Searcharxiv⌕ Search

arXiv subjects

Farid Bozorgnia

Publications and source records attributed to Farid Bozorgnia.

At least 19 recordsLinked to original sources
arXiv2026

Fractional Heat Kernel for Semi-Supervised Graph Learning with Small Training Sample Size

We develop a source-driven fractional heat-kernel framework for semi-super\-vised graph learning that combines nonlocal propagation with sustained label information. A fixed nonzero label source compatible with the Laplacian null space prevents asymptotic collapse into that space, providing a mechanism for mitigating oversmoothing at long diffusion times. The fractional order controls the relative modal attenuation and the spectral weighting of the sustained response, while the diffusion time sets the propagation horizon. We characterize conservation laws and equilibria on normalized and disconnected graphs, develop a null-space deflation, and analyze the approximation of the propagators. On Two-Moon, fractional orders improve source-free propagation, while compatible source-driven diffusion exceeds $96\%$ mean accuracy with one training label per class at orders $0.8$ and $1$. On Cora and CiteSeer, the source-driven pipeline improves mean accuracy over GAT by $9.2$ and $8.0$ percentage points at one training label per class, with model selection on $500$ labeled validation nodes and closely comparable classical and fractional pipeline configurations; it matches GAT on PubMed and trails it at ten and twenty labels per class on Cora. Within GraphHeat, validation-based exponent selection at a common diffusion time yields a paired gain of $1.45$ percentage points at one label per class.

cs.LG↗
arXiv2026

Numerical Algorithms for Partially Segregated Elliptic Systems

We develop numerical methods for elliptic systems governed by a ternary partial-segregation constraint, in which three nonnegative components are required to have a vanishing pointwise product throughout the domain. This condition forces at least one component to vanish at every spatial location while still permitting coexistence of the other two components, and therefore produces a highly nonconvex admissible set. In particular, the limiting constrained Dirichlet minimization problem need not have a unique minimizer, as we show by an explicit one-dimensional example with two distinct global minimizers. We propose two complementary computational frameworks. The first is a strong-competition penalty method solved by a relaxed Picard iteration and a sequential semi-implicit variant, combined with continuation in the penalty parameter. For this formulation, we establish existence of penalized minimizers and weak solutions for every fixed penalty parameter, monotone bracketing properties for the fixed-point iteration, geometric convergence under an explicit contraction condition, and interior exponential estimates in regions of uniformly positive competition. The second framework is a discrete projected-gradient method, together with a safeguarded inertial variant, based on an explicit pointwise projection onto the ternary partial-segregation set. Numerical experiments for several discrete boundary configurations show that, when started on a common solution branch, the two methods produce the same interfaces, and that apparent differences between them reflect the selection of different branches by their default initializations.

math.NA↗
arXiv2026

Inverse Robin Spectral Problem for the p-Laplace Operator

We study an inverse Robin spectral problem for the $p$-Laplacian with mixed Dirichlet--Robin boundary conditions. The unknown coefficient lies on an inaccessible Robin portion, and the data are the principal eigenvalue and nonlinear conormal flux on an accessible Dirichlet patch. For a coating of thickness $\varepsilonρ$ and conductivity $\varepsilon^{p-1}$, we prove convergence of the principal eigenvalue and strong $W^{1,p}$ convergence of the eigenfunctions to an effective Robin problem with coefficient $h=ρ^{-(p-1)}$. The Rayleigh characterization gives a two-sided comparison inequality, ordered uniqueness and weighted $L^1$ stability, and Fréchet differentiability of the eigenvalue map for $1 2$; together with a quantitative linearized estimate, it yields conditional local stability. On finite-dimensional classes, branch differentiability and linearized injectivity yield local Lipschitz stability (with branch differentiability automatic when $p=2$), while monotone one-parameter families are identifiable from the principal eigenvalue.

math.AP↗
arXiv2026

Rate of convergence of the $p$-torsion function to the distance function

We prove that the Dirichlet $p$-torsion function $u_p$ converges to the distance to the boundary $d$ at the rate $O(1/p)$ in the uniform norm, on every bounded domain and with a constant depending only on the dimension and the diameter. The upper bound comes from a radial barrier with its pole at a boundary point, which is an admissible comparison function for $p>n$ and needs no boundary regularity; the lower bound comes from an inscribed ball. On the ball the error equals $(1+\log n)/p+O(p^{-2})$, so the order $1/p$ cannot be improved in general. Under a uniform exterior ball condition an annular barrier gives $u_p\le K_Ω^{1/(p-1)}d$ for every $p>1$, with $K_Ω$ explicit in the dimension, the diameter and the exterior radius; on convex sets the constant is the diameter, and a multiple of $d$ is a supersolution whenever $-Δd$ has a positive distributional lower bound. On $C^2$ domains, the same barrier and the classical gradient maximum principle give $\norm{\nabla u_p}_{L^\infty}^{p-1}\le K_Ω$. When $-Δd$ is a measure of finite total variation, which we prove for $C^1$ domains with a uniform exterior ball, the gradients converge in $L^q$ at the rate $O(p^{-1/2})$ for $q\le2$ and $O(p^{-1/q})$ for $q\ge2$; these exponents are not claimed to be sharp. The explicit ball profile shows that the gradients do not converge uniformly.

