persistence exponents and scaling in two dimensional xy model and a nematic model the persistence exponents associated with the t 0 quenching dynamics of the two dimensional xy model and a two dimensional uniaxial spin nematic model have been evaluated using a numerical simulation the site persistence or the probability that the sign of a local spin component does not change starting from initial time t 0 up to a certain time t is found to decay as l t l t is the linear domain length scale with 0 305 0 020 for the two dimensional xy model and 0 199 0 009 for the two dimensional uniaxial spin nematic model we have also investigated the scaling at the late time of phase ordering associated with the correlated persistent sites in both models the persistence correlation length was found to grow in the same way as l t

dynamic scaling in miscible viscous fingering we consider dynamic scaling in gravity driven miscible viscous fingering we prove rigorous one sided bounds on bulk transport and coarsening in regimes of physical interest the analysis relies on comparison with solutions to one dimensional conservation laws and new scale invariant estimates our bounds on the size of the mixing layer are of two kinds a naive bound that is sharp in the absence of diffusion and a more careful bound that accounts for diffusion as a selection criterion in the limit of vanishingly small diffusion the naive bound is simple and robust but does not yield the experimental speed of transport in a reduced model derived by wooding 20 we prove a sharp upper bound on the size of the mixing layer in accordance with his experiments wooding s model also provides an example of a scalar conservation law where the entropy condition is not the physically appropriate selection criterion

accurate vessel segmentation with constrained b snake we describean active contour framework with accurate shape and size constraints on the vessel cross sectional planes to produce the vessel segmentation it starts with a multiscale vessel axis tracing in a 3d computed tomography ct data followed by vessel boundary delineation on the cross sectional planes derived from the extracted axis the vessel boundary surface is deformed under constrained movements on the cross sections and is voxelized to produce the final vascular segmentation the novelty of this paper lies in the accurate contour point detection of thin vessels based on the ct scanning model in the efficient implementation of missing contour points in the problematic regions and in the active contour model with accurate shape and size constraints the main advantage of our framework is that it avoids disconnected and incomplete segmentation of the vessels in the problematic regions that contain touching vessels vessels in close proximity to each other diseased portions pathologic structure attached to a vessel and thin vessels it is particularly suitable for accurate segmentation of thin and low contrast vessels our method is evaluated and demonstrated on ct data sets from our partner site and its results are compared with three related methods our method is also tested on two publicly available databases and its results are compared with the recently published method the applicability of the proposed method to some challenging clinical problems the segmentation of the vessels in the problematic regions is demonstrated with good results on both quantitative and qualitative experimentations our segmentation algorithm can delineate vessel boundaries that have level of variability similar to those obtained manually

bounds on coarsening rates for the lifschitz slyozov wagner equation this paper is concerned with the large time behavior of solutions to the lifschitz slyozov wagner lsw system of equations point wise in time upper and lower bounds on the rate of coarsening are obtained for solutions with fairly general initial data these bounds complement the time averaged upper bounds obtained by dai and pego and the point wise in time upper and lower bounds obtained by niethammer and velasquez for solutions with initial data close to a self similar solution

spiral crystal growth abstract we numerically study the spiral mode of crystal growth using a theory developed by burton cabrera and frank using a level set method this method is novel in that it can handle not only closed curves but open curves as well we use our method to compute interacting spirals and make estimates of growth rates we also propose a possible coarsening mechanism for a large number of interacting spirals

convergence of a fast explicit operator splitting method for the epitaxial growth model with slope selection a fast explicit operator splitting method for the epitaxial growth model with slope selection has been presented in cheng et al j comput phys 303 2015 pp 45 65 the original problem is split into linear and nonlinear subproblems for the linear part the pseudospectral method is adopted for the nonlinear part a 33 point difference scheme is constructed here we give a compact center difference scheme involving fewer points for the nonlinear subproblem in addition we analyze the convergence rate of the algorithm the global error order mathcal o tau 2 h 4 in discrete l 2 norm is proved theoretically and verified numerically some numerical experiments show the robustness of the algorithm for small coefficients of the fourth order term for the one dimensional case in addition coarsening dynamics are simulated in large domains and the 1 3 power laws are observed for the two dimensional case

