arXiv subjects

Nicola Guglielmi

Publications and source records attributed to Nicola Guglielmi.

At least 37 records · Page 2Linked to original sources

arXiv2024

Approximating the closest structured singular matrix polynomial

Consider a matrix polynomial $P \left( λ\right)= A_0 + λA_1 + \ldots + λ^d A_d$, with $A_0,\ldots, A_d$ complex (or real) matrices with a certain structure. In this paper we discuss an iterative method to numerically approximate the closest structured singular matrix polynomial $\widetilde P\left( λ\right)$, using the distance induced by the Frobenius norm. An important peculiarity of the approach we propose is the possibility to include different types of structural constraints. The method also allows us to limit the perturbations to just a few matrices and also to include additional structures, such as the preservation of the sparsity pattern of one or more matrices $A_i$, and also collective-like properties, like a palindromic structure. The iterative method is based on the numerical integration of the gradient system associated with a suitable functional which quantifies the distance to singularity of a matrix polynomial.

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arXiv2024

Nonuniqueness phenomena in discontinuous dynamical systems and their regularizations

In a recent paper by Guglielmi and Hairer (SIADS 2015), an analysis in the $\varepsilon\to 0$ limit was proposed of regularized discontinuous ODEs in codimension-2 switching domains; this was obtained by studying a certain 2-dimensional system describing the so-called hidden dynamics. In particular, the existence of a unique limit solution was not proved in all cases, a few of which were labeled as ambiguous, and it was not clear whether or not the ambiguity could be resolved. In this paper, we show that it cannot be resolved in general. A first contribution of this paper is an illustration of the dependence of the limit solution on the form of the switching function. Considering the parameter dependence in the ambiguous class of discontinuous systems, a second contribution is a bifurcation analysis, revealing a range of possible behaviors. Finally, we investigate the sensitivity of solutions in the transition from codimension-2 domains to codimension-3 when there is a limit cycle in the hidden dynamics.

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arXiv2024

Stabilization of a matrix via a low rank-adaptive ODE

Let $A$ be a square matrix with a given structure (e.g. real matrix, sparsity pattern, Toeplitz structure, etc.) and assume that it is unstable, i.e. at least one of its eigenvalues lies in the complex right half-plane. The problem of stabilizing $A$ consists in the computation of a matrix $B$, whose eigenvalues have negative real part and such that the perturbation $Δ=B-A$ has minimal norm. The structured stabilization further requires that the perturbation preserves the structural pattern of $A$. We solve this non-convex problem by a two-level procedure which involves the computation of the stationary points of a matrix ODE. We exploit the low rank underlying features of the problem by using an adaptive-rank integrator that follows slavishly the rank of the solution. We show the benefits derived from the low rank setting in several numerical examples, which also allow to deal with high dimensional problems.

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arXiv2024

Cholesky-like Preconditioner for Hodge Laplacians via Heavy Collapsible Subcomplex

Techniques based on $k$-th order Hodge Laplacian operators $L_k$ are widely used to describe the topology as well as the governing dynamics of high-order systems modeled as simplicial complexes. In all of them, it is required to solve a number of least square problems with $L_k$ as coefficient matrix, for example in order to compute some portions of the spectrum or integrate the dynamical system. In this work, we introduce the notion of optimal collapsible subcomplex and we present a fast combinatorial algorithm for the computation of a sparse Cholesky-like preconditioner for $L_k$ that exploits the topological structure of the simplicial complex. The performance of the preconditioner is tested for conjugate gradient method for least square problems (CGLS) on a variety of simplicial complexes with different dimensions and edge densities. We show that, for sparse simplicial complexes, the new preconditioner reduces significantly the condition number of $L_k$ and performs better than the standard incomplete Cholesky factorization.

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arXiv2024

Applying stiff integrators for ODEs and DDEs to problems with distributed delays

There exist excellent codes for an efficient numerical treatment of stiff and differential-algebraic problems. Let us mention {\sc Radau5} which is based on the $3$-stage Radau IIA collocation method, and its extension to problems with discrete delays {\sc Radar5}. The aim of the present work is to present a technique that permits a direct application of these codes to problems having a right-hand side with an additional distributed delay term (which is a special case of an integro-differential equation). Models with distributed delays are of increasing importance in pharmacodynamics and pharmacokinetics for the study of the interaction between drugs and the body. The main idea is to approximate the distribution kernel of the integral term by a sum of exponential functions or by a quasi-polynomial expansion, and then to transform the distributed (integral) delay term into a set of ordinary differential equations. This set is typically stiff and, for some distribution kernels (e.g., Pareto distribution), it contains discrete delay terms with constant delay. The original equations augmented by this set of ordinary differential equations can have a very large dimension, and a careful treatment of the solution of the arising linear systems is necessary. The use of the codes {\sc Radau5} and {\sc Radar5} is illustrated at three examples (two test equations and one problem taken from pharmacodynamics). The driver programs for these examples are publicly available from the homepages of the authors.