math.AP↗
arXiv2026

Optimal Control in Hilbert Complex Spaces with Finite Element Exterior Calculus

We develop a framework for PDE-constrained optimal control on Hilbert complexes and its structure-preserving discretization by finite element exterior calculus (FEEC), with emphasis on problems where nontrivial topology creates physically meaningful global modes. We separate the gauge-fixed local state from harmonic components representing global circulation or flux, and introduce finite-dimensional topological actuators whose reachable modes are determined jointly by domain topology and actuator rank. We establish well-posedness, weak optimality conditions, reachability results, and FEEC error estimates preserving the harmonic structure. Numerical experiments on contractible and multiply connected domains demonstrate circulation and cavity- flux control, rank-dependent reachability, mesh-independent optimization, and topology-induced singular convergence. We further extend the framework to time-dependent Maxwell control, where harmonic electric circulation and magnetic flux become dynamical states exhibiting explicit conservation and reachability properties.

math.OC↗
arXiv2026

Coincidence criteria and junction obstructions for two partially segregated elliptic systems

We compare two singularly perturbed elliptic systems with the same partial-segregation constraint but different limiting mechanisms. System~A is determined by harmonic-difference identities and admits an explicit lower-envelope limit, whereas System~B converges, along subsequences, to a constrained Dirichlet-energy minimizer. We show that the System~A profile need not be stationary for the limiting variational problem. We establish conditional coincidence criteria, give explicit data for which the interfaces differ, and derive a planar triple-junction obstruction. In particular, coincidence with the standard variational cone requires the triangle formed by the harmonic gradients to be equilateral. This condition is necessary but not sufficient.

math.AP↗
arXiv2025

Gamma Convergence of Partially Segregated Elliptic Systems

We study partially segregated elliptic systems through the use of penalized energy functionals. These systems arise from the minimization of Gross-Pitaevskii-type energies that capture the behavior of multi-component ultracold gas mixtures and other systems involving multiple interacting fluid or gas species. In the case when the domain is planar, i.e., in $\mathbb{R}^2$, our main result is the Gamma convergence of penalized energy to the constrained Dirichlet energy with strict segregation. The proof combines lower semicontinuity arguments with a recovery sequence construction based on geometric decompositions near interfaces and triple junctions. This establishes a rigorous variational link between the penalized and constrained formulations.

math.AP↗
arXiv2025

Enhancement of Quantum Semi-Supervised Learning via Improved Laplacian and Poisson Methods

This paper develops a hybrid quantum approach for graph-based semi-supervised learning to enhance performance in scenarios where labeled data is scarce. We introduce two enhanced quantum models, the Improved Laplacian Quantum Semi-Supervised Learning (ILQSSL) and the Improved Poisson Quantum Semi-Supervised Learning (IPQSSL), that incorporate advanced label propagation strategies within variational quantum circuits. These models utilize QR decomposition to embed graph structure directly into quantum states, thereby enabling more effective learning in low-label settings. We validate our methods across four benchmark datasets like Iris, Wine, Heart Disease, and German Credit Card -- and show that both ILQSSL and IPQSSL consistently outperform leading classical semi-supervised learning algorithms, particularly under limited supervision. Beyond standard performance metrics, we examine the effect of circuit depth and qubit count on learning quality by analyzing entanglement entropy and Randomized Benchmarking (RB). Our results suggest that while some level of entanglement improves the model's ability to generalize, increased circuit complexity may introduce noise that undermines performance on current quantum hardware. Overall, the study highlights the potential of quantum-enhanced models for semi-supervised learning, offering practical insights into how quantum circuits can be designed to balance expressivity and stability. These findings support the role of quantum machine learning in advancing data-efficient classification, especially in applications constrained by label availability and hardware limitations.

quant-ph↗
arXiv2024

Graph-Based Semi-Supervised Segregated Lipschitz Learning

This paper presents an approach to semi-supervised learning for the classification of data using the Lipschitz Learning on graphs. We develop a graph-based semi-supervised learning framework that leverages the properties of the infinity Laplacian to propagate labels in a dataset where only a few samples are labeled. By extending the theory of spatial segregation from the Laplace operator to the infinity Laplace operator, both in continuum and discrete settings, our approach provides a robust method for dealing with class imbalance, a common challenge in machine learning. Experimental validation on several benchmark datasets demonstrates that our method not only improves classification accuracy compared to existing methods but also ensures efficient label propagation in scenarios with limited labeled data.

