arXiv subjects

Maria K. Cameron

Publications and source records attributed to Maria K. Cameron.

14 recordsLinked to original sources

arXiv2025

Theory and algorithms for clusters of cycles in graphs for material networks

Analysis of complex networks, particularly material networks such as the carbon skeleton of hydrocarbons generated in hydrocarbon pyrolysis in carbon-rich systems, is essential for effectively describing, modeling, and predicting their features. An important and the most challenging part of this analysis is the extraction and effective description of cycles, when many of them coalesce into complex clusters. A deterministic minimum cycle basis (MCB) is generally non-unique and biased to the vertex enumeration. The union of all MCBs, called the set of relevant cycles, is unique, but may grow exponentially with the graph size. To resolve these issues, we propose a method to sample an MCB uniformly at random. The output MCB is statistically well-defined, and its size is proportional to the number of edges. We review and advance the theory of graph cycles from previous works of Vismara, Gleiss et al., and Kolodzik et al. In particular, we utilize the polyhedron-interchangeability (pi) and short loop-interchangeability (sli) classes to partition the relevant cycles. We introduce a postprocessing step forcing pairwise intersections of relevant cycles to consist of a single path. This permits the definition of a dual graph whose nodes are cycles and edges connect pairs of intersecting cycles. The pi classes identify building blocks for crystalline structures. The sli classes group together sets of large redundant cycles. We present the application to an amorphous hydrocarbon network, where we (i) theorize how the number of relevant cycles may explode with system size and (ii) observe small polyhedral structures related to diamond.

math.CO↗

arXiv2025

Learning collective variables that preserve transition rates

Collective variables (CVs) play a crucial role in capturing rare events in high-dimensional systems, motivating the continual search for principled approaches to their design. In this work, we revisit the framework of quantitative coarse graining and identify the orthogonality condition from Legoll and Lelievre (2010) as a key criterion for constructing CVs that accurately preserve the statistical properties of the original process. We establish that satisfaction of the orthogonality condition enables error estimates for both relative entropy and pathwise distance to scale proportionally with the degree of scale separation. Building on this foundation, we introduce a general numerical method for designing neural network-based CVs that integrates tools from manifold learning with group-invariant featurization. To demonstrate the efficacy of our approach, we construct CVs for butane and achieve a CV that reproduces the anti-gauche transition rate with less than ten percent relative error. Additionally, we provide empirical evidence challenging the necessity of uniform positive definiteness in diffusion tensors for transition rate reproduction and highlight the critical role of light atoms in CV design for molecular dynamics.

math.NA↗

arXiv2025

A predator-prey model with age-structured role reversal

We propose a predator-prey model with an age-structured predator population that exhibits a functional role reversal. The structure of the predator population in our model embodies the ecological concept of an "ontogenetic niche shift," in which a species' functional role changes as it grows. This structure adds complexity to our model but increases its biological relevance. The time evolution of the age-structured predator population is motivated by the Kermack-McKendrick Renewal Equation (KMRE). Unlike KMRE, the predator population's birth and death rate functions depend on the prey population's size. We establish the existence, uniqueness, and positivity of the solutions to the proposed model's initial value problem. The dynamical properties of the proposed model are investigated via Latin Hypercube Sampling in the 15-dimensional space of its parameters. Our Linear Discriminant Analysis suggests that the most influential parameters are the maturation age of the predator and the rate of consumption of juvenile predators by the prey. We carry out a detailed study of the long-term behavior of the proposed model as a function of these two parameters. In addition, we reduce the proposed age-structured model to ordinary and delayed differential equation (ODE and DDE) models. The comparison of the long-term behavior of the ODE, DDE, and the age-structured models with matching parameter settings shows that the age structure promotes the instability of the Coexistence Equilibrium and the emergence of the Coexistence Periodic Attractor.

