arXiv subjects

Alan E. Lindsay

Publications and source records attributed to Alan E. Lindsay.

18 recordsLinked to original sources

arXiv2026

String kernel representations in elastostatics

In this paper we present a new boundary integral equation formulation for the solution of the elastostatic traction boundary value problem in two and three dimensions. The approach relies on the introduction of new layer potentials, called string kernels, which are based on modifications of the Boussinesq-Cerruti family of half-space solutions. We prove that the resulting integral equations are second-kind integral equations, and show that they are well-behaved in the incompressible limit. We illustrate the performance of the method with several numerical examples.

math.NA↗

arXiv2026

Single-cell directional sensing at ultra-low chemoattractant concentrations from extreme first-passage events

We investigate single-cell directional sensing from diffusing chemoattractant signals released by a localized source. We focus on the low-concentration regime in which receptor activity is discrete and cellular decisions are made on timescales far shorter than those required for steady-state concentration profiles or receptor saturation to emerge. We derive analytic expressions for the joint distribution of receptor binding times and binding locations, conditional on the position of the source. We show that early binding events carry disproportionately more information about source directionality than later arrivals. Motivated by this observation, we propose and analyze several source-localization estimates that exploit early receptor binding statistics. For a directional estimate formed from an equally weighted average of the earliest impacts, we demonstrate that this estimate is accurate even from a small number of impacts, and derive an expression for the number of binding events that minimizes the mean squared error. Our results demonstrate that cells possess sufficient information to rapidly and accurately infer the directionality of sparse signals from a localized source through a small number of discrete receptor engagements, well before a steady-state gradient has been established.

q-bio.CB↗

arXiv2026

High-order integral methods for the Neumann Green's function: applications to capture and signaling problems in two dimensions

We present a high order numerical method for the solution of the Neumann Green's function in two dimensions. For a general closed planar curve, our computational method resolves both the interior and exterior Green's functions with the source placed either in the bulk or on the surface -- yielding four distinct functions. Our method exactly represents the singular nature of the Green's function by decomposing the singular and regular components. In the case of the interior function, we exactly prescribe an integral constraint which is necessary to obtain a unique solution given the arbitrary constant solution associated with Neumann boundary conditions. Our implementation is based on a fast integral method for the regular part of the Green's function which allows for a rapid and high order discretization for general domains. We demonstrate the accuracy of our method for simple geometries such as disks and ellipses where closed form solutions are available. To exhibit the usefulness of these new routines, we demonstrate several applications to open problems in the capture of Brownian particles, specifically, how the small traps or boundary windows should be configured to maximize the capture rate of Brownian particles.

math.NA↗

arXiv2026

The three-dimensional Neumann Green's function for general surfaces: singular asymptotics and boundary integral methods

We present an asymptotic analysis and high-order boundary integral method for the three-dimensional Neumann Green's function in general closed and smooth geometries. The Neumann Green's function is a fundamental quantity which arises in numerous fields of science and engineering. In the application of singular perturbation methods to strongly localized reactions and diffusive transport, the Green's function plays the key role in mediating global dynamics. However, this essential quantity can only be determined in closed-form for a limited set of geometries. The Green's function for the Laplacian is an elliptic problem with a Dirac forcing term. Accurate resolution of the solution requires a careful decomposition into a singular and a regular part. The bulk scenario is where the source is placed off surface and the singularity is given by the free-space function. In the surface case, where the source is placed at a curved point on the boundary, we use asymptotic analysis to determine a three-term singularity structure. With explicit knowledge of these singularities, we develop a high-order boundary integral method for the determination of the remaining regular part. To resolve the singular boundary data, our integral method uses a custom discretization with Duffy patches near the source. We validate our method using several test cases in which closed-form solutions can be developed, including spheres, prolate spheroids and constructed domains. We demonstrate the applicability of our method to address some open problems in narrow capture theory.

