arXiv subjects

Elaine Crooks

Publications and source records attributed to Elaine Crooks.

9 recordsLinked to original sources

arXiv2022

Fast reaction limit of reaction diffusion systems with nonlinear diffusion

In this paper, we present an approach to characterising fast-reaction limits of systems with nonlinear diffusion, when there are either two reaction-diffusion equations, or one reaction-diffusion equation and one ordinary differential equation, on unbounded domains. Here, we replace the terms of the form uxx in usual reaction-diffusion equation, which represent linear diffusion, by terms of form phi(u)xx, representing nonlinear diffusion. We prove the convergence as k tends to infinity to a limit that is determined by the unique solution of a certain scalar nonlinear diffusion limit problem.

math.AP↗

arXiv2022

Self-similar solutions of fast reaction limit problem with nonlinear diffusion

In this paper, we present an approach to characterising self-similar fast-reaction limits of systems with nonlinear diffusion. For appropriate initial data, in the fast-reaction limit as k tends to infinithy,spatial segregation results in the two components of the original systems converging to the positive and negative parts of a self-similar limit profile f (n), where n=xt^(1/2) that satisfies one of four ordinary differential systems. The existence of these self-similar solutions of the k tends to infinity limit problems is proved by using shooting methods which focus on a, the position of the free boundary which separates the regions where the solution is positive and where it is negative, and g, the derivative of -phi(f) at n = a. The position of the free boundary gives us intuition about how one substance penetrates into the other, and for specific forms of nonlinear diffusion, the relationship between the given form of the nonlinear diffusion and the position of the free boundary is also studied.

math.AP↗

arXiv2021

Compensated Convexity on Bounded Domains, Mixed Moreau Envelopes and Computational Methods

Compensated convex transforms have been introduced for extended real-valued functions defined over $\mathbb{R}^n$. In their application to image processing, interpolation, and shape interrogation, where one deals with functions defined over a bounded domain, one was making the implicit assumption that the function coincides with its transform at the boundary of the data domain. In this paper, we introduce local compensated convex transforms for functions defined in bounded open convex subsets $Ω$ of $\mathbb{R}^n$ by making specific extensions of the function to the whole space, and establish their relations to globally defined compensated convex transforms via the mixed critical Moreau envelopes. We find that the compensated convex transforms of such extensions coincide with the local compensated convex transforms in the closure of $Ω$. We also propose a numerical scheme for computing Moreau envelopes, establishing convergence of the scheme with the rate of convergence depending on the regularity of the original function. We give an estimate of the number of iterations needed for computing the discrete Moreau envelope. We then apply the local compensated convex transforms to image processing and shape interrogation. Our results are compared with those obtained by using schemes based on computing the convex envelope from the original definition of compensated convex transforms.

math.NA↗

arXiv2020

Compensated Convex Based Transforms for Image Processing and Shape Interrogation

This paper reviews some recent applications of the theory of the compensated convex transforms or of the proximity hull as developed by the authors to image processing and shape interrogation with special attention given to the Hausdorff stability and multiscale properties. The paper contains also numerical experiments that demonstrate the performance of our methods compared to the state-of-art ones.

eess.IV↗

arXiv2018

Compensated Convexity Methods for Approximations and Interpolations of Sampled Functions in Euclidean Spaces: Applications to Contour Lines, Sparse Data and Inpainting

This paper is concerned with applications of the theory of approximation and interpolation based on compensated convex transforms developed in [K. Zhang, E. Crooks, A. Orlando, Compensated convexity methods for approximations and interpolations of sampled functions in Euclidean spaces: Theoretical Foundations. SIAM Journal on Mathematical Analysis 48 (2016) 4126-4154]. We apply our methods to $(i)$ surface reconstruction starting from the knowledge of finitely many level sets (or `contour lines'); $(ii)$ scattered data approximation; $(iii)$ image inpainting. For $(i)$ and $(ii)$ our methods give interpolations. For the case of finite sets (scattered data), in particular, our approximations provide a natural triangulation and piecewise affine interpolation. Prototype examples of explicitly calculated approximations and inpainting results are presented for both finite and compact sets. We also show numerical experiments for applications of our methods to high density salt and pepper noise reduction in image processing, for image inpainting and for approximation and interpolations of continuous functions sampled on finitely many level sets and on scattered points.

math.MG↗

arXiv2017

Individual variability in dispersal and invasion speed

We model the growth, dispersal and mutation of two phenotypes of a species using reaction-diffusion equations, focusing on the biologically realistic case of small mutation rates. After verifying that the addition of a small linear mutation rate to a Lotka-Volterra system limits it to only two steady states in the case of weak competition, an unstable extinction state and a stable coexistence state, we prove that under some biologically reasonable condition on parameters the spreading speed of the system is linearly determinate. Using this result we show that the spreading speed is a non-increasing function of the mutation rate and hence that greater mixing between phenotypes leads to slower propagation. Finally, we determine the ratio at which the phenotypes occur at the leading edge in the limit of vanishing mutation.