rayleigh benard convection improved bounds on the nusselt number we consider rayleigh benard convection as modelled by the boussinesq equations in the infinite prandtl number limit we are interested in the scaling of the average upward heat transport the nusselt number nu in terms of the non dimensionalized temperature forcing the rayleigh number ra experiments asymptotics and heuristics suggest that nu ra1 3 this work is mostly inspired by two earlier rigorous work on upper bounds of nu in terms of ra 1 the work of constantin and doering establishing nu ra1 3ln 2 3ra with help of a logarithmically failing maximal regularity estimate in l on the level of the stokes equation 2 the work of doering reznikoff and the first author establishing nu ra1 3ln 1 3ra with help of the background field method the paper contains two results 1 the background field method can be slightly modified to yield nu ra1 3ln 1 15ra 2 the estimates behind the background field method can be combined with the maximal regularity in l to yield nu ra1 3ln 1 3ln ra

upper bounds on coarsening rates we consider two standard models of surface energy driven coarsening a constant mobility cahn hilliard equation whose large time behavior corresponds to mullins sekerka dynamics and a degenerate mobility cahn hilliard equation whose large time behavior corresponds to motion by surface diffusion arguments based on scaling suggest that the typical length scale should behave as in the first case and in the second we prove a weak one sided version of this assertion showing roughly speaking that no solution can coarsen faster than the expected rate our result constrains the behavior in a time averaged sense rather than pointwise in time and it constrains not the physical length scale but rather the perimeter per unit volume the argument is simple and robust combining the basic dissipation relations with an interpolation inequality and an ode argument

on global stability for lifschitz slyozov wagner like equations this paper is concerned with the stability and asymptotic stability at large time of solutions to a system of equations which includes the lifschitz slyozov wagner lsw system in the case when the initial data has compact support the main result of the paper is a proof of weak global asymptotic stability for lsw like systems previously strong local asymptotic stability results were obtained by niethammer and velazquez for the lsw system with initial data of compact support comparison to a quadratic model plays an important part in the proof of the main theorem when the initial data is critical the quadratic model extends the linear model of carr and penrose and has a time invariant solution which decays exponentially at the edge of its support in the same way as the infinitely differentiable self similar solution of the lsw model

transient and self similar dynamics in thin film coarsening abstract we study coarsening in a simplified model of one dimensional thin films of viscous fluids on hydrophobic substrates lubrication theory shows that such films are unstable and dewet to form droplets that then aggregate over long timescales the masses and positions of the droplets can be described by a coarsening dynamical system cds consisting of odes and deletion rules we develop discrete and continuous mean field models that reproduce the well known n t o t 2 5 long time statistical power law for the number of drops a lifshitz slyozov wagner type lsw continuous model predicts the self similar distribution of drop masses matching with histograms produced by cds simulations and the discrete mean field model we also describe the distribution of drops following homogeneous versus heterogeneous dewetting and use these as initial conditions for the cds simulations that yield characteristic staircasing transients transients can also include recurring spike formation behavior in the mass distribution for idealized initial conditions we show that the transient dynamics can span the full coarsening process bypassing the power law regime entirely

quantifying 3 d vascular structures in mra images using hybrid pde and geometric deformable models the aim of this paper is to present a hybrid approach to accurate quantification of vascular structures from magnetic resonance angiography mra images using level set methods and deformable geometric models constructed with 3 d delaunay triangulation multiple scale filtering based on the analysis of local intensity structure using the hessian matrix is used to effectively enhance vessel structures with various diameters the level set method is then applied to automatically segment vessels enhanced by the filtering with a speed function derived from enhanced mra images since the goal of this paper is to obtain highly accurate vessel borders suitable for use in fluid flow simulations in a subsequent step the vessel surface determined by the level set method is triangulated using 3 d delaunay triangulation and the resulting surface is used as a parametric deformable model energy minimization is then performed within a variational setting with a first order internal energy the external energy is derived from 3 d image gradients using the proposed method vessels are accurately segmented from mra data

