arXiv subjects

Franco Zivcovich

Publications and source records attributed to Franco Zivcovich.

5 recordsLinked to original sources

arXiv2025

Sparse stabilization of mean-field agent dynamics through a three-operator splitting method

We study the sparse stabilization of nonlinear multi-agent systems within a mean-field optimal control framework. The goal is to drive large populations of interacting agents toward consensus with minimal control effort. In the mean-field limit, the dynamics are described by a Vlasov-type kinetic equation, and sparsity is enforced through an l1-l2 penalization in the cost functional. The resulting nonsmooth optimization problem is solved via a three-operator splitting (TOS) method that separately handles smooth, nonsmooth, and constraint components through gradient, shrinkage, and projection steps. A particle-based Monte Carlo discretization with random batch interactions enables scalable computation while preserving the mean-field structure. Numerical experiments on the Cucker-Smale model demonstrate effective consensus formation with sparse, localized control actions, confirming the efficiency and robustness of the proposed approach.

math.OC↗

arXiv2024

A $μ$-mode approach for exponential integrators: actions of $φ$-functions of Kronecker sums

We present a method for computing actions of the exponential-like $φ$-functions for a Kronecker sum $K$ of $d$ arbitrary matrices $A_μ$. It is based on the approximation of the integral representation of the $φ$-functions by Gaussian quadrature formulas combined with a scaling and squaring technique. The resulting algorithm, which we call PHIKS, evaluates the required actions by means of $μ$-mode products involving exponentials of the small sized matrices $A_μ$, without forming the large sized matrix $K$ itself. PHIKS, which profits from the highly efficient level 3 BLAS, is designed to compute different $φ$-functions applied on the same vector or a linear combination of actions of $φ$-functions applied on different vectors. In addition, thanks to the underlying scaling and squaring techniques, the desired quantities are available simultaneously at suitable time scales. All these features allow the effective usage of PHIKS in the exponential integration context. In fact, our newly designed method has been tested on popular exponential Runge--Kutta integrators of stiff order from one to four, in comparison with state-of-the-art algorithms for computing actions of $φ$-functions. The numerical experiments with discretized semilinear evolutionary 2D or 3D advection--diffusion--reaction, Allen--Cahn, and Brusselator equations show the superiority of the proposed $μ$-mode approach.

math.NA↗

arXiv2022

A $μ$-mode BLAS approach for multidimensional tensor-structured problems

In this manuscript, we present a common tensor framework which can be used to generalize one-dimensional numerical tasks to arbitrary dimension $d$ by means of tensor product formulas. This is useful, for example, in the context of multivariate interpolation, multidimensional function approximation using pseudospectral expansions and solution of stiff differential equations on tensor product domains. The key point to obtain an efficient-to-implement BLAS formulation consists in the suitable usage of the $μ$-mode product (also known as tensor-matrix product or mode- $n$ product) and related operations, such as the Tucker operator. Their MathWorks MATLAB/GNU Octave implementations are discussed in the paper, and collected in the package KronPACK. We present numerical results on experiments up to dimension six from different fields of numerical analysis, which show the effectiveness of the approach.

math.NA↗

arXiv2022

A second-order low-regularity correction of Lie splitting for the semilinear Klein--Gordon equation

The numerical approximation of the semilinear Klein--Gordon equation in the $d$-dimensional space, with $d=1,2,3$, is studied by analyzing the consistency errors in approximating the solution. By discovering and utilizing a new cancellation structure in the semilinear Klein--Gordon equation, a low-regularity correction of the Lie splitting method is constructed, which can have second-order convergence in the energy space under the regularity condition $(u,\partial_tu)\in L^\infty(0,T;H^{1+\frac{d}{4}}\times H^{\frac{d}{4}})$, where $d=1,2,3$ denotes the dimension of space. In one dimension, the proposed method is shown to have a convergence order arbitrarily close to $\frac53$ in the energy space for solutions in the same space, i.e. no additional regularity in the solution is required. Rigorous error estimates are presented for a fully discrete spectral method with the proposed low-regularity time-stepping scheme. Numerical examples are provided to support the theoretical analysis and to illustrate the performance of the proposed method in approximating both nonsmooth and smooth solutions of the semilinear Klein--Gordon equation.

math.NA↗

arXiv2021

A $μ$-mode integrator for solving evolution equations in Kronecker form

In this paper, we propose a $μ$-mode integrator for computing the solution of stiff evolution equations. The integrator is based on a $d$-dimensional splitting approach and uses exact (usually precomputed) one-dimensional matrix exponentials. We show that the action of the exponentials, i.e. the corresponding batched matrix-vector products, can be implemented efficiently on modern computer systems. We further explain how $μ$-mode products can be used to compute spectral transforms efficiently even if no fast transform is available. We illustrate the performance of the new integrator by solving, among the others, three-dimensional linear and nonlinear Schrödinger equations, and we show that the $μ$-mode integrator can significantly outperform numerical methods well established in the field. We also discuss how to efficiently implement this integrator on both multi-core CPUs and GPUs. Finally, the numerical experiments show that using GPUs results in performance improvements between a factor of $10$ and $20$, depending on the problem.

math.NA↗