cs.LG↗
arXiv2024

A Laplacian-based Quantum Graph Neural Network for Semi-Supervised Learning

Laplacian learning method is a well-established technique in classical graph-based semi-supervised learning, but its potential in the quantum domain remains largely unexplored. This study investigates the performance of the Laplacian-based Quantum Semi-Supervised Learning (QSSL) method across four benchmark datasets -- Iris, Wine, Breast Cancer Wisconsin, and Heart Disease. Further analysis explores the impact of increasing Qubit counts, revealing that adding more Qubits to a quantum system doesn't always improve performance. The effectiveness of additional Qubits depends on the quantum algorithm and how well it matches the dataset. Additionally, we examine the effects of varying entangling layers on entanglement entropy and test accuracy. The performance of Laplacian learning is highly dependent on the number of entangling layers, with optimal configurations varying across different datasets. Typically, moderate levels of entanglement offer the best balance between model complexity and generalization capabilities. These observations highlight the crucial need for precise hyperparameter tuning tailored to each dataset to achieve optimal performance in Laplacian learning methods.

cs.LG↗
arXiv2024

Improved Graph-based semi-supervised learning Schemes

In this work, we improve the accuracy of several known algorithms to address the classification of large datasets when few labels are available. Our framework lies in the realm of graph-based semi-supervised learning. With novel modifications on Gaussian Random Fields Learning and Poisson Learning algorithms, we increase the accuracy and create more robust algorithms. Experimental results demonstrate the efficiency and superiority of the proposed methods over conventional graph-based semi-supervised techniques, especially in the context of imbalanced datasets.

cs.LG↗
arXiv2024

The regularity with respect to domains of the additive eigenvalues of superquadratic Hamilton--Jacobi equation

We study the additive eigenvalues on changing domains, along with the associated vanishing discount problems. We consider the convergence of the vanishing discount problem on changing domains for a general scaling type $Ω_λ= (1+r(λ))Ω$ with a continuous function $r$ and a positive constant $λ$. We characterize all solutions to the ergodic problem on $Ω$ in terms of $r$. In addition, we demonstrate that the additive eigenvalue $λ\mapsto c_{Ω_λ}$ on a rescaled domain $Ω_λ= (1+λ)Ω$ possesses one-sided derivatives everywhere. Additionally, the limiting solution can be parameterized by a real function, and we establish a connection between the regularity of this real function and the regularity of $λ\mapsto c_{Ω_λ}$. We provide examples where higher regularity is achieved.

math.AP↗
arXiv2024

Optimal Control of Two-Phase Membrane Problem

We consider an optimal control problem where the state is governed by a free boundary problem called the two-phase membrane problem and the control appears in the coefficients of the characteristic function of the positivity and negativity parts of the solution. Our investigation focuses on various properties associated with the control-to-state map. Due to the non-differentiability of this map, we regularize the state equation. The existence, uniqueness, and characterization of the optimal pairs are established.

math.OC↗
arXiv2023

The Infinity Laplacian eigenvalue problem: reformulation and a numerical scheme

In this work, we present an alternative formulation of the higher eigenvalue problem associated to the infinity Laplacian, which opens the door for numerical approximation of eigenfunctions. A rigorous analysis is performed to show the equivalence of the new formulation to the traditional one. Subsequently, we present consistent monotone schemes to approximate infinity ground states and higher eigenfunctions on grids. We prove that our method converges (up to a subsequence) to a viscosity solution of the eigenvalue problem, and perform numerical experiments which investigate theoretical conjectures and compute eigenfunctions on a variety of different domains.

math.NA↗
arXiv2022

Graph Based Semi-supervised Learning Using Spatial Segregation Theory

In this work we address graph based semi-supervised learning using the theory of the spatial segregation of competitive systems. First, we define a discrete counterpart over connected graphs by using direct analogue of the corresponding competitive system. This model turns out doesn't have a unique solution as we expected. Nevertheless, we suggest gradient projected and regularization methods to reach some of the solutions. Then we focus on a slightly different model motivated from the recent numerical results on the spatial segregation of reaction-diffusion systems. In this case we show that the model has a unique solution and propose a novel classification algorithm based on it. Finally, we present numerical experiments showing the method is efficient and comparable to other semi-supervised learning algorithms at high and low label rates.

math.NA↗
arXiv2021

Decay estimate for the solution of the evolutionary damped $p$-Laplace equation

In this note, we study the asymptotic behavior, as $t$ tends to infinity, of the solution $u$ to the evolutionary damped $p$-Laplace equation \begin{equation*} u_{tt}+a\, u_t =Δ_p u \end{equation*} with Dirichlet boundary values. Let $u^*$ denote the stationary solution with same boundary values, then the $W^{1,p}$-norm of $u(t) - u^{*}$ decays for large $t$ like $t^{-\frac{1}{(p-1)p}}$, in the degenerate case $ p > 2$.

math.AP↗
arXiv2021

A degenerate elliptic-parabolic system arising in competitive contaminant transport

In this work we investigate a coupled system of degenerate and nonlinear partial differential equations governing the transport of reactive solutes in groundwater. We show that the system admits a unique weak solution provided the nonlinear adsorption isotherm associated with the reaction process satisfies certain physically reasonable structural conditions. We conclude, moreover, that the solute concentrations stay non-negative if the source term is componentwise non-negative and investigate numerically the finite speed of propagation of compactly supported initial concentrations, in a two-component test case.

math.AP↗