We introduce the anti-Gauss cubature rule for approximating integrals defined on the square whose integrand function may have algebraic singularities at the boundaries. An application of such a rule to the numerical solution of Fredholm integral equations of the second-kind is also explored. The stability, convergence, and conditioning of the proposed Nystr\"om-type method are studied. The...
We study the universal approximation property (UAP) of shallow neural networks whose activation function is defined as the flow of a neural ODE. We prove the UAP for the space of such shallow neural networks in the space of continuous functions. In particular, we also prove the UAP with the weight matrices constrained to have unit norm.
Furthermore, in [1] we are able to bound from above...
Metriplectic or GENERIC systems describe complex dynamical systems that blend Hamiltonian and dissipative dynamics, i.e. systems that conserve energy and produce entropy. Maintaining the separation between Hamiltonian and dissipative part in the construction of approximate models is a challenging task. In this work, we employ a matrix-based implementation of the Discrete Empirical...
We consider the spectral distribution of the geometric mean of two or more Hermitian positive definite (HPD) matrix-sequences, under the assumption that all input matrix-sequences belong to the same Generalized Locally Toeplitz (GLT) $*$-algebra. As expected, the numerical experiments show that the geometric mean of $k$ positive definite GLT matrix-sequences forms a new GLT matrix-sequence,...
In numerous fields of science and engineering we are faced with problems
of the form $$A{x}+{\eta}=b^\delta,$$ where $A\in\mathbb{R}^{m\times n}$
is a large and severely ill-conditioned matrix, i.e., such that its
singular values decay rapidly to zero without any significant gap
between the smallest ones, ${x}\in\mathbb{R}^n$ is the unknown quantity
we wish to recover,...
This work focuses on computing a few smallest or largest eigenvalues and their corresponding eigenvectors of a large symmetric matrix $A$.
Krylov subspace methods are among the most effective approaches for solving large-scale symmetric eigenvalue problems. Given a starting vector $\mathbf{q}_1$, the $M$-th Krylov subspace is generated by repeatedly multiplying a general square matrix $A$...
In this work, we present a low-memory variant of the Lanczos algorithm for the solution of the Lyapunov equation
\begin{equation}
AX + XA = \mathbf{c}\mathbf{c}^T, \tag{1}
\end{equation}
where $A$ is a large-scale symmetric positive-definite matrix and $\mathbf{c}$ is a vector.
The classical Lanczos method consists in building an orthonormal basis $Q_M$ for the polynomial Krylov...
The practical motivation for this work is to reconstruct electromagnetic properties of the earth superficial layer using measurements taken above the ground. We approach this problem by inverting frequency-domain electromagnetic data through a well-established linear integral model, employing three different collocation methods to approximate the solution as a linear combination of linearly...
Given two matrices $X,B\in \mathbb{R}^{n\times m}$ and a set $\mathcal{A}\subseteq \mathbb{R}^{n\times n}$, a Procrustes problem consists in finding a matrix $A \in \mathcal{A}$ such that the Frobenius norm of $AX-B$ is minimized. When $\mathcal{A}$ is the set of the matrices whose symmetric part is positive semidefinite, we obtain the so-called non-symmetric positive semidefinite Procrustes...
The Mittag-Leffler functions interpolate between the exponential and the resolvent functions, and they can be used for walk-based indices in networks, allowing to tune the weight of longer closed walks.
Here we study the Communicability based on the Mittag-Leffler function, and define Mittag-Leffler communicability distance functions, which are proved to be Euclidean and spherical. We then...
In this talk, we consider the solution of the multilinear PageRank problem, which can be reformulated as a nonlinear system of equations in which the coefficient matrix is an M-matrix. In [1], Alfa et al., demonstrated that the inverse of a nonsingular M-matrix can be determined with a high relative entrywise accuracy by using a triplet representation of the M-matrix through a method known as...
The growth of computational capabilities is ushering the era of exascale computing, presenting unprecedented opportunities and challenges for scientific applications. Sparse Linear algebra underpins a wide range of these applications: from simulating the transition to decarbonized energy in Europe to the construction of digital twins of the human body. The demands of exascale systems...
Runge-Kutta methods are affine equivariant: applying a method before or after an affine change of variables yields the same numerical trajectory. However, for some applications, one would like to perform numerical integration after a quadratic change of variables. For example, in Lie-Poisson reduction, a quadratic transformation reduces the number of variables in a Hamiltonian system, yielding...
In this talk, we analyze the variance of a stochastic estimator for computing Schatten norms of matrices. The estimator extracts information from a single sketch of the matrix, that is, the product of the matrix with a few standard Gaussian random vectors. While this estimator has been proposed and used in the literature before, the existing variance bounds are often pessimistic. Our work...
We consider the use of sketching to improve the performances and usability of Krylov tensor methods; in this talk, we will consider the case study of TT-GMRES, and describe the many possibilities that arise where sketching is helpful to improve performances and accuracy, often in somewhat unexpected ways. Our ideas build upon sketching in GMRES and streaming low-rank approximation algorithms...
In the current talk we consider sequences of matrix-sequences $\{\{B_{n,t}\}_n\}_t$ of increasing sizes depending on a parameter $n$ and equipped with an additional parameter $t>0$. For every fixed $t>0$, we assume that each $\{B_{n,t}\}_n$ possesses a canonical spectral/singular values symbol $f_t$ defined on a measurable set $D_t\subset \mathbb{R}^{d}$ of finite measure, $d\ge 1$....
Useful properties of scalar polynomials can be obtained by embedding them into the larger framework of their generalization to matrix polynomials. As an example of this, we derive explicit simple upper bounds on the magnitudes of scalar polynomial zeros that are close approximations of the Cauchy radius. Their complexity is of the order of a single polynomial evaluation.
The Cauchy radius is...
Feed-forward neural networks can be interpreted as dyscrete switched dynamical systems where the switching rule is determined by the training process. The stability of switched systems has been extensively investigated in the linear case, where the key ingredient is given by the joint spectral radius of the family of matrices that determines the dynamics of the system. On the other hand,...
The time-frequency analysis of nonlinear and nonstationary processes is, in general, a challenging task. Standard techniques, like short time Fourier transform, and wavelet transform are limited in addressing the problem. An alternative way to tackle the problem is to first decompose the signal into simpler components and then analyze them separately. This is the idea behind what is called the...