a new approach to the continuum modeling of epitaxial growth slope selection coarsening and the role of the uphill current abstract we develop a new approach to the macroscopic modeling of epitaxial growth focusing on the slope selection and coarsening observed in spiral mode growth our model distinguishes between the surface height and the surface adatom density these quantities evolve by a coupled pair of partial differential equations a hamilton jacobi equation for the height coupled to a nonlinear diffusion equation for the adatom density the influence of the ehrlich schwoebel barrier is included through an uphill current in the equation for adatom density our model predicts slope selection and coarsening thus it offers a possible mechanism for these effects the model predicts in particular that the coarsening rate depends mainly on the strength of the ehrlich schwoebel barrier

continuum theory of epitaxial crystal growth i we present various continuum limits to describe epitaxial thin film growth we consider a hierarchy of models which can take into account the diffusion of terrace adatoms attachment and detachment of edge adatoms vapor phase diffusion and the effect of multiple species the starting point is the burton cabrera frank type step flow model we have obtained partial differential equations in the form of a coupled system of diffusion equation for the adatom density and a hamilton jacobi equation for the film height function this is supplemented with appropriate boundary conditions at the continuum level to describe the growth at the peaks and valleys on the film the results here can be used in a macroscopic description of thin film growth

snakes active contour models a snake is an energy minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges snakes are active contour models they lock onto nearby edges localizing them accurately scale space continuation can be used to enlarge the capture region surrounding a feature snakes provide a unified account of a number of visual problems including detection of edges lines and subjective contours motion tracking and stereo matching we have used snakes successfully for interactive interpretation in which user imposed constraint forces guide the snake near features of interest

on the significance of normal cross sectional extension in beam theory with application to contact problems abstract the main features of contact problems of elastic beams are explored by considering a specific equilibrium problem of a beam in contact with a smooth rigid flat surface solutions of four separate linear theories namely a general theory g and three others which are constructed as constrained theories are considered these constrained theories differ from the general theory only by the degree of exclusion of one or both types of deformation usually referred to as a transverse normal strain and b transverse shear deformation thus with both a and b absent the constrained theory corresponds to the bernoulli euler beam theory be with only b absent the constrained theory n accounts for normal extensional deformation and with only a absent the constrained theory corresponds to the timoshenko beam theory t comparison of the predictions of the solutions of the three constrained theories with that of theory g shows that theory n contains the main physics of the contact problem and correctly predicts the conditions under which the beam loses contact also the contact force is continuous at the end points of the contact region in contrast neither of the other two constrained theories be or t correctly predicts these features

second order convex splitting schemes for gradient flows with ehrlich schwoebel type energy application to thin film epitaxy we construct unconditionally stable unconditionally uniquely solvable and second order accurate in time schemes for gradient flows with energy of the form int omega f nabla phi bf x frac epsilon 2 2 delta phi bf x 2 d bf x the construction of the schemes involves the appropriate combination and extension of two classical ideas i appropriate convex concave decomposition of the energy functional and ii the secant method as an application we derive schemes for epitaxial growth models with slope selection f bf y frac14 bf y 2 1 2 or without slope selection f bf y frac12 ln 1 bf y 2 two types of unconditionally stable uniquely solvable second order schemes are presented the first type inherits the variational structure of the original continuous in time gradient flow while the second type does not preserve the variational structure we present numerical simulations for the case with slope selection which verify well known physical scaling laws for the long time coarsening process

