arXiv subjects

Paola Ferrari

Publications and source records attributed to Paola Ferrari.

13 recordsLinked to original sources

arXiv2025

When algebra twinks system biology: a conjecture on the structure of Gr\"obner bases in complex chemical reaction networks

We address the challenge of identifying all real positive steady states in chemical reaction networks (CRNs) governed by mass-action kinetics. Traditional numerical methods often require specific initial guesses and may fail to find all the solutions in systems exhibiting multistability. Gr\"obner bases offer an algebraic framework that systematically transforms polynomial equations into simpler forms, facilitating comprehensive solution enumeration. In this work, we propose a conjecture that CRNs with at most pairwise interactions yield Gr\"obner bases possessing a near-"triangular" structure, under appropriate assumptions. We illustrate this phenomenon using examples from a gene regulatory network and the Wnt signaling pathway, where the Gr\"obner basis approach reliably captures all real positive solutions. Our computational experiments reveal the potential of Gr\"obner bases to overcome limitations of local numerical methods for finding the steady states of complex biological systems, making them a powerful tool for understanding dynamical processes across diverse biochemical models.

q-bio.MN↗

arXiv2024

Dense cell-by-cell systems of PDEs: approximation, spectral analysis, and preconditioning

In the present study, we consider the Extra-Membrane-Intra model (EMI) for the simulation of excitable tissues at the cellular level. We provide the (possibly large) system of partial differential equations (PDEs), equipped with ad hoc boundary conditions, relevant to model portions of excitable tissues, composed of several cells. In particular, we study two geometrical settings: computational cardiology and neuroscience. The Galerkin approximations to the considered system of PDEs lead to large linear systems of algebraic equations, where the coefficient matrices depend on the number $N$ of cells and the fineness parameters. We give a structural and spectral analysis of the related matrix-sequences with $N$ fixed and with fineness parameters tending to zero. Based on the theoretical results, we propose preconditioners and specific multilevel solvers. Numerical experiments are presented and critically discussed, showing that a monolithic multilevel solver is efficient and robust with respect to all the problem and discretization parameters. In particular, we include numerical results increasing the number of cells $N$, both for idealized geometries (with $N$ exceeding $10^5$) and for realistic, densely populated 3D tissue reconstruction.

math.NA↗

arXiv2024

Analysis on aggregation and block smoothers in multigrid methods for block Toeplitz linear systems

We present novel improvements in the context of symbol-based multigrid procedures for solving large block structured linear systems. We study the application of an aggregation-based grid transfer operator that transforms the symbol of a block Toeplitz matrix from matrix-valued to scalar-valued at the coarser level. Our convergence analysis of the Two-Grid Method (TGM) reveals the connection between the features of the scalar-valued symbol at the coarser level and the properties of the original matrix-valued one. This allows us to prove the convergence of a V-cycle multigrid with standard grid transfer operators for scalar Toeplitz systems at the coarser levels. Consequently, we extend the class of suitable smoothers for block Toeplitz matrices, focusing on the efficiency of block strategies, particularly the relaxed block Jacobi method. General conditions on smoothing parameters are derived, with emphasis on practical applications where these parameters can be calculated with negligible computational cost. We test the proposed strategies on linear systems stemming from the discretization of differential problems with $\mathbb{Q}_{d} $ Lagrangian FEM or B-spline with non-maximal regularity. The numerical results show in both cases computational advantages compared to existing methods for block structured linear systems.

math.NA↗

arXiv2024

Flipped structured matrix-sequences in image deblurring with general boundary conditions

Motivated by a recent work on a preconditioned MINRES for flipped linear systems in imaging, in this note we extend the scope of that research for including more precise boundary conditions such as reflective and anti-reflective ones. We prove spectral results for the matrix-sequences associated to the original problem, which justify the use of the MINRES in the current setting. The theoretical spectral analysis is supported by a wide variety of numerical experiments, concerning the visualization of the spectra of the original matrices in various ways. We also report numerical tests regarding the convergence speed and regularization features of the associated GMRES and MINRES methods. Conclusions and open problems end the present study.

math.NA↗

arXiv2023

Symbol based Convergence Analysis in Block Multigrid Methods with Applications for Stokes Problems

The main focus of this paper is the study of efficient multigrid methods for large linear systems with a particular saddle-point structure. Indeed, when the system matrix is symmetric, but indefinite, the variational convergence theory that is usually used to prove multigrid convergence cannot be directly applied. However, different algebraic approaches analyze properly preconditioned saddle-point problems, proving convergence of the Two-Grid method. In particular, this is efficient when the blocks of the coefficient matrix possess a Toeplitz or circulant structure. Indeed, it is possible to derive sufficient conditions for convergence and provide optimal parameters for the preconditioning of the saddle-point problem in terms of the associated generating symbols. In this paper, we propose a symbol-based convergence analysis for problems that have a hidden block Toeplitz structure. Then, they can be investigated focusing on the properties of the associated generating function f, which consequently is a matrix-valued function with dimension depending on the block size of the problem. As numerical tests we focus on the matrix sequence stemming from the finite element approximation of the Stokes problem. We show the efficiency of the methods studying the hidden 9-by-9 block multilevel structure of the obtained matrix sequence. Moreover, we propose an efficient algebraic multigrid method with convergence rate independent of the matrix size. Finally, we present several numerical tests comparing the results with state-of-the-art strategies.

