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
$$
L(W)=\text{diag}(W\textbf{1})-W,\qquad \textbf{1}=(1,\dots,1)^T\in\mathbb{R}^n,
$$
whose eigenvalues $0=\lambda_1\leq \lambda_2 \leq \dots \leq \lambda_n$ and eigenvectors are related to the $k$...
In complex network analysis it is crucial to investigate the alteration of network structures that results from the targeted removal of vertices or edges, ranked by a centrality measure [3]. Unfortunately, a sequential recalculation of centralities after each node elimination is often impractical for large networks, and using the ranking before the removal often does not approximate accurately...
In this talk, we propose a numerical strategy to solve the Vlasov-Poisson equation, based on the dynamical low rank approximation, which is extremely efficient to reduce the complexity of such model.
Previous approaches based on dynamical low rank approximation destroyed the physical structure of the problem, as physical invariants were lost.
Starting from the conservative continuous ...
We consider the numerical solution of time-dependent space-tempered fractional diffusion equations. The use of Crank-Nicolson in time and of second-order accurate tempered weighted and shifted Grunwald difference in space leads to dense (multilevel) Toeplitz-like linear systems. By exploiting the related structure, we design an ad-hoc multigrid solver and multigrid-based preconditioners, all...
We present an algorithm for the solution of Sylvester equations with right-hand side of low rank, based on projection onto a block rational Krylov subspace. Extending the convergence analysis in [2] to the block case, we link the convergence with the problem of minimizing the norm of a small rational matrix over the spectra or field-of-values of the involved matrices. This is in contrast with...
The study of the asymptotic properties of the spectrum of a matrix sequence with a specific structure and increasing order has been a rich field of research and of great interest in applications, since they arise from approximation of integro-differential equations, also of fractional order (see e.g. [4, 5, 3] and references therein). This kind of approach leads most of the times to a huge...
In this talk I will consider the approximate computation of the von Neumann entropy of a large density matrix $A$ (i.e., a symmetric positive semidefinite matrix with unit trace), defined as $\text{trace}(f(A))$, where $f(z) = -z \log(z)$, which is an important task in several fields, such as quantum information theory and network science.
The approximation of $\text{trace}(f(A))$ can be...
Given a set of matrices $A_i \in \mathbb{C}^{n \times n}$ and a set of analytic functions $f_i : \mathbb{C} \mapsto \mathbb{C}$, we consider a regular matrix-valued function $\mathcal{D}(\lambda)=\sum_{i=0}^d f_i(\lambda) A_i$, that is such that $\det \left( \mathcal{D}(\lambda) \right)$ is not identically zero for $\lambda \in \mathbb{C}$. This class of functions includes for example...
Neural networks have achieved tremendous success in a wide variety of applications. However, their memory footprint and computational demand can render them impractical in application settings with limited hardware or energy resources. At the same time, overparameterization seems to be necessary in order to overcome the highly non-convex nature of the training optimization problem. An optimal...
In this poster [1], we deal with the numerical solution of systems of oscillatory second-order differential equations which often arise from the space semi-discretization of partial differential equations. Since these differential equations exhibit (pronounced or highly) oscillatory behavior, standard numerical methods are known to perform poorly. Our approach consists in directly discretizing...
Image reconstruction problems, like image deblurring and computer tomography, are usually ill-posed and require regularization.
A popular approach to regularization is to substitute the original problem with an optimization problem that minimizes the sum of two terms, an $\ell^2$ term and an $\ell^q$ term with $0
In this talk, we are going to present how perturbations in the co-efficient matrix $A$ propagate along the solutions of n-dimensional linear ordinary differential equations
$$ \left\{ \begin{array}{l} y^\prime(t) =Ay(t),\ t\geq 0,\\ y(0)=y_0. \end{array} \right. $$ In other words we are considering the conditioning of the problem $$ (y_0,A) \mapsto e^{tA}y_0 $$
and an...
The problem of optimizing of the spectral radius over a convex set of matrices is notoriously hard. The spectral radius is neither convex nor concave. It can be non-Lipschitz either, which makes the standard gradient-type methods inapplicable. We consider the case of nonnegative matrices, when the spectral radius becomes the Perron eigenvalue. For the matrix sets with the product structure ...
Time-dependent PDEs arise very often in many scientific areas, such as mechanics, biology, economics, or chemistry, just to name a few. Of late, many researchers have devoted their effort to devising parallel-in-time methods for the numerical solution of time-dependent PDEs. As opposed to the classical approach, in which an approximation of the solution at a time $t$ is computed by a...
The spectral analysis of matrix-sequences generated by the discretization and numerical approximation of partial differential equations, in case the domain is a generic Peano–Jordan measurable set, can be performed through the lens of generalized locally Toeplitz (GLT) theory.
In fact, it is observed that such matrix-sequences often present a spectral symbol, a measurable function...
Sketching can be seen as a random dimensionality reduction able to preserving the main features of the original problem with probabilistic confidence. Such kind of techniques is emerging as one of the most promising tools to boost numerical computations and it is quite well-known by theoretical computer scientists. Nowadays, sketching is getting popularity also in the numerical linear algebra...
A finite element method is defined by specifying three objects: a mesh, a finite dimensional functional space of shape functions and a collection of degrees of freedom, namely linear functionals on shape functions. It is rather this third aspect that characterises the method.
In the last twenty years a large amount of research has been devoted to generalising existing methods. These...
The study of spectral properties of structured graphs constitutes a recent and fast growing research field, useful in various branches of applied mathematics.
Spectral properties of the graph-Laplacian turn out to be useful in the discretization and in the computation of solutions of partial differential equations (PDEs), in the study of spectral gaps, clustering, etc. ([1],[2],[3]).
In...
Liquid water, besides being fundamental for life on Earth, has long fascinated scientists due to several anomalies. Different hypotheses have been put forward to explain these peculiarities. The most accredited one foresees the presence in the supercooled region of two phases at different densities: the low-density liquid phase (LDL) and the high-density liquid phase (HDL). In a previous study...
This talk focuses on preconditioning techniques that enhance the performance of iterative regularization methods in image deblurring. The preconditioners are applied to problems with different point spread functions (PSFs) and boundary conditions [1]. More precisely, we first consider the anti-identity preconditioner [3], which symmetrizes the coefficient matrix associated to problems with...
Neural ordinary differential equations (neural ODEs) are a new family of deep neural networks.
Essentially, a neural ODE is a differential equation whose vector field is a neural network.
Having a neural ODE as a part of a machine learning model makes the model more efficient than a standard one. Indeed, it is possible to train the neural ODE block of the model using the adjoint sensitivity...
