Alberto Bucci (Charles University)
16.00 – 29 settembre 2025
Low-rank structures appear in an astonishing variety of scientific and engineering problems. From compressing images and videos, to reconstructing incomplete data in tomography, to accelerating the solution of large-scale differential equations, low-rank approximations offer a unifying principle: complex, high-dimensional data often live close to a much simpler, low-dimensional representation.
In this talk, we will introduce the concept of low-rankness and explain why it arises so naturally in applications. We will then explore how exploiting low-rank structure leads to dramatic gains in efficiency, both in terms of storage and computation, and present concrete examples from imaging, tomography, numerical PDEs, video recognition, and beyond. The goal is to convey the ubiquity and power of low-rank methods, and to provide an accessible overview of the mathematical ideas that make them such a central tool in modern computational science.
Matrix functions play a central role in many areas of scientific computing, ranging from differential equations and network analysis to quantum mechanics and data science. Common examples include the matrix exponential, logarithm, square root and sign function. In many applications, one is interested not in the full matrix function f(A), but only in its action on a given vector, i.e. the product f(A)b. Krylov subspace methods offer an efficient way to perform this task, by relying only on matrix–vector products or linear system solves, rather than forming f(A) explicitly. By projecting the problem onto a low-dimensional polynomial or rational Krylov subspace built from A and b, these methods compute accurate approximations whose convergence is closely related to the quality of polynomial and rational approximations of the scalar function f.
The goal of this talk is to provide a broad and accessible introduction to Krylov subspace methods for matrix functions.
We begin by recalling the fundamental ideas of polynomial and rational approximation for scalar functions, which form the theoretical foundation of these methods. Building on this intuition, we will explore how Krylov subspaces can be used to approximate f(A)b efficiently, and highlight how the underlying polynomial and rational approximations lead to algorithms with distinct computational and convergence properties. Throughout the talk, we will demonstrate these concepts with a series of illustrative numerical examples.
The nearest correlation matrix problem consists in finding the closest valid correlation
matrix to a given symmetric matrix that may fail to be positive semi-definite. In other
words, given a symmetric unit-diagonal matrix that is not a proper correlation matrix, one
seeks the nearest positive semi-definite matrix with unit diagonal entries.
We address the problem of finding the nearest correlation matrix to a given symmetric
unit-diagonal matrix under additional structural constraints such as sparsity, block, or band
patterns. This task arises in applications where positive semi-definiteness must be restored
without losing essential structure.
Our method combines a two-level iteration: a structured gradient flow computes feasible perturbations within the prescribed structure, while an outer Newton scheme adjusts
their magnitude to meet accuracy requirements. To handle high-dimensional settings efficiently, we replace full eigenvalue decompositions with a Rayleigh quotient approximation, focusing only on the critical invariant subspace needed to restore positive semi-definiteness.
The resulting algorithm systematically incorporates structural constraints into the nearest correlation matrix problem. Numerical experiments highlight its robustness across diverse structured scenarios, with promising applications in finance, statistics, and network
analysis.
Many networks include both allies and adversaries, excitations and inhibitions, agreements and disagreements. Signed networks offer a formal way to represent these interactions, but negative edges introduce many challenges in standard notions like communities, distances, and centralities. In this talk, I will present the most common concepts and methodologies used in the study of signed networks, with a focus on structural balance and mesoscale structures. I will then introduce the Gremban expansion, a method that transforms signed networks into unsigned ones without information loss. This framework enables a principled separation between communities and factions, addressing the limitations of existing algorithms (which typically recover only factions).
Toeplitz matrices appear in many applications such as signal processing, time series analysis, and the discretization of differential and integral equations. Their defining feature-constant diagonals-implies that they contain only linear information. However, when computing functions of Toeplitz matrices, this structure appears to be lost.
In this talk, we present an approach for efficiently approximating f(T) by uncovering hidden structure. The key idea is to transform the Toeplitz matrix into an equivalent representation that can be well approximated by hierarchical low-rank formats. This structure is numerically preserved by matrix functions, and its computation can be improved using low-rank update techniques.
I sistemi biologici costituiti da cellule ed elementi del citoscheletro possono organizzarsi in una fase nematica, in cui fibre allungate si allineano parallelamente tra loro, generando un ordine orientazionale parziale. Questo ordine può essere descritto da un campo macroscopico di direttori, che riflette l’orientamento locale delle fibre. I difetti topologici nell’ordine nematico corrispondono a singolarità di questo campo e sono piuttosto comuni; essi vengono classificati in base alla loro carica topologica.
L’emergere dell’ordine nematico da una disposizione inizialmente isotropa o disordinata delle fibre su superfici chiuse e deformabili gioca un ruolo cruciale nella morfogenesi della materia biologica attiva. Un esempio significativo è la rigenerazione di Hydra, un piccolo invertebrato d’acqua dolce: un frammento asportato si richiude inizialmente in una superficie sferica in uno stato approssimativamente isotropo e solo successivamente le fibre di actina del tessuto epiteliale organizzano un ordine nematico. Su una superficie con topologia sferica, la formazione di difetti topologici è inevitabile, e la carica totale dei difetti deve essere pari a +2.
In questo lavoro presento un modello continuo che accoppia il tensore d’ordine bidimensionale di Landau-de Gennes, che descrive l’ordine nematico planar, con la meccanica di una membrana sferica deformabile e a massa conservata. Studiando la transizione di fase isotropico-nematico, indotta da una diminuzione di temperatura che simula l’attivazione naturale dell’ordine nematico nelle fibre actomiosiniche, realizziamo sia un’analisi lineare sia un’analisi di biforcazione debolmente non lineare. Mostriamo che l’insorgenza dell’ordine nematico rompe spontaneamente la simmetria sferica, generando morfologie di equilibrio distinte che dipendono dalla deformabilità della membrana.
Le configurazioni assialsimmetriche, caratterizzate da due difetti topologici +1 ai poli, emergono tramite una biforcazione discontinua (transcritica), dando luogo a una forma prolata stabile, con allineamento meridiano dei direttori, e a una forma oblata metastabile. Le configurazioni non assialsimmetriche, invece, presentano quattro difetti +1/2 disposti in geometria quadrata e emergono tramite una biforcazione continua (supercritica). La morbidezza della membrana determina la natura di primo ordine (discontinua) della transizione; nel limite di rigidità infinita, infatti, tutte le biforcazioni diventano continue. Inoltre, i difetti interi si accoppiano fortemente con la ridistribuzione locale di massa, causando assottigliamento o ispessimento della membrana, mentre i difetti semi-interi non inducono deformazioni locali. Questi risultati forniscono un quadro puramente meccanico per comprendere la formazione dell’asse corporeo, che determina la futura posizione di testa e piede dell’ Hydra, e la morfogenesi mediata dai difetti nelle vescicole biologiche.
