arXiv subjects

Nathan Albin

Publications and source records attributed to Nathan Albin.

At least 19 recordsLinked to original sources

arXiv2026

Discontinuity Analysis and Semi-Analytic Spectral Approximation for the Nonlocal Poisson Equation

We study a nonlocal Poisson problem with discontinuous source term and analyze how the regularity of the integral kernel determines the discontinuity structure of the corresponding solution. Under general assumptions on compactly supported integrable kernels, we show that jump discontinuities in the source term are inherited by the solution. We then identify two principal mechanisms governing higher-order regularity: singular behavior of the kernel at the origin and jump discontinuities of the kernel, or of its derivatives, at the horizon endpoints. Singularities at the origin lead to blow-up of certain derivatives of the solution at the source discontinuity, while jumps at the horizon generate cascades of derivative discontinuities at translated locations. These phenomena occur for kernels commonly used in peridynamic-type models. By contrast, compactly supported $C^\infty$ kernels do not generate derivative blow-up or cascading losses of regularity, and in this case the source term and the solution have equivalent piecewise smooth regularity. Motivated by this analysis, we develop a semi-analytic spectral method for the accurate numerical treatment of discontinuous nonlocal problems. The method uses successive smoothing transformations and explicitly constructed correction functions to convert the original problem into an auxiliary problem with improved regularity. A spectral solver is then applied to the smoothed problem, and the approximation to the original solution is recovered by adding back the analytic corrections. Numerical experiments show substantial gains in accuracy and convergence, demonstrating that the method effectively mitigates the loss of accuracy caused by discontinuities and Gibbs oscillations while retaining the efficiency of spectral methods.

math.NA↗

arXiv2026

Neural Network Based Molecular Structure Retrieval from Coulomb Explosion Imaging Data

Determining the structure and following the structural evolution of molecules undergoing chemical reactions is one of the key goals of ultrafast molecular physics and chemistry. Recently, Coulomb explosion imaging has emerged as a promising technique for imaging the evolving structure of individual molecules in the gas phase. However, its practical application for structure determination is hampered by the lack of suitable algorithms to directly retrieve the molecular structure from the measured fragment-ion momentum data. Here, we propose a scheme to solve the underlying inverse problem by employing neural networks to infer the initial atomic positions from the final ion momenta on an event-by-event basis. Using this scheme, we retrieve the structure of several polyhalomethane isomers from simulated Coulomb explosion imaging data with an average per-atom position error of approximately 0.1 atomic units, i.e., to within 5% of the typical bond lengths. This development paves the way for an automated structure retrieval from Coulomb explosion data one molecule at a time, making it ideally suitable for analyzing pump-probe experiments where several products are formed that need to be distinguished.

physics.chem-ph↗

arXiv2025

Stability under lamination and polycrystalline effective conductivity

We prove the stability under lamination of a set of real, symmetric 3$\times$3 matrices that can be viewed as a subset of the effective conductivities of a polycrystal. Constructed in a companion paper, such set in combination with several previous constructions provides the best inner bound known so far on the $G$-closure of a three dimensional polycrystal.

math.AP↗

arXiv2025

A Fast Algorithm for Finding Minimum Weight Cycles in Mining Cyclic Graph Topologies

Cyclic structures are fundamental topological features in graphs, playing critical roles in network robustness, information flow, community structure, and various dynamic processes. Algorithmic tools that can efficiently probe and analyze these cyclic topologies are increasingly vital for tasks in graph mining, network optimization, bioinformatics, and social network analysis. A core primitive for quantitative analysis of cycles is finding the Minimum Weight Cycle (MWC), representing the shortest cyclic path in a weighted graph. However, computing the MWC efficiently remains a challenge, particularly compared to shortest path computations. This paper introduces a novel deterministic algorithm for finding the MWC in general weighted graphs. Our approach adapts the structure of Dijkstra's algorithm by introducing and minimizing a \textit{composite distance} metric, effectively translating the global cycle search into an iterative node-centric optimization. We provide a rigorous proof of correctness based on loop invariants. We detail two mechanisms for accelerating the search: a provable node discarding technique based on intermediate results, and a highly effective graph pruning heuristic. This heuristic dynamically restricts the search to relevant subgraphs, leveraging the principle of locality often present in complex networks to achieve significant empirical speedups, while periodic resets ensure global optimality is maintained. The efficiency of the proposed MWC algorithm enables its use as a core component in more complex analyses focused on cyclic properties. We illustrate this through a detailed application case study: accelerating the computation of the Loop Modulus, a measure of cycle richness used in advanced network characterization. Our algorithm dramatically reduces the runtime of the iterative constraint-finding bottleneck in this computation.

