration-based model reduction of parameterized two-dimensional conservation laws

We propose a nonlinear registration-based model reduction procedure for rapid and reliable solution of parameterized two-dimensional steady conservation laws. This class of problems is challenging for model reduction techniques due to the presence of nonlinear terms in the equations and also due to the presence of parameter-dependent discontinuities that cannot be adequately represented through linear approximation spaces. Our approach builds on a general (i.e., independent of the underlying equation) registration procedure for the computation of a mapping $Φ$ that tracks moving features of the solution field and on an hyper-reduced least-squares Petrov-Galerkin reduced-order model for the rapid and reliable computation of the solution coefficients. The contributions of this work are twofold. First, we investigate the application of registration-based methods to two-dimensional hyperbolic systems. Second, we propose a multi-fidelity approach to reduce the offline costs associated with the construction of the parameterized mapping and the reduced-order model. We discuss the application to an inviscid supersonic flow past a parameterized bump, to illustrate the many features of our method and to demonstrate its effectiveness.

math.NA↗