computational geometry for patient specific reconstruction and meshing of blood vessels from mr and ct angiography investigation of three dimensional 3 d geometry and fluid dynamics in human arteries is an important issue in vascular disease characterization and assessment thanks to recent advances in magnetic resonance mr and computed tomography ct it is now possible to address the problem of patient specific modeling of blood vessels in order to take into account interindividual anatomic variability of vasculature generation of models suitable for computational fluid dynamics is still commonly performed by semiautomatic procedures in general based on operator dependent tasks which cannot be easily extended to a significant number of clinical cases in this paper we overcome these limitations making use of computational geometry techniques in particular 3 d modeling was carried out by means of 3 d level sets approach model editing was also implemented ensuring harmonic mean curvature vectors distribution on the surface and model geometric analysis was performed with a novel approach based on solving eikonal equation on voronoi diagram this approach provides calculation of central paths maximum inscribed sphere estimation and geometric characterization of the surface generation of adaptive thickness boundary layer finite elements is finally presented the use of the techniques presented here makes it possible to introduce patient specific modeling of blood vessels at clinical level

exact exponent λ of the autocorrelation function for a soluble model of coarsening the exponent lambda that describes the decay of the autocorrelation function a t in a phase ordering system a t sim l d lambda where d is the dimension and l the characteristic length scale at time t is calculated exactly for the time dependent ginzburg landau equation in d 1 we find lambda 0 399 383 5 ldots we also show explicitly that a small bias of positive domains over negative gives a magnetization which grows in time as m t sim l mu and prove that for the 1d ginzburg landau equation mu lambda exemplifying a general result

three dimensional vibration analysis of a cantilevered parallelepiped exact and approximate solutions the paper investigates the hypothetical assumption of neglecting transverse normal stress in vibration analysis for cantilevered thick plates and rectangular parallelepiped the analysis solves the three dimensional elasticity energy functional including as well as excluding transverse normal stress and obtains free vibration solutions for a cantilevered parallelepiped although it is widely accepted the omission of transverse normal stress is well justified in kirchhoff love thin plate theory and higher order thick plate models the transverse normal stress effects and thickness extent to which the thick plate models apply as the thickness increases are practically unknown the inconsistency of assuming constant transverse normal displacement through thickness for thick plate models is also addressed the paper concludes that for a rectangular parallelepiped with thickness exceeding a certain limit there is considerable discrepancy if transverse normal stress is neglected

level set methods for the simulation of epitaxial phenomena we introduce a model for epitaxial phenomena based on the motion of island boundaries which is described by the level set method our model treats the growing film as a continuum in the lateral direction but retains atomistic discreteness in the growth direction an example of such an island dynamics model using the level set method is presented and compared with the corresponding rate equation description extensions of our methodology to more general settings are then discussed s1063 651x 98 50212 7

grain boundary motion arising from the gradient flow of the aviles giga functional abstract this paper considers the singular limit of the equation t 2 1 2 1 grain boundaries limiting discontinuities in form networks that coarsen over time a matched asymptotic analysis is used to derive a free boundary problem consisting of curve motion coupled along hyperbolic characteristics and junction conditions an intermediate boundary layer near extrema junctions is discovered along with the relevant nonlocal junction conditions the limiting dynamics can be viewed in the context of a gradient flow of the sharp interface energy on an attracting manifold dynamic scaling of the long time coarsening process can be explained by dimensional analysis of the reduced problem

phase segregation dynamics in particle systems with long range interactions ii interface motion we study properties of the solutions of a family of second order integrodifferential equations which describe the large scale dynamics of a class of microscopic phase segregation models with particle conserving dynamics we first establish existence and uniqueness as well as some properties of the instantonic solutions then we concentrate on formal asymptotic sharp interface limits we argue that the obtained interface evolution laws a stefan like problem and the mullins sekerka solidification model coincide with the ones which can be obtained in the analogous limits from the cahn hilliard equation the fourth order pde which is the standard macroscopic model for phase segregation with one conservation law

convergence of the cahn hilliard equation to the hele shaw model we prove that level surfaces of solutions to the cahn hilliard equation tend to solutions of the hele shaw problem under the assumption that classical solutions of the latter exist the method is based on a new matched asymptotic expansion for solutions a spectral analysis for linearizd operators and an estimate for the difference between the true solutions and certain approximate ones