cs.DS↗

arXiv2025

Fourier Spectral Method for Nonlocal Equations on Bounded Domains

This work introduces efficient and accurate spectral solvers for nonlocal equations on bounded domains. These spectral solvers exploit the fact that integration in the nonlocal formulation transforms into multiplication in Fourier space and that nonlocality is decoupled from the grid size, allowing fast and accurate solutions to the nonlocal problems. Our approach extends the spectral solvers developed by Alali and Albin (2020) for periodic domains by incorporating the two-dimensional Fourier continuation (2D-FC) algorithm introduced by Bruno and Paul (2022). We evaluate the performance of the proposed methods on two-dimensional nonlocal Poisson and nonlocal diffusion equations defined on bounded domains. While the regularity of solutions to these equations in bounded settings remains an open problem, we conduct numerical experiments to explore this issue, particularly focusing on studying discontinuities.

math.NA↗

arXiv2024

Regularity of Solutions for Peridynamics Equilibrium and Evolution Equations on Periodic Distributions

Results on the peridynamics equilibrium and evolution equations over the space of periodic vector-distributions in multi-spatial dimensions are presented. The associated operator considered is the linear state-based peridynamic operator for a homogeneous material. Results for weakly singular (integrable) as well as singular integral kernels are developed. The asymptotic behavior of the eigenvalues of the peridynamic operator's Fourier multipliers and eigenvalues are characterized explicitly in terms of the nonlocality (peridynamic horizon), the integral kernel singularity, and the spatial dimension. We build on the asymptotic analysis to develop regularity of solutions results for the peridynamic equilibrium as well as the peridynamic evolution equations over periodic distribution. The regularity results are presented explicitly in terms of the data, the integral kernel singularity, and the spatial dimension. Nonlocal-to-local convergence results are presented for the eigenvalues of the peridynamic operator and for the solutions of the equilibrium and evolution equations. The local limiting behavior is shown for two types of limits as the peridynamic horizon (nonlocality) vanishes or as the integral kernel becomes hyper-singular.

math.AP↗

arXiv2024

Regularity of Solutions for the Nonlocal Wave Equation on Periodic Distributions

This work addresses the regularity of solutions for a nonlocal wave equation over the space of periodic distributions. The spatial operator for the nonlocal wave equation is given by a nonlocal Laplace operator with a compactly supported integral kernel. We follow a unified approach based on the Fourier multipliers of the nonlocal Laplace operator, which allows the study of regular as well as distributional solutions of the nonlocal wave equation, integrable as well as singular kernels, in any spatial dimension. In addition, the results extend beyond operators with singular kernels to nonlocal-pseudo differential operators. We present results on the spatial and temporal regularity of solutions in terms of regularity of the initial data or the forcing term. Moreover, solutions of the nonlocal wave equation are shown to converge to the solution of the classical wave equation for two types of limits: as the spatial nonlocality vanishes or as the singularity of the integral kernel approaches a certain critical singularity that depends on the spatial dimension.

math.AP↗

arXiv2024

Minimizing the determinant of the graph Laplacian

In this paper, we study extremal values for the determinant of the weighted graph Laplacian under simple nondegeneracy conditions on the weights. We derive necessary and sufficient conditions for the determinant of the Laplacian to be bounded away from zero and for the existence of a minimizing set of weights. These conditions are given both in terms of properties of random spanning trees and in terms of a type of density on graphs. These results generalize and extend the work of [7].