q-bio.PE↗

arXiv2023

A Finite Expression Method for Solving High-Dimensional Committor Problems

Transition path theory (TPT) is a mathematical framework for quantifying rare transition events between a pair of selected metastable states $A$ and $B$. Central to TPT is the committor function, which describes the probability to hit the metastable state $B$ prior to $A$ from any given starting point of the phase space. Once the committor is computed, the transition channels and the transition rate can be readily found. The committor is the solution to the backward Kolmogorov equation with appropriate boundary conditions. However, solving it is a challenging task in high dimensions due to the need to mesh a whole region of the ambient space. In this work, we explore the finite expression method (FEX, Liang and Yang (2022)) as a tool for computing the committor. FEX approximates the committor by an algebraic expression involving a fixed finite number of nonlinear functions and binary arithmetic operations. The optimal nonlinear functions, the binary operations, and the numerical coefficients in the expression template are found via reinforcement learning. The FEX-based committor solver is tested on several high-dimensional benchmark problems. It gives comparable or better results than neural network-based solvers. Most importantly, FEX is capable of correctly identifying the algebraic structure of the solution which allows one to reduce the committor problem to a low-dimensional one and find the committor with any desired accuracy.

math.NA↗

arXiv2023

The effect of scatter of polymer chain length on strength

A polymer network fractures by breaking covalent bonds, but the experimentally measured strength of the polymer network is orders of magnitude lower than the strength of covalent bonds. We investigate the effect of statistical variation of the number of links in polymer chains on strength using a parallel chain model. Each polymer chain is represented by a freely-jointed chain, with a characteristic J-shaped force-extension curve. The chain carries entropic forces for most of the extension and carries covalent forces only for a narrow range of extension. The entropic forces are orders of magnitude lower than the covalent forces. Chains with a statistical distribution of the number of links per chain are pulled between two rigid parallel plates. Chains with fewer links attain covalent forces and rupture at smaller extensions, while chains with more links still carry entropic forces. We compute the applied force on the rigid plates as a function of extension and define the strength of the parallel chain model by the maximum force divided by the total number of chains. With the J-shaped force-extension curve of each chain, even a small scatter in the number of links per chain greatly reduces the strength of the parallel chain model. We further show that the strength of the parallel chain model relates to the scatter in the number of links per chain according to a power law.

cond-mat.soft↗

arXiv2023

Numerical geometric acoustics: an eikonal-based approach for modeling sound propagation in 3D environments

We present algorithms for solving high-frequency acoustic scattering problems in complex domains. The eikonal and transport partial differential equations from the WKB/geometric optic approximation of the Helmholtz equation are solved recursively to generate boundary conditions for a tree of eikonal/transport equation pairs, describing the phase and amplitude of a geometric optic wave propagating in a complicated domain, including reflection and diffraction. Edge diffraction is modeled using the uniform theory of diffraction. For simplicity, we limit our attention to domains with piecewise linear boundaries and a constant speed of sound. The domain is discretized into a conforming tetrahedron mesh. For the eikonal equation, we extend the jet marching method to tetrahedron meshes. Hermite interpolation enables second order accuracy for the eikonal and its gradient and first order accuracy for its Hessian, computed using cell averaging. To march the eikonal on an unstructured mesh, we introduce a new method of rejecting unphysical updates by considering Lagrange multipliers and local visibility. To handle accuracy degradation near caustics, we introduce several fast Lagrangian initialization algorithms. We store the dynamic programming plan uncovered by the marcher in order to propagate auxiliary quantities along characteristics. We introduce an approximate origin function which is computed using the dynamic programming plan, and whose 1/2-level set approximates the geometric optic shadow and reflection boundaries. We also use it to propagate geometric spreading factors and unit tangent vector fields needed to compute the amplitude and evaluate the high-frequency edge diffraction coefficient. We conduct numerical tests on a semi-infinite planar wedge to evaluate the accuracy of our method. We also show an example with a more realistic building model with challenging architectural features.