math.NA↗

arXiv2023

Modeling excitable cells with the EMI equations: spectral analysis and iterative solution strategy

In this work, we are interested in solving large linear systems stemming from the Extra-Membrane-Intra (EMI) model, which is employed for simulating excitable tissues at a cellular scale. After setting the related systems of partial differential equations (PDEs) equipped with proper boundary conditions, we provide numerical approximation schemes for the EMI PDEs and focus on the resulting large linear systems. We first give a relatively complete spectral analysis using tools from the theory of Generalized Locally Toeplitz matrix sequences. The obtained spectral information is used for designing appropriate (preconditioned) Krylov solvers. We show, through numerical experiments, that the presented solution strategy is robust w.r.t. problem and discretization parameters, efficient and scalable.

math.NA↗

arXiv2022

Symmetrization Techniques in Image Deblurring

This paper presents a couple of preconditioning techniques that can be used to enhance the performance of iterative regularization methods applied to image deblurring problems with a variety of point spread functions (PSFs) and boundary conditions. More precisely, we first consider the anti-identity preconditioner, which symmetrizes the coefficient matrix associated to problems with zero boundary conditions, allowing the use of MINRES as a regularization method. When considering more sophisticated boundary conditions and strongly nonsymmetric PSFs, the anti-identity preconditioner improves the performance of GMRES. We then consider both stationary and iteration-dependent regularizing circulant preconditioners that, applied in connection with the anti-identity matrix and both standard and flexible Krylov subspaces, speed up the iterations. A theoretical result about the clustering of the eigenvalues of the preconditioned matrices is proved in a special case. The results of many numerical experiments are reported to show the effectiveness of the new preconditioning techniques, including when considering the deblurring of sparse images.

math.NA↗

arXiv2022

Symbol based convergence analysis in multigrid methods for saddle point problems

Saddle point problems arise in a variety of applications, e.g., when solving the Stokes equations. They can be formulated such that the system matrix is symmetric, but indefinite, so the variational convergence theory that is usually used to prove multigrid convergence cannot be applied. In a 2016 paper in Numerische Mathematik Notay has presented a different algebraic approach that analyzes properly preconditioned saddle point problems, proving convergence of the Two-Grid method. In the present paper we analyze saddle point problems where the blocks are circulant within this framework. We are able to derive sufficient conditions for convergence and provide optimal parameters for the preconditioning of the saddle point problem and for the point smoother that is used. The analysis is based on the generating symbols of the circulant blocks. Further, we show that the structure can be kept on the coarse level, allowing for a recursive application of the approach in a W- or V-cycle and proving the "level independency" property. Numerical results demonstrate the efficiency of the proposed method in the circulant and the Toeplitz case.

math.NA↗

arXiv2021

Asymptotic spectra of large (grid) graphs with a uniform local structure (part II): numerical applications

In the current work we are concerned with sequences of graphs having a grid geometry, with a uniform local structure in a bounded domain $Ω\subset {\mathbb R}^d$, $d\ge 1$. When $Ω=[0,1]$, such graphs include the standard Toeplitz graphs and, for $Ω=[0,1]^d$, the considered class includes $d$-level Toeplitz graphs. In the general case, the underlying sequence of adjacency matrices has a canonical eigenvalue distribution, in the Weyl sense, and it has been shown in the theoretical part of this work that we can associate to it a symbol $\boldsymbol{\mathfrak{f}}$. The knowledge of the symbol and of its basic analytical features provides key information on the eigenvalue structure in terms of localization, spectral gap, clustering, and global distribution. In the present paper, many different applications are discussed and various numerical examples are presented in order to underline the practical use of the developed theory. Tests and applications are mainly obtained from the approximation of differential operators via numerical schemes such as Finite Differences (FDs), Finite Elements (FEs), and Isogeometric Analysis (IgA). Moreover, we show that more applications can be taken into account, since the results presented here can be applied as well to study the spectral properties of adjacency matrices and Laplacian operators of general large graphs and networks, whenever the involved matrices enjoy a uniform local structure.

math.NA↗

arXiv2021

A symbol based analysis for multigrid methods for Block-Circulant and Block-Toeplitz Systems

