Fabio Durastante

Fabio Durastante, Beatrice Meini, Luca Saluzzi

Time-evolving networks, or temporal networks, play a crucial role in modeling dynamic interactions across various domains, including biology, social sciences, and information technology. Unlike static networks, these systems undergo continuous changes in topology and edge weights, influencing processes such as information flow, transportation efficiency, and neural activity. Understanding and controlling these networks are essential for predicting future behavior and optimizing dynamic processes. This work focuses on the problem of dynamic centrality, a measure of node importance in time-dependent networks. Specifically, we address how to steer network centrality to a desired state by making minimal modifications to the network structure. This problem is formulated as an optimal control problem for an ordinary differential equation, either matrix- or vector-based, where the control acts on network edges. The proposed framework generalizes centrality control problems studied in static networks and leverages the Pontryagin Maximum Principle for efficient solutions. For large-scale problems, the required matrix-function actions are approximated by Krylov-type techniques, avoiding the explicit formation of dense matrix functions. Numerical experiments on synthetic and real temporal networks show that the proposed framework can effectively steer receive centrality under prescribed control constraints.

Francesca Arrigo, Fabio Durastante

We develop a novel framework for modeling diffusion on complex networks by constructing Laplacian-like operators based on walks around a graph. Our approach introduces a parametric family of walk-based Laplacians that naturally incorporate memory effects by excluding or downweighting backtracking trajectories, where walkers immediately revisit nodes. The framework includes: (i) walk-based Laplacians that count all traversals in the network; (ii) nonbacktracking variants that eliminate immediate reversals; and (iii) backtrack-downweighted variants that provide a continuous interpolation between these two regimes. We establish that these operators extend the definition of the standard Laplacian and also preserve some of its properties. We present efficient algorithms using Krylov subspace methods for computing them, ensuring applicability of our proposed framework to large networks. Extensive numerical experiments on real-world networks validate the modeling flexibility of our approach and demonstrate the computational efficiency of the proposed algorithms, including GPU acceleration.

Pasqua D'Ambra, Fabio Durastante, Salvatore Filippone

Isogeometric analysis (IgA) offers enhanced approximation capabilities for the discretization of elliptic boundary-value problems, yet it results in large, sparse, and increasingly ill-conditioned linear systems due to higher interconnectivity among degrees of freedom. In particular, the discretization with tensor-product B-splines or NURBS of degree $p$ on a mesh with $n$ elements per parametric direction leads to symmetric positive-definite systems of the form $K\mathbf{u} = \mathbf{F}$, where the matrix bandwidth and condition number scale unfavorably with both $p$ and spatial dimension $d$. To address the computational challenges posed by such systems, especially in three-dimensional or high-order scenarios, Krylov subspace methods with specialized preconditioners become essential. This paper investigates the efficacy of algebraic multigrid (AMG) preconditioners tailored for IgA-based discretizations, with a focus on performance in modern high-performance computing (HPC) environments. Leveraging the Parallel Sparse Computation Toolkit (PSCToolkit), we explore distributed-memory and GPU-accelerated strategies for solving large-scale problems. The study assesses algorithmic efficiency and scalability across a range of benchmark tests. The results demonstrate that AMG preconditioners can achieve robust and scalable performance, confirming their potential as practical solvers for large IgA systems in engineering and scientific applications.

Stefano Cipolla, Fabio Durastante, Miryam Gnazzo et al.

Reversibility is a key property of Markov chains, central to algorithms such as Metropolis-Hastings and other MCMC methods. Yet many applications yield non-reversible chains, motivating the problem of approximating them by reversible ones with minimal modification. We formulate this task as a matrix nearness problem and focus on the practically relevant case of sparse transition matrices. The resulting optimization problem is a quadratic programming problem, and numerical experiments illustrate the effectiveness of the approach. This framework provides a principled way to enforce reversibility and sparsity patterns in Markov chains with applications in MCMC, computational chemistry, and data-driven modeling.

Fabio Durastante, Mariarosa Mazza

We study energy-conserving Hamiltonian Boundary Value Methods (HBVMs) for Hamiltonian systems, which arise in applications where long-term preservation of energy and symplecticity is essential. HBVMs are multi-stage schemes whose stage equations reformulate as matrix equations with a low-rank right-hand side. For linear systems, we exploit this structure directly via Krylov projection solvers. For nonlinear systems, we leverage it within simplified Newton iterations and as a preconditioner in a Newton--Krylov framework, combined with adaptive time-stepping for robust convergence. Numerical experiments on semi-discretized wave equations demonstrate the efficiency and robustness of the proposed approach.

Alberto Bucci, Fabio Durastante

In this paper we present an equivalence result between a randomized version of Anderson Acceleration and of randomized GMRES for linear problems. Namely, we extend the classical result of Walker and Ni (2011) to the case in which the least-squares problem in Anderson Acceleration is solved in a sketched space defined by a random projection. This equivalence suggests potential avenues for further research in the design and analysis of randomized acceleration methods.

Fabio Durastante, Miryam Gnazzo, Beatrice Meini

We address the algorithmic problem of determining the reversible Markov chain $\tilde X$ that is closest to a given Markov chain $X$, with an identical stationary distribution. More specifically, $\tilde X$ is the reversible Markov chain with the closest transition matrix, in the Frobenius norm, to the transition matrix of $X$. To compute the transition matrix of $\tilde X$, we propose a novel approach based on Riemannian optimization. Our method introduces a modified multinomial manifold endowed with a prescribed stationary vector, while also satisfying the detailed balance conditions, all within the framework of the Fisher metric. We evaluate the performance of the proposed approach in comparison with an existing quadratic programming method and demonstrate its effectiveness through a series of synthetic experiments, as well as in the construction of a reversible Markov chain from transition count data obtained via direct estimation from a stochastic differential equation.

Luca Lombardo, Fabio Durastante

Sparse matrix kernels form the computational backbone of scientific computing, traditionally relying on C/C++ and Fortran implementations that prioritize performance over memory safety. This work evaluates Rust as a systems-level alternative for sparse linear algebra by implementing and benchmarking three core workloads: sparse matrix-vector multiplication (SpMV), Lanczos-based Krylov methods, and matrix-exponential evaluation. We compare native Rust code against established baselines (Intel oneMKL, Eigen, PETSc, and PSBLAS) across a suite of representative matrices. Our results show that Rust's sparse kernels achieve performance comparable to Eigen and PSBLAS, tracking the state-of-the-art for CSC formats, while trailing PETSc's advanced blocked CSR optimizations. By analyzing compile-time monomorphization, SIMD vectorization, and FFI boundaries, we assess the practical impact of Rust's safety model and ecosystem readiness. The study provides concrete, evidence-based guidance for modernizing high-performance numerical software stacks.