sharp interface limits of the cahn hilliard equation with degenerate mobility in this work sharp interface limits for the degenerate cahn hilliard equation with a polynomial double well free energy and a mobility that vanishes at the minima of the double well are derived for the choice of a quadratic mobility the leading order sharp interface motion is not governed by pure surface diffusion as has been previously claimed in the literature but contains a contribution from nonlinear porous medium type bulk diffusion at the same order our analysis reveals that there are two subcases one where the solution for the order parameter is bounded between the minima proven to exist for the first mobility by elliott and garcke siam j math anal 27 1996 pp 404 423 and one where it converges to the classical stationary solution of the cahn hilliard equation consistent treatment of the bulk diffusion requires the matching of exponentially large and small terms in combination with multiple inner layers moreover the leading order sharp interface motion depends sensitively on the choice of mobility the asymptotic analysis shows that for example with a biquadratic mobility the leading order sharp interface motion is driven only by surface diffusion the sharp interface models are corroborated by comparing relaxation rates of perturbations to a radially symmetric stationary state with those obtained by the phase field model

interpolation inequalities in pattern formation abstract we prove some interpolation inequalities which arise in the analysis of pattern formation in physics they are the strong version of some already known estimates in weak form that are used to give a lower bound of the energy in many contexts coarsening and branching in micromagnetics and superconductors the main ingredient in the proof of our inequalities is a geometric construction which was first used by choksi conti kohn and one of the authors in 4 in the study of branching in superconductors

minimal surfaces based object segmentation a geometric approach for 3d object segmentation and representation is presented the segmentation is obtained by deformable surfaces moving towards the objects to be detected in the 3d image the model is based on curvature motion and the computation of surfaces with minimal areas better known as minimal surfaces the space where the surfaces are computed is induced from the 3d image volumetric data in which the objects are to be detected the model links between classical deformable surfaces obtained via energy minimization and intrinsic ones derived from curvature based flows the new approach is stable robust and automatically handles changes in the surface topology during the deformation

derivation of the lsw theory for ostwald ripening by homogenization methods the coarsening of a solid phase in an undercooled liquid is described by a stefan problem with surface tension the solid phase is characterized by a large number of balls with small volume fraction and small capacity we solve the quasi static equation as well as the parabolic problem and construct approximate solutions by means of comparison principles the framework of young measures is used to pass to the limit of infinitely many particles we obtain that the capacity is the crucial quantity to get the conservation law for the particle size distribution which is studied in the classical theory for ostwald ripening by lifshitz slyozov wagner lsw

upper bounds for coarsening for the deep quench obstacle problem the deep quench obstacle problem models phase separation at low temperatures during phase separation domains of high and low concentration are formed then coarsen or grow in average size of interest is the time dependence of the dominant length scales of the system relying on recent results by novick cohen and shishkov discrete contin dyn syst b 25 251 272 2009 we demonstrate upper bounds for coarsening for the deep quench obstacle problem with either constant or degenerate mobility for the case of constant mobility we obtain upper bounds of the form t1 3 at early times as well as at times t for which e t le frac 1 overline u 2 4 where e t denotes the free energy for the case of degenerate mobility we get upper bounds of the form t1 3 or t1 4 at early times depending on the value of e 0 as well as bounds of the form t1 4 whenever e t le frac 1 overline u 2 4

geometric motion for a degenerate allen cahn cahn hilliard system the partial wetting case abstract using formal asymptotics we demonstrate that in a low temperature coarsening limit a degenerate allen cahn cahn hilliard system yields a geometric problem in which small particles whose shape evolves according to surface diffusion move along a surface where the chemical potential is quasi static which itself moves by motion by mean curvature the degenerate allen cahn cahn hilliard system was developed in j w cahn a novick cohen evolution equations for phase separation and ordering in binary alloys j stat phys 76 1994 877 909 to describe simultaneous ordering and phase separation and within this context the particles which contain a minor disordered phase are embedded along grain boundaries which partition the system into two ordered phase variants the limiting problem though can also be viewed as a diffuse interface approximation for various problems in materials science in which surface diffusion and motion by mean curvature are coupled see for example j kanel a novick cohen a vilenkin a numerical study of grain boundary motion in bicrystals acta mat 53 2004 227 235 a novick cohen order disorder and phase separation modeling grain sintering in e inan k z markov eds proceedings of the ninth international symposium cmds9 istanbul turkey world scientific publishing co 1999 pp 56 68 the present analysis extends a previous study a novick cohen triple junction motion for an allen cahn cahn hilliard system physica d 137 2000 1 24 which focused on the complete wetting limit and on motions in the plane here we treat the partial wetting case and our analysis accommodates motion in three dimensions