math.CO↗

arXiv2024

Differential inclusions and polycrystals

We study the differential inclusion $Du\in K$, where $K$ is an unbounded and rotationally invariant subset of the real symmetric $3\times 3$ matrices. We exhibit a subset of all possible average fields. The corresponding microgeometries are laminates of infinite rank. The problem originated in the search for the effective conductivity of polycrystalline composites. In the latter context, our result is an improvement of the previously known bounds established by Nesi $\&$ Milton, hence proving the optimality of a new full-measure class of microgeometries.

math.AP↗

arXiv2023

Modulus of edge covers and stars

This paper explores the modulus (discrete $p$-modulus) of the family of edge covers on a discrete graph. This modulus is closely related to that of the larger family of fractional edge covers; the modulus of the latter family is guaranteed to approximate the modulus of the former within a multiplicative factor. The bounds on edge cover modulus can be computed efficiently using a duality result that relates the fractional edge covers to the family of stars.

math.CO↗

arXiv2023

Asymptotic Analysis for the Eigenvalues of Peridynamic Operators

Explicit representations of the eigenvalues of the peridynamic operator have been recently derived in [5]. These representations are given in terms of generalized hypergeometric functions. Asymptotic analysis of the hypergeometric functions is utilized to identify the asymptotic behavior of the eigenvalues. We show that the eigenvalues are bounded when the kernel is integrable and diverge when the kernel is singular. The bounds and decay rates are presented explicitly in terms of the spatial dimension, the integral kernel and the peridynamic horizon.

math-ph↗

arXiv2022

Regularity of Solutions for the Nonlocal Diffusion Equation on Periodic Distributions

This work addresses the regularity of solutions for a nonlocal diffusion equation over the space of periodic distributions. The spatial operator for the nonlocal diffusion equation is given by a nonlocal Laplace operator with a compactly supported integral kernel. We follow a unified approach based on the Fourier multipliers of the nonlocal Laplace operator, which allows the study of regular as well as distributional solutions of the nonlocal diffusion equation, integrable as well as singular kernels, in any spatial dimension. In addition, the results extend beyond operators with singular kernels to nonlocal super-diffusion operators. We present results on the spatial and temporal regularity of solutions in terms of regularity of the initial data or the diffusion source term. Moreover, solutions of the nonlocal diffusion equation are shown to converge to the solution of the classical diffusion equation for two types of limits: as the spatial nonlocality vanishes or as the singularity of the integral kernel approaches a certain critical singularity that depends on the spatial dimension. Furthermore, we show that, for the case of integrable kernels, discontinuities in the initial data propagate and persist in the solution of the nonlocal diffusion equation. The magnitude of a jump discontinuity is shown to decay overtime.

math.AP↗

arXiv2022

Linear peridynamics Fourier multipliers and eigenvalues

A characterization for the Fourier multipliers and eigenvalues of linear peridynamic operators is provided. The analysis is presented for state-based peridynamic operators for isotropic homogeneous media in any spatial dimension. We provide explicit formulas for the eigenvalues in terms of the space dimension, the nonlocal parameters, and the material properties. The approach we follow is based on the Fourier multiplier analysis developed for the nonlocal Laplacian. The Fourier multipliers of linear peridynamic operators are second-order tensor fields, which are given through integral representations. It is shown that the eigenvalues of the peridynamic operators can be derived directly from the eigenvalues of the Fourier multiplier tensors. We reveal a simple structure for the Fourier multipliers in terms of hypergeometric functions, which allows for providing integral representations as well as hypergeometric representations of the eigenvalues. These representations are utilized to show the convergence of the eigenvalues of linear peridynamics to the eigenvalues of the Navier operator of linear elasticity in the limit of vanishing nonlocality. Moreover, the hypergeometric representation of the eigenvalues is utilized to compute the spectrum of linear peridynamic operators.