math.NA↗

arXiv2026

Mean first passage times of higher-dimensional velocity jump processes

First passage phenomena arise across physics, biology, and finance when stochastic processes first reach a threshold, triggering downstream events. Examples include the irreversible exit from a domain, a biochemical reaction, a financial selloff. While typical formulations involve diffusive motion, many stochastic processes are better described as velocity jump processes, characterized by persistent motion interrupted by stochastic velocity changes. Despite their ubiquity, first-passage properties of velocity jump processes remain underdeveloped in higher dimensions, especially under directional bias. We present a general framework to estimate the mean first passage time (MFPT) and higher moments of the survival probability for fixed-speed velocity jump processes where possible reorientations range from strong alignment to full angular anisotropy. For low Knudsen numbers, when the mean free path is small compared to the distance to the target, we derive a universal form for the MFPT in which two bias functions encode broad classes of angular distributions, including von Mises-Fisher, wrapped Cauchy, and elliptical families. In the narrow capture limit of a vanishingly small target, directional persistence induces anomalous scaling, including regimes where the MFPT remains finite whereas standard diffusion would predict divergence. Finally, we obtain a Langevin representation that accurately reproduces first-passage statistics. Analytical predictions are confirmed by numerical simulations.

cond-mat.stat-mech↗

arXiv2026

The lightning method for the heat equation

This paper introduces a new method for solving the planar heat equation based on the Lightning Method. The lightning method is a recent development in the numerical solution of linear PDEs which expresses solutions using sums of polynomials and rational functions, or more generally as sums of fundamental solutions. The method is particularly well suited to handle domains with sharp corners where solution singularities are present. Boundary conditions are formed on a set of collocation points which is then solved as an overdetermined linear system. The approach of the present work is to utilize the Laplace transform to obtain a modified Helmholtz equation which is solved by an application of the lightning method. The numerical inversion of the Laplace transform is then performed by means of Talbot integration. Our validation of the method against existing results and multiple challenging test problems shows the method attains spectral accuracy with root-exponential convergence while being robust across a wide range of time intervals and adaptable to a variety of geometric scenarios.

math.NA↗

arXiv2025

The effect of target orientation on the mean first passage time of a Brownian particle to a small elliptical absorber

We develop a high order asymptotic expansion for the mean first passage time (MFPT) of the capture of Brownian particles by a small elliptical trap in a bounded two dimensional region. This new result describes the effect that trap orientation plays on the capture rate and extends existing results that give information only on the role of trap position on the capture rate. Our results are validated against numerical simulations which confirm the accuracy of the asymptotic approximation. In the case of the unit disk domain, we identify a bifurcation such that the high order correction to the global MFPT (GMFPT) is minimized when the trap is orientated in the radial direction for traps centered at $0<r<r_c :=\sqrt{2-\sqrt{2}}$. When centered at position $r_c<r<1$, the GMFPT correction is minimized by orientating the trap in the angular direction. In the scenario of a general two-dimensional geometry, we identify the orientation that minimizes the GMFPT in terms of the regular part of the Neumann Green's function. This theory is demonstrated on several regular domains such as disks, ellipses and rectangles.

math.AP↗

arXiv2024

Kinetic Monte Carlo methods for three-dimensional diffusive capture problems in exterior domains

Cellular scale decision making is modulated by the dynamics of signalling molecules and their diffusive trajectories from a source to small absorbing sites on the cellular surface. Diffusive capture problems are computationally challenging due to the complex geometry and the applied boundary conditions together with intrinsically long transients that occur before a particle is captured. This paper reports on a particle-based Kinetic Monte Carlo (KMC) method that provides rapid accurate simulation of arrival statistics for (i) a half-space bounded by a surface with a finite collection of absorbing traps and (ii) the domain exterior to a convex cell again with absorbing traps. We validate our method by replicating classical results and in addition, newly developed boundary homogenization theories and matched asymptotic expansions on capture rates. In the case of non-spherical domains, we describe a new shielding effect in which geometry can play a role in sharpening cellular estimates on the directionality of diffusive sources.