math.AP↗

arXiv2016

Compensated Convex Transforms and Geometric Singularity Extraction from Semiconvex Functions

We apply upper and lower compensated convex transforms, which are `tight' one-sided approximations of a given function, to the extraction of fine geometric singularities from semiconvex/semiconcave functions and DC-functions in $\mathbb{R}^n$ (difference of convex functions). Well-known examples of (locally) semiconcave functions include the Euclidean distance and squared distance functions. For a locally semiconvex function $f$ with general modulus, we show that `locally' a point is a singular (non-differentiable) point if and only if it is a scale $1$-valley point, and if $x$ is a singular point, then locally the limit of the scaled valley transform exists at every point $x$ and $ \lim_{λ\to \infty}λV_λ(f)(x)=r_x^2/4$, where $r_x$ is the radius of the minimal bounding sphere of the (Fréchet) subdifferential $\partial_- f(x)$ and $V_λ(f)(x)$ is the valley transform at $x$. Thus the limit function $\mathcal{V}_\infty(f)(x):=\lim_{λ\to+\infty}λV_λ(f)(x)=r_x^2/4$ gives a `scale $1$-valley landscape function' of the singular set for a locally semiconvex function $f$, and also provides an asymptotic expansion of the upper transform $C^u_λ(f)(x)$ when $λ\to \infty$. For a locally semiconvex function $f$ with linear modulus we show that the limit of the gradient of the upper compensated convex transform $\lim_{λ\to+\infty}\nabla C^u_λ(f)(x)$ exists and equals the centre of the minimal bounding sphere of $\partial_- f(x$, and that for a DC-function $f=g-h$, the scale $1$-edge transform satisfies $\liminf_{λ\to+\infty}λE_λ(f)(x)\geq (r_{g,x}-r_{h,x})^2/4$, where $r_{g,x}$ and $r_{h,x}$ are the radii of the minimal bounding spheres of the subdifferentials $\partial_- g$ and $\partial_- h$ of the convex functions $g$ and $h$ at $x$ respectively.

math.OC↗

arXiv2016

Compensated Convexity Methods for Approximations and Interpolations of Sampled Functions in Euclidean Spaces: Theoretical Foundations

We introduce Lipschitz continuous and $C^{1,1}$ geometric approximation and interpolation methods for sampled bounded uniformly continuous functions over compact sets and over complements of bounded open sets in $\mathbb{R}^n$ by using compensated convex transforms. Error estimates are provided for the approximations of bounded uniformly continuous functions, of Lipschitz functions, and of $C^{1,1}$ functions. We also prove that our approximation methods, which are differentiation and integration free and not sensitive to sample type, are stable with respect to the Hausdorff distance between samples.

math.MG↗

arXiv2015

Compensated Convexity, Multiscale Medial Axis Maps and Sharp Regularity of the Squared Distance Function

We introduce a new stable mathematical model for locating and measuring the medial axis of geometric objects, called the quadratic multiscale medial axis map of scale $λ$, and prove a sharp regularity result for the squared-distance function to any closed non-empty subset $K$ of $\mathbb{R}^n$. Our results exploit properties of the function $C^l_λ(dist^2(\cdot;\, K))$ obtained by applying the quadratic lower compensated convex transform of parameter $λ$ to $dist^2(\cdot;\, K)$, the Euclidean squared-distance function to $K$. Using an estimate for the tight approximation of $dist^2(\cdot;\, K)$ by $C^l_λ(dist^2(\cdot;\, K))$, we prove $C^{1,1}$-regularity of $dist^2(\cdot;\, K)$ outside a neighbourhood of the closure of the medial axis $M_K$ of $K$, and give an asymptotic formula for $C^l_λ(dist^2(\cdot;\, K))$ in terms of the scaled squared distance to $K$ and to the convex hull of the set of points that realize the minimum distance to $K$. The multiscale medial axis map, $M_λ(\cdot;\, K)$, is a family of non-negative functions whose limit as $λ\to \infty$ exists and is called the multiscale medial axis landscape map, $M_{\infty}(\cdot;\, K)$. We show $M_{\infty}(\cdot;\, K)$ is strictly positive on the medial axis $M_K$ and zero elsewhere. We give conditions to ensure $M_λ(\cdot;\, K)$ keeps a constant height along parts of $M_K$ generated by two-point subsets with the height dependent on the distance between the generating points, so giving a hierarchy between different parts of $M_K$ that enables subsets of $M_K$ to be selected by thresholding. Given a compact subset $K$ of $\mathbb{R}^n$, while it is well known that $M_K$ is not Hausdorff stable, we prove $M_λ(\cdot;\, K)$ is stable under Hausdorff distance, and deduce implications for localization of the stable parts of $M_K$. Examples are included.

math.MG↗