math.AP↗

arXiv2021

Convergence of the Probabilistic Interpretation of Modulus

Given a Jordan domain $\Omega\subset\mathbb{C}$ and two disjoint arcs $A, B$ on $\partial\Omega$, the modulus $m$ of the curve family connecting $A$ and $B$ in $\Omega$ is equal to the modulus of the curve family connecting the vertical sides in the rectangle $R=[0,1]\times[0,m]$. Also, $m>0$ is the unique value such that there is a conformal map $\psi$ mapping $\Omega$ to ${\rm int}(R)$ so that $\psi$ extends continuously to a homeomorphism of $\partial \Omega$ onto $\partial R$ and the arcs $A$ and $B$ are sent to the vertical sides of $R$. Moreover, in the case of the rectangle the family of horizontal segments connecting the two sides has the same modulus as the entire connecting family. Pulling these segments back to $\Omega$ via $\psi$ yields a family of extremal curves (also known as horizontal trajectories) connecting $A$ to $B$ in $\Omega$. In this paper, we show that these extremal curves can be approximated by some discrete paths arising from an orthodiagonal approximation of $\Omega$. Moreover, we show that there is a natural probability mass function (pmf) on these paths, deriving from the theory of discrete modulus, which converges to the transverse measure on the set of extremal curves. The key ingredient is an algorithm that, for an embedded planar graph, takes the current flow between two sets of nodes, and produces a unique path decomposition with non-crossing paths. Moreover, some care was taken to adapt recent results for harmonic convergence on orthodiagonal maps, due to Gurel-Gurevich, Jerison, and Nachmias, to our context. Finally, we generalize a result of N.~Alrayes from the square grid setting to the orthodiagonal setting, and prove that the discrete modulus of the approximating non-crossing paths converges to the continuous modulus.

math.CV↗

arXiv2021

The scaling limit of fair Peano paths

We study random Peano paths on planar square grids that arise from fair random spanning trees. These are trees that are sampled in such a way as to have the same (if possible) edge probabilities. In particular, we are interested in identifying the scaling limit as the mesh-size of the grid tends to zero. It is known \cite{lawler-schramm-werner2002} that if the trees are sampled uniformly, then the scaling limit exists and equals ${\rm SLE}_8$. We show that if we simply follow the same steps as in \cite{lawler-schramm-werner2002}, then fair Peano paths have a deterministic scaling limit.

math.PR↗

arXiv2021

Fourier Continuation Discontinuous Galerkin Methods for Linear Hyperbolic Problems

Fourier continuation is an approach used to create periodic extensions of non-periodic functions in order to obtain highly-accurate Fourier expansions. These methods have been used in PDE-solvers and have demonstrated high-order convergence and spectrally accurate dispersion relations in numerical experiments. Discontinuous Galerkin (DG) methods are increasingly used for solving PDEs and, as all Galerkin formulations, come with a strong framework for proving stability and convergence. Here we propose the use of Fourier continuation in forming a new basis for the DG framework.

math.NA↗

arXiv2020

An exact-arithmetic algorithm for spanning tree modulus

Spanning tree modulus is a generalization of effective resistance that is closely related to graph strength and fractional arboricity. The optimal edge density associated with spanning tree modulus is known to produce two hierarchical decompositions of arbitrary graphs, one based on strength and the other on arboricity. Here we introduce an exact-arithmetic algorithm for spanning tree modulus and the strength-based decomposition using Cunningham's algorithm for graph vulnerability. The algorithm exploits an interesting connection between spanning tree modulus and critical edge sets from the vulnerability problem. This paper introduces the new algorithm, describes a practical means for implementing it using integer arithmetic, and presents some examples and computational time scaling tests.

math.CO↗

arXiv2020

Modulus of time-respecting paths

On a static graph, the p-modulus of a family of paths reflects both the lengths of these paths as well as their diversity; a family of many short, disjoint paths has larger modulus than a family of a few long overlapping paths. In this work, we define a version of p-modulus for time-respecting paths on temporal graphs. This formulation makes use of a time penalty function as a means of discounting paths that take a relatively long time to traverse, thus allowing modulus to capture temporal information about the family as well. By means of a transformation, we show that this temporal p-modulus can be recognized as a p-modulus problem on a static graph and, therefore, that much of the known theory of p-modulus of families of objects can be translated to the case of temporal paths. We demonstrate some properties of temporal modulus on examples.

math.OC↗