math.NA↗

arXiv2024

Mean First Passage Times for Transport Equations

Many transport processes in ecology, physics and biochemistry can be described by the average time to first find a site or exit a region, starting from an initial position. Typical mathematical treatments are based on formulations that allow for various diffusive forms and geometries but where only initial and final positions are taken into account. Here, we develop a general theory for the mean first passage time (MFPT) for velocity jump processes. For random walkers, both position and velocity are tracked and the resulting Fokker-Planck equation takes the form of a kinetic transport equation. Starting from the forward and backward formulations we derive a general elliptic integro-PDE for the MFPT of a random walker starting at a given location with a given velocity. We focus on two scenarios that are relevant to biological modelling; the diffusive case and the anisotropic case. For the anisotropic case we also perform a parabolic scaling, leading to a well known anisotropic MFPT equation. To illustrate the results we consider a two-dimensional circular domain under radial symmetry, where the MFPT equations can be solved explicitly. Furthermore, we consider the MFPT of a random walker in an ecological habitat that is perturbed by linear features, such as wolf movement in a forest habitat that is crossed by seismic lines.

math.AP↗

arXiv2024

Boundary Integral Methods for Particle Diffusion in Complex Geometries: Shielding, Confinement, and Escape

We present a numerical method for the solution of diffusion problems in unbounded planar regions with complex geometries of absorbing and reflecting bodies. Our numerical method applies the Laplace transform to the parabolic problem, yielding a modified Helmholtz equation which is solved with a boundary integral method. Returning to the time domain is achieved by quadrature of the inverse Laplace transform by deforming along the so-called Talbot contour. We demonstrate the method for various complex geometries formed by disjoint bodies of arbitrary shape on which either uniform Dirichlet or Neumann boundary conditions are applied. The use of the Laplace transform bypasses constraints with traditional time-stepping methods and allows for integration over the long equilibration timescales present in diffusion problems in unbounded domains. Using this method, we demonstrate shielding effects where the complex geometry modulates the dynamics of capture to absorbing sets. In particular, we show examples where geometry can guide diffusion processes to particular absorbing sites, obscure absorbing sites from diffusing particles, and even find the exits of confining geometries, such as mazes.

math.NA↗

arXiv2024

Numerical bifurcation analysis of post-contact states in mathematical models of Micro-Electromechanical Systems

This paper is a computational bifurcation analysis of a non-linear partial differential equation (PDE) characterizing equilibrium configurations in Micro electromechanical Systems (MEMS). MEMS are engineering systems that utilize electrostatic forces to actuate elastic surfaces. The potential equilibrium states of MEMS are described by solutions of a singularly perturbed elliptic nonlinear PDE. We develop a numerical method which couples a finite element approximation with mesh refinement to a pseudo arc-length continuation algorithm to numerically obtain bifurcation diagrams in the physically relevant two dimensional scenario. Several geometries, including a unit disk, square, and annulus, are studied to understand the behavior of the system over a range of domains and parameter regimes. We find that solution multiplicity, and importantly the potential for bistability in the system, depends sensitively on the parameters. In the annulus domain, symmetry breaking bifurcations are located and asymmetric solution branches are tracked. This work significantly extends the envelope for numerical characterization of equilibrium states in microscopic electrostatic contact problems relating to MEMS.

math.DS↗

arXiv2021

Trapping of Planar Brownian Motion: Full First Passage Time Distributions by Kinetic Monte-Carlo, Asymptotic and Boundary Integral Methods

We consider the problem of determining the arrival statistics of unbiased planar random walkers to complex target configurations. In contrast to problems posed in finite domains, simple moments of the distribution, such as the mean (MFPT) and variance, are not defined and it is necessary to obtain the full arrival statistics. We describe several methods to obtain these distributions and other associated quantities such as splitting probabilities. One approach combines a Laplace transform of the underlying parabolic equation with matched asymptotic analysis followed by numerical transform inversion. The second approach is similar, but uses a boundary integral equation method to solve for the Laplace transformed variable. To validate the results of this theory, and to obtain the arrival time statistics in very general configurations of absorbers, we introduce an efficient Kinetic Monte Carlo (KMC) method that describes trajectories as a combination of large but exactly solvable projection steps. The effectiveness of these methodologies is demonstrated on a variety of challenging examples highlighting the applicability of these methods to a variety of practical scenarios, such as source inference. A particularly useful finding arising from these results is that homogenization theories, in which complex configurations are replaced by equivalent simple ones, are remarkably effective at describing arrival time statistics.