math.NA↗

arXiv2022

Computing committors via Mahalanobis diffusion maps with enhanced sampling data

The study of phenomena such as protein folding and conformational changes in molecules is a central theme in chemical physics. Molecular dynamics (MD) simulation is the primary tool for the study of transition processes in biomolecules, but it is hampered by a huge timescale gap between the processes of interest and atomic vibrations which dictate the time step size. Therefore, it is imperative to combine MD simulations with other techniques in order to quantify the transition processes taking place on large timescales. In this work, the diffusion map with Mahalanobis kernel, a meshless approach for approximating the Backward Kolmogorov Operator (BKO) in collective variables, is upgraded to incorporate standard enhanced sampling techniques such as metadynamics. The resulting algorithm, which we call the "target measure Mahalanobis diffusion map" (tm-mmap), is suitable for a moderate number of collective variables in which one can approximate the diffusion tensor and free energy. Imposing appropriate boundary conditions allows use of the approximated BKO to solve for the committor function and utilization of transition path theory to find the reactive current delineating the transition channels and the transition rate. The proposed algorithm, tm-mmap, is tested on the two-dimensional Moro-Cardin two-well system with position-dependent diffusion coefficient and on alanine dipeptide in two collective variables where the committor, the reactive current, and the transition rate are compared to those computed by the finite element method (FEM). Finally, tm-mmap is applied to alanine dipeptide in four collective variables where the use of finite elements is infeasible.

physics.comp-ph↗

arXiv2022

Computing committors in collective variables via Mahalanobis diffusion maps

The study of rare events in molecular and atomic systems such as conformal changes and cluster rearrangements has been one of the most important research themes in chemical physics. Key challenges are associated with long waiting times rendering molecular simulations inefficient, high dimensionality impeding the use of PDE-based approaches, and the complexity or breadth of transition processes limiting the predictive power of asymptotic methods. Diffusion maps are promising algorithms to avoid or mitigate all these issues. We adapt the diffusion map with Mahalanobis kernel proposed by Singer and Coifman (2008) for the SDE describing molecular dynamics in collective variables in which the diffusion matrix is position-dependent and, unlike the case considered by Singer and Coifman, is not associated with a diffeomorphism. We offer an elementary proof showing that one can approximate the generator for this SDE discretized to a point cloud via the Mahalanobis diffusion map. We use it to calculate the committor functions in collective variables for two benchmark systems: alanine dipeptide, and Lennard-Jones-7 in 2D. For validating our committor results, we compare our committor functions to the finite-difference solution or by conducting a "committor analysis" as used by molecular dynamics practitioners. We contrast the outputs of the Mahalanobis diffusion map with those of the standard diffusion map with isotropic kernel and show that the former gives significantly more accurate estimates for the committors than the latter.

math.NA↗

arXiv2021

Jet Marching Methods for Solving the Eikonal Equation

We develop a family of compact high-order semi-Lagrangian label-setting methods for solving the eikonal equation. These solvers march the total 1-jet of the eikonal, and use Hermite interpolation to approximate the eikonal and parametrize characteristics locally for each semi-Lagrangian update. We describe solvers on unstructured meshes in any dimension, and conduct numerical experiments on regular grids in two dimensions. Our results show that these solvers yield at least second-order convergence, and, in special cases such as a linear speed of sound, third-order of convergence for both the eikonal and its gradient. We additionally show how to march the second partials of the eikonal using cell-based interpolants. Second derivative information computed this way is frequently second-order accurate, suitable for locally solving the transport equation. This provides a means of marching the prefactor coming from the WKB approximation of the Helmholtz equation. These solvers are designed specifically for computing a high-frequency approximation of the Helmholtz equation in a complicated environment with a slowly varying speed of sound, and, to the best of our knowledge, are the first solvers with these properties. We provide a link to a package online providing the solvers, and from which the results of this paper can be reproduced easily.

math.NA↗

arXiv2019

Ordered Line Integral Methods for Solving the Eikonal Equation

We present a family of fast and accurate Dijkstra-like solvers for the eikonal equation and factored eikonal equation which compute solutions on a regular grid by solving local variational minimization problems. Our methods converge linearly but compute significantly more accurate solutions than competing first order methods. In 3D, we present two different families of algorithms which significantly reduce the number of FLOPs needed to obtain an accurate solution to the eikonal equation. One method employs a fast search using local characteristic directions to prune unnecessary updates, and the other uses the theory of constrained optimization to achieve the same end. The proposed solvers are more efficient than the standard fast marching method in terms of the relationship between error and CPU time. We also modify our method for use with the additively factored eikonal equation, which can be solved locally around point sources to maintain linear convergence. We conduct extensive numerical simulations and provide theoretical justification for our approach. A library that implements the proposed solvers is available on GitHub.

