arXiv subjects

Mattia Manucci

Publications and source records attributed to Mattia Manucci.

11 recordsLinked to original sources

arXiv2026

Contour integral methods and model order reduction for parametric linear control systems

This paper introduces a contour integral method (CIM) for efficiently computing outputs of parametric linear systems in control form over specified time intervals and to a user-prescribed accuracy. The CIM approximates the inverse Laplace transform via a quadrature rule applied along a modified integration contour. For parametric systems, we show how CIM integrates effectively with projection-based model order reduction (MOR) where a greedy algorithm builds the projection spaces following an error estimate we derive for this setting. We additionally demonstrate that the developed projection framework naturally enforces Hermite interpolation conditions. This combination substantially lowers the cost of evaluating the input-output relations across the parameter domain, for a wide range of input functions, and for initial conditions well captured by a low-dimensional subspace.. We demonstrate the accuracy and efficiency of the approach on benchmark non-parametric and parametric control systems, comparing against state-of-the-art projection-based MOR methods.

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arXiv2026

Contour integral methods and structured perturbations for linear differential-algebraic equations

We generalize the contour integral methods (CIM) framework to the time integration of linear dynamical systems that are subject to algebraic constraints at all times during their evolution. The proposed approach relies on applying the Laplace transform to the Cauchy problem associated with a linear system of differential-algebraic equations (DAE), and subsequently reconstructing the time-domain solution by approximating the inverse Laplace transform via a suitable quadrature rule. This procedure yields an efficient and accurate alternative to classical Runge-Kutta schemes, which are well known to exhibit order reduction in accuracy when applied to DAE. In the second part of the paper, we address linear parametric DAE and propose an efficient strategy for tuning the integration contour in the CIM framework using suitable structured-unstructured pseudospectral computations. This allows the identification of a single integration profile capable of approximating an entire family of parametric solutions, thereby facilitating the efficient application of model order reduction techniques. Finally, numerical experiments are presented to validate the proposed methodology and support the theoretical findings.

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arXiv2026

Model Reduction for Switched Linear Systems via Generalized Lyapunov Equations

In this work, we study projection-based model order reduction (MOR) for switched linear systems (SLS) in control form, where the projection matrices are obtained from the solutions of generalized Lyapunov equations (GLEs). We investigate how numerical inaccuracies in solving the GLEs propagate through the MOR process and impact the accuracy and reliability of the resulting reduced-order model. This highlights the importance of accounting for such inaccuracies, motivating the introduction of a novel error bound to quantify and control the error in the approximation of the GLE solution. Moreover, classical balanced truncation error estimates for SLS are neither theoretically sound nor practically applicable, as they rely on restrictive assumptions requiring several linear matrix inequalities (LMIs) to be satisfied exactly by numerically computed GLE solutions. To address these limitations, we propose a new MOR framework for SLS, termed piecewise balanced reduction (PBR). The approach is based on solving multiple GLEs and constructing projection matrices that are piecewise constant in time. By extending the standard balanced truncation error bound for SLS, we show that the PBR framework effectively controls errors arising from inexact LMI satisfaction. In addition, the proposed error bound captures the influence of the piecewise constant in time projection matrices. Altogether, this makes the PBR approach applicable to a broad and flexible class of switched linear systems. Numerical experiments are presented to support the theoretical results.

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arXiv2025

Certified Model Order Reduction for parametric Hermitian eigenproblems

This article deals with the efficient and certified numerical approximation of the smallest eigenvalue and the associated eigenspace of a large-scale parametric Hermitian matrix. For this aim, we rely on projection-based model order reduction (MOR), i.e., we approximate the large-scale problem by projecting it onto a suitable subspace and reducing it to one of a much smaller dimension. Such a subspace is constructed by means of weak greedy-type strategies. After detailing the connections with the reduced basis method for source problems, we introduce a novel error estimate for the approximation error related to the eigenspace associated with the smallest eigenvalue. Since the difference between the second smallest and the smallest eigenvalue, the so-called spectral gap, is crucial for the reliability of the error estimate, we propose efficiently computable upper and lower bounds for higher eigenvalues and for the spectral gap, which enable the assembly of a subspace for the MOR approximation of the spectral gap. Based on that, a second subspace is then generated for the MOR approximation of the eigenspace associated with the smallest eigenvalue. We also provide efficiently computable conditions to ensure that the multiplicity of the smallest eigenvalue is fully captured in the reduced space. This work is motivated by a specific application: the repeated identifications of the states with minimal energy, the so-called ground states, of parametric quantum spin system models.

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arXiv2024

Certified Model Predictive Control for Switched Evolution Equations using Model Order Reduction

We present a model predictive control (MPC) framework for linear switched evolution equations arising from a parabolic partial differential equation (PDE). First-order optimality conditions for the resulting finite-horizon optimal control problems are derived. The analysis allows for the incorporation of convex control constraints and sparse regularization. Then, to mitigate the computational burden of the MPC procedure, we employ Galerkin reduced-order modeling (ROM) techniques to obtain a low-dimensional surrogate for the state-adjoint systems. We derive recursive a-posteriori estimates for the ROM feedback law and the ROM-MPC closed-loop state and show that the ROM-MPC trajectory evolves within a neighborhood of the true MPC trajectory, whose size can be explicitly computed and is controlled by the quality of the ROM. Such estimates are then used to formulate two ROM-MPC algorithms with closed-loop certification.