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arXiv2024

Transient dynamics under structured perturbations: bridging unstructured and structured pseudospectra

The structured $\varepsilon$-stability radius is introduced as a quantity to assess the robustness of transient bounds of solutions to linear differential equations under structured perturbations of the matrix. This applies to general linear structures such as complex or real matrices with a given sparsity pattern or with restricted range and corange, or special classes such as Toeplitz matrices. The notion conceptually combines unstructured and structured pseudospectra in a joint pseudospectrum, allowing for the use of resolvent bounds as with unstructured pseudospectra and for structured perturbations as with structured pseudospectra. We propose and study an algorithm for computing the structured $\varepsilon$-stability radius. This algorithm solves eigenvalue optimization problems via suitably discretized rank-1 matrix differential equations that originate from a gradient system. The proposed algorithm has essentially the same computational cost as the known rank-1 algorithms for computing unstructured and structured stability radii. Numerical experiments illustrate the behavior of the algorithm.

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arXiv2023

Quantifying the structural stability of simplicial homology

The homology groups of a simplicial complex reveal fundamental properties of the topology of the data or the system and the notion of topological stability naturally poses an important yet not fully investigated question. In the current work, we study the stability in terms of the smallest perturbation sufficient to change the dimensionality of the corresponding homology group. Such definition requires an appropriate weighting and normalizing procedure for the boundary operators acting on the Hodge algebra's homology groups. Using the resulting boundary operators, we then formulate the question of structural stability as a spectral matrix nearness problem for the corresponding higher-order graph Laplacian. We develop a bilevel optimization procedure suitable for the formulated matrix nearness problem and illustrate the method's performance on a variety of synthetic quasi-triangulation datasets and transportation networks.

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arXiv2023

A low rank ODE for spectral clustering stability

Spectral clustering is a well-known technique which identifies $k$ clusters in an undirected graph with weight matrix $W\in\mathbb{R}^{n\times n}$ by exploiting its graph Laplacian $L(W)$, whose eigenvalues $0=λ_1\leq λ_2 \leq \dots \leq λ_n$ and eigenvectors are related to the $k$ clusters. Since the computation of $λ_{k+1}$ and $λ_k$ affects the reliability of this method, the $k$-th spectral gap $λ_{k+1}-λ_k$ is often considered as a stability indicator. This difference can be seen as an unstructured distance between $L(W)$ and an arbitrary symmetric matrix $L_\star$ with vanishing $k$-th spectral gap. A more appropriate structured distance to ambiguity such that $L_\star$ represents the Laplacian of a graph has been proposed by Andreotti et al. (2021). Slightly differently, we consider the objective functional $ F(Δ)=λ_{k+1}\left(L(W+Δ)\right)-λ_k\left(L(W+Δ)\right)$, where $Δ$ is a perturbation such that $W+Δ$ has non-negative entries and the same pattern of $W$. We look for an admissible perturbation $Δ_\star$ of smallest Frobenius norm such that $F(Δ_\star)=0$. In order to solve this optimization problem, we exploit its low rank underlying structure. We formulate a rank-4 symmetric matrix ODE whose stationary points are the optimizers sought. The integration of this equation benefits from the low rank structure with a moderate computational effort and memory requirement, as it is shown in some illustrative numerical examples.

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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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arXiv2022

Rank-$1$ matrix differential equations for structured eigenvalue optimization

A new approach to solving eigenvalue optimization problems for large structured matrices is proposed and studied. The class of optimization problems considered is related to computing structured pseudospectra and their extremal points, and to structured matrix nearness problems such as computing the structured distance to instability or to singularity. The structure can be a general linear structure and includes, for example, large matrices with a given sparsity pattern, matrices with given range and co-range, and Hamiltonian matrices. Remarkably, the eigenvalue optimization can be performed on the manifold of complex (or real) rank-1 matrices, which yields a significant reduction of storage and in some cases of the computational cost. The method relies on a constrained gradient system and the projection of the gradient onto the tangent space of the manifold of complex rank-$1$ matrices. It is shown that near a local minimizer this projection is very close to the identity map, and so the computationally favorable rank-1 projected system behaves locally like the %computationally expensive gradient system.

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arXiv2021

Identification of Time Delays in COVID-19 Data

COVID-19 data released by public health authorities features the presence of notable time-delays, corresponding to the difference between actual time of infection and identification of infection. These delays have several causes, including the natural incubation period of the virus, availability and speed of testing facilities, population demographics, and testing center capacity, among others. Such delays have important ramifications for both the mathematical modeling of COVID-19 contagion and the design and evaluation of intervention strategies. In the present work, we introduce a novel optimization technique for the identification of time delays in COVID-19 data, making use of a delay-differential equation model. The proposed method is general in nature and may be applied not only to COVID-19, but for generic dynamical systems in which time delays may be present.