upper bounds on the coarsening rate of discrete ill posed nonlinear diffusion equations we prove a weak upper bound on the coarsening rate of the discrete in space version of an ill posed nonlinear diffusion equation the continuum version of the equation violates parabolicity and lacks a complete well posedness theory in particular numerical simulations indicate very sensitive dependence on initial data nevertheless models based on its discrete in space version which we study are widely used in a number of applications including population dynamics chemotactic movement of bacteria granular flow formation of shear bands and computer vision image denoising and segmentation our bounds have implications for all three applications 2008 wiley periodicals inc

stability analysis of large time stepping methods for epitaxial growth models numerical methods for solving the continuum model of the dynamics of the molecular beam epitaxy mbe require very large time simulation and therefore large time steps become necessary the main purpose of this work is to construct and analyze highly stable time discretizations which allow much larger time steps than those of a standard implicit explicit approach to this end an extra term which is consistent with the order of the time discretization is added to stabilize the numerical schemes then the stability properties of the resulting schemes are established rigorously numerical experiments are carried out to support the theoretical claims the proposed methods are also applied to simulate the mbe models with large solution times the power laws for the coarsening process are obtained and are compared with previously published results

a compactness result in the gradient theory of phase transitions in the plane as 0 this functional favours jraj 1 and penalizes singularities where jrraj concentrates our main result is a compactness theorem if fe a g 0 is uniformly bounded then fra g 0 is compact in l thus in the limit 0 a solves the eikonal equation jraj 1 almost everywhere our analysis uses entropy relations and the div curl lemma adopting tartar s approach to the interaction of linear di erential equations and nonlinear algebraic relations

a multi phase mullins sekerka system matched asymptotic expansions and an implicit time discretisation for the geometric evolution problem we propose a generalisation of the mullins sekerka problem to model phase separation in multi component systems the model includes equilibrium equations in bulk the gibbs thomson relation on the interfaces young s law at triple junctions together with a dynamic law of stefan type using formal asymptotic expansions we establish the relationship to a transition layer model known as the cahn hilliard system we introduce a notion of weak solutions for this sharp interface model based on integration by parts on manifolds together with measure theoretical tools through an implicit time discretisation we construct approximate solutions by stepwise minimisation under the assumption that there is no loss of area as the time step tends to zero we show the existence of a weak solution

upper bound on the coarsening rate for an epitaxial growth model we study a specific example of energy driven coarsening in two space dimensions the energy is u 2 1 u 2 2 the evolution is the fourth order pde representing steepest descent this equation has been proposed as a model of epitaxial growth for systems with slope selection numerical simulations and heuristic arguments indicate that the standard deviation of u grows like t1 3 and the energy per unit area decays like t 1 3 we prove a weak one sided version of the latter statement the time averaged energy per unit area decays no faster than t 1 3 our argument follows a strategy introduced by kohn and otto in the context of phase separation combining i a dissipation relation ii an isoperimetric inequality and iii an ode lemma the interpolation inequality is new and rather subtle our proof is by contradiction relying on recent compactness results for the aviles giga energy 2003 wiley periodicals inc

on the three dimensional vibrations of the cantilevered rectangular parallelepiped a solution is presented for the three dimensional problem of determining the free vibration frequencies and mode shapes for a rectangular parallelepiped which is completely fixed on one face and free on the other five faces this problem apparently is previously unsolved in the published literature the ritz method is used with displacements assumed in the form of algebraic polynomials convergence is studied numerical results are given for the first five frequencies of each of the four symmetry classes of vibration for five thick parallelepiped configurations including the cube contour plots are exhibited for the modal displacements of the cube the effects of varying poisson s ratio are also observed