In the literature, there exist several studies on symbol-based multigrid methods for the solution of linear systems having structured coefficient matrices. In particular, the convergence analysis for such methods has been obtained in an elegant form in the case of Toeplitz matrices generated by a scalar-valued function. In the block-Toeplitz setting, that is, in the case where the matrix entries are small generic matrices instead of scalars, some algorithms have already been proposed regarding specific applications and a first rigorous convergence analysis has been performed in [7]. However, with the existent symbol-based theoretical tools, it is still not possible to prove the convergence of many multigrid methods known in the literature. This paper aims to generalize the previous results giving more general sufficient conditions on the symbol of the grid transfer operators.In particular, we treat matrix-valued trigonometric polynomials which can be non-diagonalizable and singular at all points and we express the new conditions in terms of the eigenvectors associated with the ill-conditioned subspace. Moreover, we extend the analysis to the V-cycle method proving a linear convergence rate under stronger conditions, which resemble those given in the scalar case. In order to validate our theoretical findings, we present a classical block structured problem stemming from a FEM approximation of a second order differential problem. We focus on two multigrid strategies that use the geometric and the standard bisection grid transfer operators and we prove that both fall into the category of projectors satisfying the proposed conditions. In addition, using a tensor product argument, we provide a strategy to construct efficient V-cycle procedures in the block multilevel setting.

math.NA↗

arXiv2020

Multilevel symmetrized Toeplitz structures and spectral distribution results for the related matrix-sequences

In recent years, motivated by computational purposes, the singular value and spectral features of the symmetrization of Toeplitz matrices generated by a Lebesgue integrable function have been studied. Indeed, under the assumptions that $f$ belongs to $L^1([-π,π])$ and it has real Fourier coefficients, the spectral and singular value distribution of the matrix-sequence $\{Y_nT_n[f]\}_n$ has been identified, where $n$ is the matrix-size, $Y_n$ is the anti-identity matrix, and $T_n[f]$ is the Toeplitz matrix generated by $f$. In this note, we consider the multilevel Toeplitz matrix $T_{\bf n}[f]$ generated by $f\in L^1([-π,π]^k)$, $\bf n$ being a multi-index identifying the matrix-size, and we prove spectral and singular value distribution results for the matrix-sequence $\{Y_{\bf n}T_{\bf n}[f]\}_{\bf n}$ with $Y_{\bf n}$ being the corresponding tensorization of the anti-identity matrix.

math.NA↗

arXiv2019

Multigrid methods for block-Toeplitz linear systems: convergence analysis and applications

In the past decades, multigrid methods for linear systems having multilevel Toeplitz coefficient matrices with scalar entries have been largely studied. On the other hand, only few papers have investigated the case of block entries, where the entries are small generic matrices instead of scalars. In that case the efforts of the researchers have been mainly devoted to specific applications, focusing on algorithmic proposals but with very marginal theoretical results. In this paper, we propose a general two-grid convergence analysis proving an optimal convergence rate independent of the matrix size, in the case of positive definite block Toeplitz matrices with generic blocks. In particular, the proof of the approximation property has not a straightforward generalization of the scalar case and in fact we have to require a specific commutativity condition on the block symbol of the grid transfer operator. Furthermore, we define a class of grid transfer operators satisfying the previous theoretical conditions and we propose a strategy to insure fast multigrid convergence even for more than two grids. Among the numerous applications that lead to the block Toeplitz structure, high order Lagrangian finite element methods and staggered discontinuous Galerkin methods are considered in the numerical results, confirming the effectiveness of our proposal and the correctness of the proposed theoretical analysis.

math.NA↗

arXiv2018

The eigenvalue distribution of special $2$-by-$2$ block matrix sequences, with applications to the case of symmetrized Toeplitz structures

Given a Lebesgue integrable function $f$ over $[0,2\pi]$, we consider the sequence of matrices $\{Y_nT_n[f]\}_n$, where $T_n[f]$ is the $n$-by-$n$ Toeplitz matrix generated by $f$ and $Y_n$ is the flip permutation matrix, also called the anti-identity matrix. Because of the unitary character of $Y_n$, the singular values of $T_n[f]$ and $Y_n T_n[f]$ coincide. However, the eigenvalues are affected substantially by the action of the matrix $Y_n$. Under the assumption that the Fourier coefficients are real, we prove that $\{Y_nT_n[f]\}_n$ is distributed in the eigenvalue sense as \[ \phi_g(\theta)=\left\{ \begin{array}{cc} g(\theta), & \theta\in [0,2\pi], -g(-\theta), & \theta\in [-2\pi,0), \end{array} \right.\, \] with $g(\theta)=|f(\theta)|$. We also consider the preconditioning introduced by Pestana and Wathen and, by using the same arguments, we prove that the preconditioned sequence is distributed in the eigenvalue sense as $\phi_1$, under the mild assumption that $f$ is sparsely vanishing. We emphasize that the mathematical tools introduced in this setting have a general character and in fact can be potentially used in different contexts. A number of numerical experiments are provided and critically discussed.

math.NA↗