math.NA↗

arXiv2018

Computing the quasipotential for nongradient SDEs in 3D

Nongradient SDEs with small white noise often arise when modeling biological and ecological time-irreversible processes. If the governing SDE were gradient, the maximum likelihood transition paths, transition rates, expected exit times, and the invariant probability distribution would be given in terms of its potential function. The quasipotential plays a similar role for nongradient SDEs. Unfortunately, the quasipotential is the solution of a functional minimization problem that can be obtained analytically only in some special cases. We propose a Dijkstra-like solver for computing the quasipotential on regular rectangular meshes in 3D. This solver results from a promotion and an upgrade of the previously introduced ordered line integral method with the midpoint quadrature rule for 2D SDEs. The key innovations that have allowed us to keep the CPU times reasonable while maintaining good accuracy are $(i)$ a new hierarchical update strategy, $(ii)$ the use of Karush-Kuhn-Tucker theory for rejecting unnecessary simplex updates, and $(iii)$ pruning the number of admissible simplexes and a fast search for them. An extensive numerical study is conducted on a series of linear and nonlinear examples where the quasipotential is analytically available or can be found at transition states by other methods. In particular, the proposed solver is applied to Tao's examples where the transition states are hyperbolic periodic orbits, and to a genetic switch model by Lv et al. (2014). The C source code implementing the proposed algorithm is available at M. Cameron's web page.

math.NA↗

arXiv2015

QPot: An R Package for Stochastic Differential Equation Quasi-Potential Analysis

QPot is an R package for analyzing two-dimensional systems of stochastic differential equations. It provides users with a wide range of tools to simulate, analyze, and visualize the dynamics of these systems. One of QPot's key features is the computation of the quasi-potential, an important tool for studying stochastic systems. Quasi-potentials are particularly useful for comparing the relative stabilities of equilibria in systems with alternative stable states. This paper describes QPot's primary functions, and explains how quasi-potentials can yield insights about the dynamics of stochastic systems. Three worked examples guide users through the application of QPot's functions.

q-bio.QM↗

arXiv2015

Metastability, Spectra, and Eigencurrents of the Lennard-Jones-38 Network

We develop computational tools for spectral analysis of stochastic networks representing energy landscapes of atomic and molecular clusters. Physical meaning and some properties of eigenvalues, left and right eigenvectors, and eigencurrents are discussed. We propose an approach to compute a collection of eigenpairs and corresponding eigencurrents describing the most important relaxation processes taking place in the system on its way to the equilibrium. It is suitable for large and complex stochastic networks where pairwise transition rates, given by the Arrhenius law, vary by orders of magnitude. The proposed methodology is applied to the network representing the Lennard-Jones-38 cluster created by Wales's group. Its energy landscape has a double funnel structure with a deep and narrow face-centered cubic funnel and a shallower and wider icosahedral funnel. Contrary to the expectations, there is no appreciable spectral gap separating the eigenvalue corresponding to the escape from the icosahedral funnel. We provide a detailed description of the escape process from the icosahedral funnel using the eigencurrent and demonstrate a superexponential growth of the corresponding eigenvalue. The proposed spectral approach is compared to the methodology of the Transition Path Theory. Finally, we discuss whether the Lennard-Jones-38 cluster is metastable from the points of view of a mathematician and a chemical physicist, and make a connection with experimental works.

math.SP↗

arXiv2014

Computing the Asymptotic Spectrum for Networks Representing Energy Landscapes using the Minimal Spanning Tree

The concept of metastability has caused a lot of interest in recent years. The spectral decomposition of the generator matrix of a stochastic network exposes all of the transition processes in the system. The assumption of the existence of a low lying group of eigenvalues separated by a spectral gap, leading to factorization of the dynamics, has become a popular theme. We consider stochastic networks representing potential energy landscapes where the states and the edges correspond to local minima and transition states respectively, and the pairwise transition rates are given by the Arrhenuis formula. Using the minimal spanning tree, we construct the asymptotics for eigenvalues and eigenvectors of the generator matrix starting from the low lying group. This construction gives rise to an efficient algorithm for computing the asymptotic spectrum suitable for large and complex networks. We apply it to Wales's Lennard-Jones-38 network with 71887 states and 119853 edges where the underlying potential energy landscape has a double-funnel structure. Our results demonstrate that the concept of metastability should be applied with care to this system. In particular, for the full network, there is no significant spectral gap separating the eigenvalue corresponding to the exit from the wider and shallower icosahedral funnel at any reasonable temperature range.

math.SP↗