math.NA↗

arXiv2020

Pattern Formation in a Coupled Membrane-Bulk Reaction-Diffusion Model for Intracellular Polarization and Oscillations

Reaction-diffusion systems have been widely used to study spatio-temporal phenomena in cell biology, such as cell polarization. Coupled bulk-surface models naturally include compartmentalization of cytosolic and membrane-bound polarity molecules. Here we study the distribution of the polarity protein Cdc42 in a mass-conserved membrane-bulk model, and explore the effects of diffusion and spatial dimensionality on spatio-temporal pattern formation. We first analyze a 1-D model for Cdc42 oscillations in fission yeast, consisting of two diffusion equations in the bulk domain coupled to nonlinear ODEs for binding kinetics at each end of the cell. In 1-D, our analysis reveals the existence of symmetric and asymmetric steady states, as well as anti-phase relaxation oscillations typical of slow-fast systems. We then extend our analysis to a 2-D model with circular bulk geometry, for which species can either diffuse inside the cell or become bound to the membrane and undergo a nonlinear reaction-diffusion process. We also consider a nonlocal system of PDEs approximating the dynamics of the 2-D membrane-bulk model in the limit of fast bulk diffusion. In all three model variants we find that mass conservation selects perturbations of spatial modes that simply redistribute mass. In 1-D, only anti-phase oscillations between the two ends of the cell can occur, and in-phase oscillations are excluded. In higher dimensions, no radially symmetric oscillations are observed. Instead, the only instabilities are symmetry-breaking, either corresponding to stationary Turing instabilities, leading to the formation of stationary patterns, or to oscillatory Turing instabilities, leading to traveling and standing waves. Codimension-two Bogdanov--Takens bifurcations occur when the two distinct instabilities coincide, causing traveling waves to slow down and to eventually become stationary patterns.

nlin.PS↗

arXiv2018

Moving Mesh simulation of contact sets in two dimensional models of elastic-electrostatic deflection problems

Numerical and analytical methods are developed for the investigation of contact sets in electrostatic-elastic deflections modeling micro-electro mechanical systems. The model for the membrane deflection is a fourth-order semi-linear partial differential equation and the contact events occur in this system as finite time singularities. Primary research interest is in the dependence of the contact set on model parameters and the geometry of the domain. An adaptive numerical strategy is developed based on a moving mesh partial differential equation to dynamically relocate a fixed number of mesh points to increase density where the solution has fine scale detail, particularly in the vicinity of forming singularities. To complement this computational tool, a singular perturbation analysis is used to develop a geometric theory for predicting the possible contact sets. The validity of these two approaches are demonstrated with a variety of test cases.

math.NA↗

arXiv2017

Numerical approximation of diffusive capture rates by planar and spherical surfaces with absorbing pores

In 1977 Berg & Purcell published a landmark paper entitled "Physics of Chemoreception" which examined how a bacterium can sense a chemical attractant in the fluid surrounding it. At small scales the attractant molecules move by Brownian motion and diffusive processes dominate. This example is the archetype of diffusive signaling problems where an agent moves via a random walk until it either strikes or eludes a target. Berg & Purcell modeled the target as a sphere with a set of small circular targets (pores) that can capture a diffusing agent. They argued that, in the limit of small radii and wide spacing, each pore could be modeled independently as a circular pore on an infinite plane. Using a known exact solution, they showed the capture rate to be proportional to the combined perimeter of the pores. In this paper we study how to improve this approximation by including inter-pore competition effects and verify this result numerically for a finite collection of pores on a plane or sphere. Asymptotically we have found the corrections to the Berg-Purcell formula that account for enhancement of capture due to the curvature of the spherical target and inhibition of capture due to spatial interactions of the pores. Numerically we develop a spectral boundary element method for the exterior mixed Neumann-Dirichlet boundary value problem. Our formulation reduces the problem to a linear integral equation, specifically a Neumann to Dirichlet map, which is supported only on the individual pores. The difficulty is that both the kernel and the flux are singular, a notorious obstacle in such problems. A judicious choice of singular boundary elements allows us to resolve the flux singularity at the edge of the pore. In biological systems there can be thousands of receptors whose radii are 0.1% the cell radius. Our numerics resolve this realistic limit with an accuracy of roughly one part in 10^8.

