We introduce a definition of long range dependence of random processes and fields on an (unbounded) index space $T\subseteq \mathbb{R}^d$ in terms of integrability of the covariance of indicators that a random function exceeds any given level. This definition is specifically designed to cover the case of random functions with infinite variance. We show the value of this new definition and its connection to limit theorems via some examples including subordinated Gaussian as well as random volatility fields and time series.

Amethod for non-rigid image registration that is suitable for large deformations is presented. Conventional registration methods embed the image in a B-spline object, and the image is evolved by deforming the B-spline object. In this work, we represent the image using B-spline and deform the image using a composition approach. We also derive a computationally efficient algorithm for calculating the B-spline coefficients and gradients of the image by adopting ideas from signal processing using image filters. We demonstrate the application of our method on several different types of 2D and 3D images and compare it with existing methods.

In this research, the effect of the surface inclination on the hydrodynamics and heat transfer of droplets impinging on very hot surfaces is studied. The applied numerical algorithm is based on the accurate calculation of the vaporization rate in the simulation process using a combination of the level set and ghost fluid methods. Also a mesh clustering technique is utilized to create sufficient mesh resolution near the surface in order to take into account the effect of the thin vapor layer between droplet and very hot surface. The results are verified against available experiments. The effect of the surface inclination on the droplet maximum spreading radius, droplet contact time and total heat removal from the surface is considered. Results show that for the studied regime, the maximum spreading radius of the droplet is decreased with an increase in the surface inclination while the droplet contact time on the surface is independent from the surface inclination. For inclinations greater than 45°, the total heat removal is decreased considerably with an increase in the inclination angle. For smaller inclinations, the dependency of the total heat removal on the surface inclination is not strong.

We present a solver for the Poisson-Boltzmann equation and demonstrate its applicability for biomolecular electrostatics computation. The solver uses a level set framework to represent sharp, complex interfaces in a simple and robust manner. It also uses non-graded, adaptive octree grids which, in comparison to uniform grids, drastically decrease memory usage and runtime without sacrificing accuracy. The basic solver was introduced in earlier works [16,27], and here is extended to address biomolecular systems. First, a novel approach of calculating the solvent excluded and the solvent accessible surfaces is explained; this allows to accurately represent the location of the molecule’s surface. Next, a hybrid finite difference/finite volume approach is presented for discretizing the nonlinear Poisson-Boltzmann equation and enforcing the jump boundary conditions at the interface. Since the interface is implicitly represented by a level set function, imposing the jump boundary conditions is straightforward and efficient.

In this article, we detail the methodology developed to construct arbitrarily high order schemes — linear and WENO — on 3D mixed-element unstructured meshes made up of general convex polyhedral elements. The approach is tailored specifically for the solution of scalar level set equations for application to incompressible two-phase flow problems. The construction of WENO schemes on 3D unstructured meshes is notoriously difficult, as it involves a much higher level of complexity than 2D approaches. This due to the multiplicity of geometrical considerations introduced by the extra dimension, especially on mixed-element meshes. Therefore, we have specifically developed a number of algorithms to handle mixed-element meshes composed of convex polyhedra with convex polygonal faces. The contribution of this work concerns several areas of interest: the formulation of an improved methodology in 3D, the minimisation of computational runtime in the implementation through the maximum use of pre-processing operations, the generation of novel methods to handle complex 3D mixed-element meshes and finally the application of the method to the transport of a scalar level set.

Most image segmentation techniques efficiently segment images with prominent edges, but are less efficient for some images with low frequencies and overlapping regions of homogeneous intensities. A recently proposed selective segmentation model often works well, but not for such challenging images. In this paper, we introduce a new model using the coefficient of variation as a fidelity term, and our test results show it performs much better in these challenging cases.

We present a three dimensional preconditioned implicit free-surface capture scheme on tetrahedral grids. The current scheme improves our recently reported method [10] in several aspects. Specifically, we modified the original eigensystem by applying a preconditioning matrix so that the new eigensystem is virtually independent of density ratio, which is typically large for practical two-phase problems. Further, we replaced the explicit multi-stage Runge-Kutta method by a fully implicit Euler integration scheme for the Navier-Stokes (NS) solver and the Volume of Fluids (VOF) equation is now solved with a second order Crank-Nicolson implicit scheme to reduce the numerical diffusion effect. The preconditioned restarted Generalized Minimal RESidual method (GMRES) is then employed to solve the resulting linear system. The validation studies show that with these modifications, the method has improved stability and accuracy when dealing with large density ratio two-phase problems.

Asymptotic independence of the components of random vectors is a concept used in many applications. The standard criteria for checking asymptotic independence are given in terms of distribution functions (DFs). DFs are rarely available in an explicit form, especially in the multivariate case. Often we are given the form of the density or, via the shape of the data clouds, we can obtain a good geometric image of the asymptotic shape of the level sets of the density. In this paper we establish a simple sufficient condition for asymptotic independence for light-tailed densities in terms of this asymptotic shape. This condition extends Sibuya's classic result on asymptotic independence for Gaussian densities.

Cell motility is an integral part of a diverse set of biological processes. The quest formathematical models of cell motility has prompted the development of automated approachesfor gathering quantitative data on cell morphology, and the distribution of molecularplayers involved in cell motility. Here we review recent approaches for quantifying cellmotility, including automated cell segmentation and tracking. Secondly, we present our ownnovel method for tracking cell boundaries of moving cells, the Electrostatic ContourMigration Method (ECMM), as an alternative to the generally accepted level set method(LSM). ECMM smoothly tracks regions of the cell boundary over time to compute localmembrane displacements using the simple underlying concept of electrostatics. It offerssubstantial speed increases and reduced computational overheads in comparison to the LSM.We conclude with general considerations regarding boundary tracking in the context ofmathematical modelling.

Assessing the number of clusters of a statistical population is one of the essential issues of unsupervised learning. Given n independent observations X1,...,Xn drawn from an unknown multivariate probability density f, we propose a new approach to estimate the number of connected components, or clusters, of the t-level set $\mathcal L(t)=\{x:f(x) \geq t\}$ . The basic idea is to form a rough skeleton of the set $\mathcal L(t)$ using any preliminary estimator of f, and to count the number of connected components of the resulting graph. Under mild analytic conditions on f, and using tools from differential geometry, we establish the consistency of our method.