Nicola Guglielmi

Eleonora Andreotti, Dominik Edelmann, Nicola Guglielmi et al.

Given a connected undirected weighted graph, we are concerned with problems related to partitioning the graph. First of all we look for the closest disconnected graph (the minimum cut problem), here with respect to the Euclidean norm. We are interested in the case of constrained minimum cut problems, where constraints include cardinality or membership requirements, which leads to NP-hard combinatorial optimization problems. Furthermore, we are interested in ambiguity issues, that is in the robustness of clustering algorithms that are based on Fiedler spectral partitioning. The above-mentioned problems are restated as matrix nearness problems for the weight matrix of the graph. A key element in the solution of these matrix nearness problems is the use of a constrained gradient system of matrix differential equations.

Paolo Buttà, Nicola Guglielmi, Silvia Noschese

The computation of the structured pseudospectral abscissa and radius (with respect to the Frobenius norm) of a Toeplitz matrix is discussed and two algorithms based on a low rank property to construct extremal perturbations are presented. The algorithms are inspired by those considered in [SIAM J. Matrix Anal. Appl., 32 (2011), pp. 1166-1192] for the unstructured case, but their extension to structured pseudospectra and analysis presents several difficulties. Natural generalizations of the algorithms, allowing to draw significant sections of the structured pseudospectra in proximity of extremal points are also discussed. Since no algorithms are available in the literature to draw such structured pseudospectra, the approach we present seems promising to extend existing software tools (Eigtool, Seigtool) to structured pseudospectra representation for Toeplitz matrices. We discuss local convergence properties of the algorithms and show some applications to a few illustrative examples.

Raphael M. Jungers, Nicola Guglielmi, Antonio Cicone

We describe new methods for deciding the stability of switching systems. The methods build on two ideas previously appeared in the literature: the polytope norm iterative construction, and the lifting procedure. Moreover, the combination of these two ideas allows us to introduce a pruning algorithm which can importantly reduce the computational burden. We prove several appealing theoretical properties of our methods like a finiteness computational result which extends a known result for unlifted sets of matrices, and provide numerical examples of their good behaviour.

Nicola Guglielmi, Mutti-Ur Rehman, Daniel Kressner

A novel method for approximating structured singular values (also known as mu-values) is proposed and investigated. These quantities constitute an important tool in the stability analysis of uncertain linear control systems as well as in structured eigenvalue perturbation theory. Our approach consists of an inner-outer iteration. In the outer iteration, a Newton method is used to adjust the perturbation level. The inner iteration solves a gradient system associated with an optimization problem on the manifold induced by the structure. Numerical results and comparison with the well-known Matlab function mussv, implemented in the Matlab Control Toolbox, illustrate the behavior of the method.

Emanuele Zangrando, Piero Deidda, Simone Brugiapaglia et al.

Recent work in deep learning has shown strong empirical and theoretical evidence of an implicit low-rank bias: weight matrices in deep networks tend to be approximately low-rank. Moreover, removing relatively small singular values during training, or from available trained models, may significantly reduce model size while maintaining or even improving model performance. However, the majority of the theoretical investigations around low-rank bias in neural networks deal with oversimplified models, often not taking into account the impact of nonlinearity. In this work, we first of all quantify a link between the phenomenon of deep neural collapse and the emergence of low-rank weight matrices for a general class of feedforward networks with nonlinear activation. In addition, for the general class of nonlinear feedforward and residual networks, we prove the global optimality of deep neural collapsed configurations and the practical absence of a loss barrier between interpolating minima and globally optimal points, offering a possible explanation for its common occurrence. As a byproduct, our theory also allows us to forecast the final global structure of singular values before training. Our theoretical findings are supported by a range of experimental evaluations illustrating the phenomenon.

Arturo De Marinis, Davide Murari, Elena Celledoni et al.

We study the approximation properties of shallow neural networks whose activation function is defined as the flow map of a neural ordinary differential equation (neural ODE) at the final time of the integration interval. We prove the universal approximation property (UAP) of such shallow neural networks in the space of continuous functions. Furthermore, we investigate the approximation properties of shallow neural networks whose parameters satisfy specific constraints. In particular, we constrain the Lipschitz constant of the neural ODE's flow map and the norms of the weights to increase the network's stability. We prove that the UAP holds if we consider either constraint independently. When both are enforced, there is a loss of expressiveness, and we derive approximation bounds that quantify how accurately such a constrained network can approximate a continuous function.

Pietro Sittoni, Emanuele Zangrando, Angelo A. Casulli et al.

Deep learning-based methods have shown remarkable effectiveness in solving PDEs, largely due to their ability to enable fast simulations once trained. However, despite the availability of high-performance computing infrastructure, many critical applications remain constrained by the substantial computational costs associated with generating large-scale, high-quality datasets and training models. In this work, inspired by studies on the structure of Green's functions for elliptic PDEs, we introduce Neural-HSS, a parameter-efficient architecture built upon the Hierarchical Semi-Separable (HSS) matrix structure that is provably data-efficient for a broad class of PDEs. We theoretically analyze the proposed architecture, proving that it satisfies exactness properties even in very low-data regimes. We also investigate its connections with other architectural primitives, such as the Fourier neural operator layer and convolutional layers. We experimentally validate the data efficiency of Neural-HSS on the three-dimensional Poisson equation over a grid of two million points, demonstrating its superior ability to learn from data generated by elliptic PDEs in the low-data regime while outperforming baseline methods. Finally, we demonstrate its capability to learn from data arising from a broad class of PDEs in diverse domains, including electromagnetism, fluid dynamics, and biology.

Nicola Guglielmi, Maria López-Fernández, Giancarlo Nino

In this paper a novel contour integral method is proposed for linear convection-diffusion equations. The method is based on the inversion of the Laplace transform and makes use of a contour given by an elliptic arc joined symmetrically to two half-lines. The trapezoidal rule is the chosen integration method for the numerical inversion of the Laplace transform, due to its well-known fast convergence properties when applied to analytic functions. Error estimates are provided as well as careful indications about the choice of several involved parameters. The method selects the elliptic arc in the integration contour by an algorithmic strategy based on the computation of pseudospectral level sets of the discretized differential operator. In this sense the method is general and can be applied to any linear convection-diffusion equation without knowing any a priori information about its pseudospectral geometry. Numerical experiments performed on the Black-Scholes ($1D$) and Heston ($2D$) equations show that the method is competitive with other contour integral methods available in the literature.