arXiv subjects

Alfio Borzì

Publications and source records attributed to Alfio Borzì.

14 recordsLinked to original sources

arXiv2026

Analysis and Optimal Design of Equilibria in the Vlasov-Poisson System

The optimal design of equilibrium particle distributions generated by external electric fields is investigated in the framework of stationary Maxwellian equilibria of the Vlasov-Poisson system. The resulting normalized Poisson-Boltzmann equation is analyzed, and existence and uniqueness of the self-consistent equilibrium potential are established by two complementary approaches. A Schauder fixed-point argument based on uniform estimates for the normalized nonlinear source and a variational approach based on a strictly convex free-energy functional are developed. Building upon these analytical results, a unified optimal design framework is formulated for equilibrium densities. Valley-potential, quadratic tracking, and Kullback-Leibler design criteria are treated within the same optimization framework. First-order optimality conditions are derived, and a projected nonlinear conjugate-gradient algorithm is proposed for the numerical solution of the resulting optimization problems.

math.OC↗

arXiv2026

A Low-Rank Symplectic Gradient Adjustment Method for Computing Nash Equilibria

This work presents a theoretical and numerical investigation of the symplectic gradient adjustment (SGA) method and of a low-rank SGA (LRSGA) method for efficiently solving revviolet optimization problems arising from two-player Nash games. The SGA method outperforms the gradient method by including second-order mixed derivatives computed at each iterate, which requires considerably larger computational effort. For this reason, an LRSGA method is proposed where the approximation to second-order mixed derivatives is obtained by rank-one updates. The theoretical analysis presented in this work focuses on novel convergence estimates for the SGA and LRSGA methods, including parameter bounds. The numerical experiments complement the theory by studying the behavior of LRSGA on explicit deterministic games with known equilibria and by evaluating its computational advantage over exact SGA on a CLIP-inspired neural-network training task, where LRSGA achieves comparable loss values lower CPU time than SGA with explicitly assembled mixed-derivative blocks.

math.OC↗

arXiv2026

A Moser-Type Construction for the Liouville Equation

A Moser-type construction for the kinetic Liouville equation is proposed, which is based on a characteristic-adapted interpolation. For Hamiltonian accelerations, the construction is reduced to a family of weighted elliptic problems in the velocity variable. The corresponding kinetic compatibility condition is derived.

math.OC↗

arXiv2026

MultiLRSGA: A method for multi-player differentiable games

We propose MultiLRSGA, an $h$-player extension of LRSGA for the computation of stable Nash equilibria in differentiable games. The method originates from the decomposition of the game Jacobian into symmetric and antisymmetric components, which motivates symplectic corrections designed to attenuate the rotational part of the dynamics. In the two-player setting, LRSGA replaces mixed second-order blocks with low-rank secant approximations. The passage to the multi-player case, however, is not a mere blockwise reformulation: the antisymmetric correction is no longer determined by a single pair of cross-interactions, but by a block antisymmetric operator collecting all pairwise couplings among the players. On this basis, we formulate MultiLRSGA by constructing, for each player, a low-rank approximation of the Jacobian of the partial gradient and extracting from it the blocks required to define an approximate antisymmetric correction. Under standard local assumptions around a stable Nash equilibrium, we prove local linear convergence of the method. The key technical ingredient is a lemma controlling the distance between the exact antisymmetric correction and its secant approximation in the $h$-player setting, thereby extending to the multi-player framework the convergence mechanism previously available for LRSGA. The proposed formulation preserves the computational advantages of low-rank symplectic corrections and is naturally suited to numerical validation on differentiable games with explicit payoffs and more than two agents.

math.OC↗

arXiv2025

The Pontryagin Maximum Principle for Training Convolutional Neural Networks

A novel batch sequential quadratic Hamiltonian (bSQH) algorithm for training convolutional neural networks (CNNs) with $L^0$-based regularization is presented. This methodology is based on a discrete-time Pontryagin maximum principle (PMP). It uses forward and backward sweeps together with the layerwise approximate maximization of an augmented Hamiltonian function, where the augmentation parameter is chosen adaptively. A technique for determining this augmentation parameter is proposed, and the loss-reduction and convergence properties of the bSQH algorithm are analysed theoretically and validated numerically. Results of numerical experiments in the context of image classification with a sparsity enforcing $L^0$-based regularizer demonstrate the effectiveness of the proposed method in full-batch and mini-batch modes.

math.OC↗

arXiv2024

Full- and low-rank exponential Euler integrators for the Lindblad equation

The Lindblad equation is a widely used quantum master equation to model the dynamical evolution of open quantum systems whose states are described by density matrices. These solution matrices are characterized by semi-positiveness and trace preserving properties, which must be guaranteed in any physically meaningful numerical simulation. In this paper, novel full- and low-rank exponential Euler integrators are developed for approximating the Lindblad equation that preserve positivity and trace unconditionally. Theoretical results are presented that provide sharp error estimates for the two classes of exponential integration methods. Results of numerical experiments are discussed that illustrate the effectiveness of the proposed schemes, beyond present state-of-the-art capabilities.

math.NA↗

arXiv2024

Optimal design of equilibrium solutions of the Vlasov-Poisson system by an external electric field