math.AP↗

arXiv2017

A boundary integral equation method for mode elimination and vibration confinement in thin plates with clamped points

We consider the bi-Laplacian eigenvalue problem for the modes of vibration of a thin elastic plate with a discrete set of clamped points. A high-order boundary integral equation method is developed for efficient numerical determination of these modes in the presence of multiple localized defects for a wide range of two-dimensional geometries. The defects result in eigenfunctions with a weak singularity that is resolved by decomposing the solution as a superposition of Green's functions plus a smooth regular part. This method is applied to a variety of regular and irregular domains and two key phenomena are observed. First, careful placement of clamping points can entirely eliminate particular eigenvalues and suggests a strategy for manipulating the vibrational characteristics of rigid bodies so that undesirable frequencies are removed. Second, clamping of the plate can result in partitioning of the domain so that vibrational modes are largely confined to certain spatial regions. This numerical method gives a precision tool for tuning the vibrational characteristics of thin elastic plates.

math.NA↗

arXiv2016

Hybrid asymptotic-numerical approach for estimating first passage time densities of the two-dimensional narrow capture problem

A hybrid asymptotic-numerical method is presented for obtaining the full probability distribution of capture times of a random walker by multiple small traps located inside a bounded two-dimensional domain with reflective boundaries. As motivation for this study, we calculate the variance in the capture time of a random walker by a single interior trap and determine this quantity to be comparable in magnitude to the mean. This implies that the mean is not necessarily reflective of typical capture times and that the full density must be determined. To solve the underlying diffusion equation, the method of Laplace transforms is used to obtain an elliptic problem of modified Helmholtz type. In the limit of vanishing trap sizes, each trap is represented as a Dirac point source which permits the solution of the transform equation to be represented as a superposition of Helmholtz Green's functions. Using this solution, we construct asymptotic short time solutions of the first passage time density which captures peaks associated with rapid capture by the absorbing traps. When numerical evaluation of the Helmholtz Green's function is employed followed by numerical inversion of the Laplace transform, the method reproduces the density for larger times. We demonstrate the accuracy of our solution technique with comparison to statistics obtained from a time-dependent solution of the diffusion equation and discrete particle simulations. In particular, we demonstrate that the method is capable of capturing the multimodal behavior in the capture time density that arises when the traps are strategically arranged. The hybrid method presented can be applied to scenarios involving both arbitrary domains and trap shapes.

cond-mat.stat-mech↗

arXiv2011

Analysis of the singular solution branch of a prescribed mean curvature equation with singular nonlinearity

The existence and multiplicity of solutions to a quasilinear, elliptic partial differential equation (PDE) with singular non-linearity is analyzed. The PDE is a recently derived variant of a canonical model used in the modeling of Micro-Electro Mechanical Systems (MEMS). It is observed that the bifurcation curve of solutions terminates at single dead-end point, beyond which no classical solutions exist. A necessary condition for the existence of solutions is developed which reveals that this dead-end point corresponds to a blow-up in the solution derivative at a point internal to the domain. Using asymptotic analysis, an accurate prediction of this dead end point is obtained. An arc-length parameterization of the solution curve can be employed to continue solutions beyond the dead end point, however, all extra solutions are found to be multi-valued.

math.AP↗