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arXiv2021

Computation of the nearest structured matrix triplet with common null space

We study computational methods for computing the distance to singularity, the distance to the nearest high index problem, and the distance to instability for linear differential-algebraic systems (DAEs) with dissipative Hamiltonian structure. While for general unstructured DAEs the characterization of these distances is very difficult, and partially open, it has been recently shown that for dissipative Hamiltonian systems and related matrix pencils there exist explicit characterizations. We will use these characterizations for the development of computational methods to compute these distances via methods that follow the flow of a differential equation converging to the smallest perturbation that destroys the property of regularity, index one or stability.

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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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arXiv2021

Finding the nearest passive or non-passive system via Hamiltonian eigenvalue optimization

We propose and study an algorithm for computing a nearest passive system to a given non-passive linear time-invariant system (with much freedom in the choice of the metric defining `nearest', which may be restricted to structured perturbations), and also a closely related algorithm for computing the structured distance of a given passive system to non-passivity. Both problems are addressed by solving eigenvalue optimization problems for Hamiltonian matrices that are constructed from perturbed system matrices. The proposed algorithms are two-level methods that optimize the Hamiltonian eigenvalue of smallest positive real part over perturbations of a fixed size in the inner iteration, using a constrained gradient flow. They optimize over the perturbation size in the outer iteration, which is shown to converge quadratically in the typical case of a defective coalescence of simple eigenvalues approaching the imaginary axis. For large systems, we propose a variant of the algorithm that takes advantage of the inherent low-rank structure of the problem. Numerical experiments illustrate the behavior of the proposed algorithms.

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arXiv2021

Delay differential equations for the spatially-resolved simulation of epidemics with specific application to COVID-19

In the wake of the 2020 COVID-19 epidemic, much work has been performed on the development of mathematical models for the simulation of the epidemic, and of disease models generally. Most works follow the susceptible-infected-removed (SIR) compartmental framework, modeling the epidemic with a system of ordinary differential equations. Alternative formulations using a partial differential equation (PDE) to incorporate both spatial and temporal resolution have also been introduced, with their numerical results showing potentially powerful descriptive and predictive capacity. In the present work, we introduce a new variation to such models by using delay differential equations (DDEs). The dynamics of many infectious diseases, including COVID-19, exhibit delays due to incubation periods and related phenomena. Accordingly, DDE models allow for a natural representation of the problem dynamics, in addition to offering advantages in terms of computational time and modeling, as they eliminate the need for additional, difficult-to-estimate, compartments (such as exposed individuals) to incorporate time delays. Here, we introduce a DDE epidemic model in both an ordinary- and partial differential equation framework. We present a series of mathematical results assessing the stability of the formulation. We then perform several numerical experiments, validating both the mathematical results and establishing model's ability to reproduce measured data on realistic problems.

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arXiv2020

A gradient system approach for Hankel structured low-rank approximation

Rank deficient Hankel matrices are at the core of several applications. However, in practice, the coefficients of these matrices are noisy due to e.g. measurements errors and computational errors, so generically the involved matrices are full rank. This motivates the problem of Hankel structured low-rank approximation. Structured low-rank approximation problems, in general, do not have a global and efficient solution technique. In this paper we propose a local optimization approach based on a two-levels iteration. Experimental results show that the proposed algorithm usually achieves good accuracy and shows a higher robustness with respect to the initial approximation, compared to alternative approaches.

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arXiv2020

Measuring the stability of spectral clustering

As an indicator of the stability of spectral clustering of an undirected weighted graph into $k$ clusters, the $k$th spectral gap of the graph Laplacian is often considered. The $k$th spectral gap is characterized in this paper as an unstructured distance to ambiguity, namely as the minimal distance of the Laplacian to arbitrary symmetric matrices with vanishing $k$th spectral gap. As a conceptually more appropriate measure of stability, the structured distance to ambiguity of the $k$-clustering is introduced as the minimal distance of the Laplacian to Laplacians of graphs with the same vertices and edges but with weights that are perturbed such that the $k$th spectral gap vanishes. To compute a solution to this matrix nearness problem, a two-level iterative algorithm is proposed that uses a constrained gradient system of matrix differential equations in the inner iteration and a one-dimensional optimization of the perturbation size in the outer iteration. The structured and unstructured distances to ambiguity are compared on some example graphs. The numerical experiments show, in particular, that selecting the number $k$ of clusters according to the criterion of maximal stability can lead to different results for the structured and unstructured stability indicators.

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arXiv2019

Switching systems with dwell time: computation of the maximal Lyapunov exponent

We study asymptotic stability of continuous-time systems with mode-dependent guaranteed dwell time. These systems are reformulated as special cases of a general class of mixed (discrete-continuous) linear switching systems on graphs, in which some modes correspond to discrete actions and some others correspond to continuous-time evolutions. Each discrete action has its own positive weight which accounts for its time-duration. We develop a theory of stability for the mixed systems; in particular, we prove the existence of an invariant Lyapunov norm for mixed systems on graphs and study its structure in various cases, including discrete-time systems for which discrete actions have inhomogeneous time durations. This allows us to adapt recent methods for the joint spectral radius computation (Gripenberg's algorithm and the Invariant Polytope Algorithm) to compute the Lyapunov exponent of mixed systems on graphs.

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