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arXiv2024

Uniform Approximation of Eigenproblems of a Large-Scale Parameter-Dependent Hermitian Matrix

We consider the uniform approximation of the smallest eigenvalue of a large parameter-dependent Hermitian matrix by that of a smaller counterpart obtained through projections. The projection subspaces are constructed iteratively by means of a greedy strategy; at each iteration the parameter where a surrogate error is maximal is computed and the eigenvectors associated with the smallest eigenvalues at the maximizing parameter value are added to the subspace. Unlike the classical approaches, such as the successive constraint method, that maximize such surrogate errors over a discrete and finite set, we maximize the surrogate error over the continuum of all permissible parameter values globally. We formally prove that the projected eigenvalue function converges to the actual eigenvalue function uniformly. In the second part, we focus on the uniform approximation of the smallest singular value of a large parameter-dependent matrix, in case it is non-Hermitian. The proposed frameworks on numerical examples, including those arising from discretizations of parametric PDEs, reduce the size of the large matrix-valued function drastically, while retaining a high accuracy over all permissible parameter values.

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arXiv2024

Reachable and observable sets for switched systems via generalized Lyapunov equations: application to switched descriptor systems

In a recent work [Manucci, Unger, ArXiv e-print 2404.10511, 2024], the authors propose using two generalized Lyapunov equations (GLEs) to derive a balancing-based model order reduction~(MOR) method for a general class of switched differential-algebraic equations (DAEs). This work explains why these GLEs provide solutions suitable for MOR by showing that the image set of the solutions of the two GLEs always encloses the reachable and observable set of a suitably defined switched system with the same input to output map of the switched DAE system.

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arXiv2024

Balancing-based model reduction for switched descriptor systems

We propose a novel projection-based model order reduction (MOR) algorithm for a broad class of switched linear descriptor systems. Our approach integrates the reformulation strategy of [Hossain & Trenn, DAE Panel, 2024], which converts the switched descriptor system into a switched ordinary differential equation featuring state- and input-dependent jumps as well as impulsive components at the switching instants in the output, with the piecewise balanced truncation framework for switched linear systems introduced in [Manucci & Unger, arXiv:2601.13039, 2026]. The central idea is to additionally reformulate the switched system with input jumps and impulsive output as a standard switching system that exhibits only state-dependent jumps, while appropriately augmenting the input and output dimensions to preserve the original input-output behavior. Based on this reformulation, we develop new stopping criteria for the stationary iterative scheme employed to approximate the solution of a generalized Lyapunov equation, a key ingredient for the computation of the projection matrices in our MOR procedure. Finally, numerical experiments are conducted to demonstrate and assess the effectiveness of the proposed reduction methodology.

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arXiv2022

Model order reduction in contour integral methods for parametric PDEs

In this paper we discuss a projection model order reduction (MOR) method for a class of parametric linear evolution PDEs, which is based on the application of the Laplace transform. The main advantage of this approach consists in the fact that, differently from time stepping methods, like Runge-Kutta integrators, the Laplace transform allows to compute the solution directly at a given instant, which can be done by approximating the contour integral associated to the inverse Laplace transform by a suitable quadrature formula. In terms of some classical MOR methodology, this determines a significant improvement in the reduction phase - like the one based on the classical proper orthogonal decomposition (POD) - since the number of vectors to which the decomposition applies is drastically reduced as it does not contain all intermediate solutions generated along an integration grid by a time stepping method. We show the effectiveness of the method by some illustrative parabolic PDEs arising from finance and also provide some evidence that the method we propose, when applied to a linear advection equation, does not suffer the problem of slow decay of singular values which instead affects time stepping methods for the numerical approximation of the Cauchy problem arising from space discretization.

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arXiv2021

Sparse data-driven quadrature rules via $\ell^p$-quasi-norm minimization

In this paper we show the use of the focal underdetermined system solver to recover sparse empirical quadrature rules for parametrized integrals from existing data, consisting of the values of given parametric functions sampled on a discrete set of points. This algorithm, originally proposed for image and signal reconstruction, relies on an approximated $\ell^p$-quasi-norm minimization. The choice of $0<p<1$ fits the nature of the constraints to which quadrature rules are subject, thus providing a more natural formulation for sparse quadrature recovery compared to the one based on $\ell^1$-norm minimization. We also extend an a priori error estimate available for the $\ell^1$-norm formulation by considering the error resulting from data compression. Finally, we present two numerical examples to illustrate some practical applications. The first concerns the fundamental solution of the linear 1D Schrödinger equation, the second example deals with the hyper-reduction of a partial differential equation modelling a nonlinear diffusion process in the framework of the reduced basis method. For both the examples we compare our method with the one based on $\ell^1$-norm minimization and the one relaying on the use of the non-negative least square method. Matlab codes related to the numerical examples and the algorithms described are provided.

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arXiv2021

Pseudospectral roaming contour integral methods for convection-diffusion equations

We generalize ideas in the recent literature and develop new ones in order to propose a general class of contour integral methods for linear convection-diffusion PDEs and in particular for those arising in finance. These methods aim to provide a numerical approximation of the solution by computing its inverse Laplace transform. The choice of the integration contour is determined by the computation of a few suitably weighted pseudo-spectral level sets of the leading operator of the equation. Parabolic and hyperbolic profiles proposed in the literature are investigated and compared to the elliptic contour originally proposed by Guglielmi, López-Fernández and Nino. In summary, the article (i) provides a comparison among three different integration profiles; (ii) proposes a new fast pseudospectral roaming method; (iii) optimizes the selection of time windows on which one may arbitrarily approximate the solution by no extra computational cost with respect to the case of a fixed time instant; (iv) focuses extensively on computational aspects and it is the reference of the MATLAB code https://github.com/MattiaManucci/Contour_Integral_Methods.git, where all algorithms described here are implemented.

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