arXiv subjects

Henry Kasumba

Publications and source records attributed to Henry Kasumba.

5 recordsLinked to original sources

arXiv2026

Physics-Informed Residuals for Adaptive Mesh Refinement in Finite-Difference PDE Solvers

Classical finite-difference solvers remain reliable tools for partial differential equations, but their efficiency depends on where mesh resolution is placed. Uniform refinement can waste degrees of freedom when solution difficulty is localised near sharp gradients, fronts, oscillations, or constraint-sensitive regions. This paper studies a hybrid strategy in which a physics-informed neural network (PINN) is used not as the final solver, but as an off-grid residual probe for adaptive mesh refinement. The PINN residual is sampled over the domain, converted into cellwise indicators, and used to guide refinement before the final approximation is computed by a finite-difference solver. The method is evaluated on three benchmarks. The main full-solver validation uses the one-dimensional viscous Burgers equation with a nonuniform finite-difference solve on the adapted meshes. PINN-threshold refinement attains final relative $L^2$ error $0.021067$ with $60$ degrees of freedom, compared with $0.022617$ for uniform refinement with $192$ degrees of freedom. At matched mesh size, PINN-threshold reduces the error by about $67.5\%$. PINN-D\"orfler refinement gives similar performance, with error $0.021264$ using $58$ degrees of freedom. A gradient indicator remains slightly more accurate, so the result supports usefulness rather than universal superiority. Manufactured 2D and 3D proxy tests, based on a nonlinear Schr\"odinger equation and an incompressible Navier--Stokes system, show that PINN residuals can organise structured refinement and improve over random refinement, although they do not consistently outperform gradient or uniform baselines. The results support PINN-guided AMR as a residual-indicator strategy for transferring physics-informed diagnostic information into finite-difference mesh adaptation while preserving the classical solver as the final approximation engine.

math.NA↗

arXiv2024

Curvature-Adaptive Perturbation and Subspace Descent for Robust Saddle Point Escape in High-Dimensional Optimization

High-dimensional non-convex optimization problems in engineering design, control, and learning are often hindered by saddle points, flat plateaus, and strongly anisotropic curvature. This paper develops a unified, curvature-adaptive framework that combines stochastic perturbations, adaptive learning rates, and randomized subspace descent to enhance escape efficiency and scalability. We show theoretically that gradient flow almost surely avoids strict saddles, with escape probability increasing exponentially in dimension. For noise-perturbed gradient descent, we derive explicit escape-time bounds that depend on local curvature and noise magnitude. Adaptive step sizes further reduce escape times by responding to local gradient variability, while randomized subspace descent preserves descent directions in low-dimensional projections and ensures global convergence with logarithmic dependence on dimension. Numerical experiments on nonlinear and constrained benchmarks validate these results, demonstrating faster escape, improved robustness to ill-conditioning, and lower total runtime compared to standard first- and second-order methods. The proposed approach offers practical tools for large-scale engineering optimization tasks where curvature, noise, and dimensionality interplay critically.

math.OC↗

arXiv2024

The time fractional order derivative for multi-class AR model

In this paper, a multi-class Aw-Rascle \textrm{(AR)} model with time fractional order derivative is presented. The conservative form of the proposed model is considered for the natural extension and generalization of equations involved. The fractional order derivative involved in the model equations is computed by applying the Caputo fractional derivative definition. An explicit difference scheme is obtained through finite difference method of discretization. The scheme is shown to be consistent, conditionally stable and convergent. Numerical flux is computed by original Roe decomposition and an entropy condition applied to the Roe decomposition. From numerical results, the effect of fractional-order derivative of time, on the traffic flow of vehicle classes is determined. Results obtained from the proposed model remain within limits therefore, they are realistic.

math.AP↗

arXiv2023

Analysis of Heterogeneous Vehicular Traffic: Using Proportional Densities

An extended multi-class Aw-Rascle (AR) model with pressure term described as a function of area occupancy defined in form of proportional densities is presented. Two vehicle classes that is; cars and motorcycles are considered based on an assumption that proportions of these form total traffic density. Qualitative properties of the proposed equilibrium velocity is established. Conditions under which the proposed model is stable are determine by linear stability analysis. To compute numerical flux, the model is discretized by the original Roe decomposition scheme, where Roe matrix, averaged data variables and wave strengths are explicitly derived. The Roe matrix is shown to be hyperbolic, consistent and conservative. From the numerical results, the effect of motorcycles proportion on the flow of vehicle classes is determined. Results obtained remain within limits therefore, the proposed model is realistic.

math.AP↗

arXiv2021

Eigenvector Centrality and Uniform Dominant Eigenvalue of Graph Components

Eigenvector centrality is one of the outstanding measures of central tendency in graph theory. In this paper we consider the problem of calculating eigenvector centrality of graph partitioned into components and how this partitioning can be used. Two cases are considered; first where the a single component in the graph has the dominant eigenvalue, secondly when there are at least two components that share the dominant eigenvalue for the graph. In the first case we implement and compare the method to the usual approach (power method) for calculating eigenvector centrality while in the second case with shared dominant eigenvalues we show some theoretical and numerical results. Keywords: Eigenvector centrality, power iteration, graph, strongly connected component.

math.NA↗