Farid Bozorgnia

Farid Bozorgnia, Jan Valdman

This paper is concerned with the two--phase obstacle problem, a type of a variational free boundary problem. We recall the basic estimates of Repin and Valdman (2015) and verify them numerically on two examples in two space dimensions. A solution algorithm is proposed for the construction of the finite element approximation to the two--phase obstacle problem. The algorithm is not based on the primal (convex and nondifferentiable) energy minimization problem but on a dual maximization problem formulated for Lagrange multipliers. The dual problem is equivalent to a quadratic programming problem with box constraints. The quality of approximations is measured by a functional a posteriori error estimate which provides a guaranteed upper bound of the difference of approximated and exact energies of the primal minimization problem. The majorant functional in the upper bound contains auxiliary variables and it is optimized with respect to them to provide a sharp upper bound. A space density of the nonlinear related part of the majorant functional serves as an indicator of the free boundary.

Farid Bozorgnia, Seyyed Abbas Mohammadi, Tomas Vejchodsky

In this paper, we study the first eigenvalue of a nonlinear elliptic system involving $p$-Laplacian as the differential operator. The principal eigenvalue of the system and the corresponding eigenfunction are investigated both analytically and numerically. An alternative proof to show the simplicity of the first eigenvalue is given. In addition, the upper and lower bounds of the first eigenvalue are provided. Then, a numerical algorithm is developed to approximate the principal eigenvalue. This algorithm generates a decreasing sequence of positive numbers and various examples numerically indicate its convergence. Further, the algorithm is generalized to a class of gradient quasilinear elliptic systems.

Avetik Arakelyan, Farid Bozorgnia

In this paper, we study a numerical approximation for a class of stationary states for reaction-diffusion system with m densities having disjoint support, which are governed by a minimization problem. We use quantitative properties of both solutions and free boundaries to derive our scheme. Furthermore, the proof of convergence of the numerical method is given in some particular cases. We also apply our numerical simulations for the spatial segregation limit of diffusive Lotka-Volterra models in presence of high competition and inhomogeneous Dirichlet boundary conditions. We discuss numerical implementations of the resulting approach and present computational tests.

Mahmoudreza Bazarganzadeh, Farid Bozorgnia

In this work, we present two numerical schemes for a free boundary problem called one phase quadrature domain. In the first method by applying the proprieties of given free boundary problem, we derive a method that leads to a fast iterative solver. The iteration procedure is adapted in order to work in the case when topology changes. The second method is based on shape reconstruction to establish an efficient Shape-Quasi-Newton-Method. Various numerical experiments confirm the efficiency of the derived numerical methods.

Hamed Gholipour, Farid Bozorgnia, Kailash Hambarde et al.

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.

Farid Bozorgnia, Yassine Belkheiri, Abderrahim Elmoataz

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.

Farid Bozorgnia, Vyacheslav Kungurtsev, Shirali Kadyrov et al.

In this work, we introduce novel algorithms for label propagation and self-training using fractional heat kernel dynamics with a source term. We motivate the methodology through the classical correspondence of information theory with the physics of parabolic evolution equations. We integrate the fractional heat kernel into Graph Neural Network architectures such as Graph Convolutional Networks and Graph Attention, enhancing their expressiveness through adaptive, multi-hop diffusion. By applying Chebyshev polynomial approximations, large graphs become computationally feasible. Motivating variational formulations demonstrate that by extending the classical diffusion model to fractional powers of the Laplacian, nonlocal interactions deliver more globally diffusing labels. The particular balance between supervision of known labels and diffusion across the graph is particularly advantageous in the case where only a small number of labeled training examples are present. We demonstrate the effectiveness of this approach on standard datasets.

Hamed Gholipour, Farid Bozorgnia, Hamzeh Mohammadigheymasi et al.

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.

Farid Bozorgnia

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.