A new optimization framework to design steady equilibrium solutions of the Vlasov-Poisson system by means of external electric fields is presented. This optimization framework requires the minimization of an ensemble functional with Tikhonov regularization of the control field under the differential constraint of a nonlinear elliptic equation that models equilibrium solutions of the Vlasov-Poisson system. Existence of optimal control fields and their characterization as solutions to first-order optimality conditions are discussed. Numerical approximations and optimization schemes are developed to validate the proposed framework.

math.OC↗

arXiv2023

Fokker-Planck analysis of superresolution microscopy images

A method for the analysis of superresolution microscopy images is presented. This method is based on the analysis of stochastic trajectories of particles moving on the membrane of a cell with the assumption that this motion is determined by the properties of this membrane. Thus, the purpose of this method is to recover the structural properties of the membrane by solving an inverse problem governed by the Fokker-Planck equation related to the stochastic trajectories. Results of numerical experiments demonstrate the ability of the proposed method to reconstruct the potential of a cell membrane by using synthetic data similar those captured by superresolution microscopy of luminescent activated proteins.

math.OC↗

arXiv2022

Instability of oscillations in the Rosenzweig-MacArthur model of one consumer and two resources

The system of two resources $R_1$, $R_2$ and one consumer $C$ is investigated within the Rosenzweig-MacArthur model with Holling type II functional response. The rates $β_i$ of consumption of resources $i=1,2$ are coupled by the condition $β_1+β_2=1$. The dynamic switching is introduced by a maximization of $C$: $dβ_1/dt=(1/τ) dC/dβ_1$, where the characteristic time $τ$ is large but finite. The space of parameters where both resources coexist is explored numerically. The results indicate that oscillations of $C$ and mutually synchronized $R_i$ which appear at $β_i=0.5$ are destabilized for $β_i$ larger or smaller. Then, the system is driven to one of fixed points where either $β_1>0.5$ and $R_1<R_2$ or the opposite. This behaviour is explained as an inability of the consumer to change the preferred resource, once it is chosen.

physics.soc-ph↗

arXiv2019

An extended model of wine fermentation including aromas and acids

The art of viticulture and the quest for making wines has a long tradition and it just started recently that mathematicians entered this field with their main contribution of modelling alcoholic fermentation. These models consist of systems of ordinary differential equations that describe the kinetics of the bio-chemical reactions occurring in the fermentation process. The aim of this paper is to present a new model of wine fermentation that accurately describes the yeast dying component, the presence of glucose transporters, and the formation of aromas and acids. Therefore the new model could become a valuable tool to predict the taste of the wine and provide the starting point for an emerging control technology that aims at improving the quality of the wine by steering a well-behaved fermentation process that is also energetically more efficient. Results of numerical simulations are presented that successfully confirm the validity of the proposed model by comparison with real data.

q-bio.MN↗

arXiv2018

A theoretical investigation of Brockett's ensemble optimal control problems

This paper is devoted to the analysis of problems of optimal control of ensembles governed by the Liouville (or continuity) equation. The formulation and study of these problems have been put forward in recent years by R.W. Brockett, with the motivation that ensemble control may provide a more general and robust control framework. Following Brockett's formulation of ensemble control, a Liouville equation with unbounded drift function, and a class of cost functionals that include tracking of ensembles and different control costs is considered. For the theoretical investigation of the resulting optimal control problems, a well-posedness theory in weighted Sobolev spaces is presented for the Liouville and transport equations. Then, a class of non-smooth optimal control problems governed by the Liouville equation is formulated and existence of optimal controls is proved. Furthermore, optimal controls are characterised as solutions to optimality systems; such a characterisation is the key to get (under suitable assumptions) also uniqueness of optimal controls.

math.AP↗

arXiv2017

Investigation of optimal control problems governed by a time-dependent Kohn-Sham model

Many application models in quantum physics and chemistry require to control multi-electron systems to achieve a desired target configuration. This challenging task appears possible in the framework of time-dependent density functional theory (TDDFT) that allows to describe these systems while avoiding the high dimensionality resulting from the multi-particle Schrödinger equation. For this purpose, the theory and numerical solution of optimal control problems governed by a Kohn-Sham TDDFT model are investigated, considering different objectives and a bilinear control mechanism. Existence of optimal control solutions and their characterization as solutions to Kohn-Sham TDDFT optimality systems are discussed. To validate this control framework, a time-splitting discretization of the optimality systems and a nonlinear conjugate gradient scheme are implemented. Results of numerical experiments demonstrate the computational capability of the proposed control approach.

math.OC↗

arXiv2017

A theoretical investigation of time-dependent Kohn-Sham equations

In this work, the existence, uniqueness and regularity of solutions to the time-dependent Kohn-Sham equations are investigated. The Kohn-Sham equations are a system of nonlinear coupled Schrödinger equations that describe multi-particle quantum systems in the framework of the time dependent density functional theory. In view of applications with control problems, the presence of a control function and of an inhomogeneity are also taken into account.

math.AP↗

arXiv2014

Novel model for wine fermentation including the yeast dying phase

This paper presents a novel model for wine fermentation including a death phase for yeast and the influence of oxygen on the process. A model for the inclusion of the yeast dying phase is derived and compared to a model taken from the literature. The modeling ability of the several models is analyzed by comparing their simulation results.

q-bio.QM↗