energy scaling approach to phase ordering growth laws we present a simple unified approach to determining the growth law for the characteristic length scale l t in the phase ordering kinetics of a system quenched from a disordered phase to within an ordered phase this approach based on a scaling assumption for pair correlations determines l t self consistently for purely dissipative dynamics by computig the time dependence of the energy in two ways we derive growth laws for conserved and nonconserved o n models including two dimensional xy models and systems with textures we demonstrate that the growth laws for other systems such as liquid crystals and potts models are determined by the type of topological defect in the order parameter field that dominates the energy we also obtain generalized porod laws for systems with topological textures

nonlinear approximation and the space bv r2 given a function 2 0 q 0 l 2 and a real number t 0 let u f t infg bv g 11 ll vgig where the infimum is taken over all functions g g bv of bounded variation on this and related extremal problems arise in several areas of mathematics such as interpolation of operators and statistical estimation as well as in digital image processing techniques for finding minimizers g for u f t based on variational calculus and nonlinear partial differential equations have been put forward by several authors dms ro ms cl the main disadvantage of these approaches is that they are numerically intensive on the other hand it is well known that more elementary methods based on wavelet shrinkage solve related extremal problems for example the above problem with bv replaced by the besov space b l i see e g cdll however since bv has no simple description in terms of wavelet coefficients it is not clear that minimizers for u f t can be realized in this way we shall show in this paper that simple methods based on haar thresholding provide near minimizers for u f t our analysis of this extremal problem brings forward many interesting relations between haar decompositions and the space bv 1 introduction nonlinear approximation has recently played an impor tant role in several problems of image processing including compression noise removal and feature extraction we have in mind techniques such as wavelet compression djl wavelet shrinkage or thresholding djkp1 wavelet packets cw and greedy algorithms mz dt there has also been an impressive contribution of techniques based on variational calculus and nonlinear partial dif ferential equations see e g dms ro ms cl especially to the problems of noise removal and image segmentation the common point between these two approaches is their ability to adapt to the composite nature of images edge tex tures and smooth regions should be treated adaptively a requirement which is certainly not fulfilled by the classical linear filtering techniques

error analysis of a mixed finite element method for the molecular beam epitaxy model this paper investigates the error analysis of a mixed finite element method with crank nicolson time stepping for simulating the molecular beam epitaxy mbe model the fourth order differential equation of the mbe model is replaced by a system of equations consisting of one nonlinear parabolic equation and an elliptic equation then a mixed finite element method requiring only continuous elements is proposed to approximate the resulting system it is proved that the semidiscrete and fully discrete versions of the numerical schemes satisfy the nonlinearity energy stability property which is important in the numerical implementation moreover detailed analysis is provided to obtain the convergence rate numerical experiments are carried out to validate the theoretical results

optimal rate convergence analysis of a second order scheme for a thin film model with slope selection abstract in this paper an energy stable second order mixed finite element scheme is proposed and analyzed for the thin film epitaxial growth model with slope selection we employ second order backward differentiation formula bdf scheme with a second order stabilized term which guarantees the long time energy stability to approximate the continuous model in terms of the convergence analysis the key difficulty to derive an optimal rate spatial estimate is associated with the appearance of the gradient operator in the nonlinear terms which may lead to a loss of optimal accuracy order to overcome this well known difficulty we make use of some auxiliary techniques over triangular elements and obtain an optimal convergence rate o h q 1 t 2 in comparison with o h q t 2 rate from a standard projection estimate furthermore we use an efficient preconditioned steepest descent psd solver for the numerical implementation a few numerical examples are presented to validate the stability and convergence

numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method abstract the molecular beam epitaxial model is derived from the variation of a free energy that consists of either a fourth order ginzburg landau double well potential or a nonlinear logarithmic potential in terms of the gradient of a height function one challenge in solving the mbe model numerically is how to develop proper temporal discretization for the nonlinear terms in order to preserve energy stability at the time discrete level in this paper we resolve this issue by developing a first and second order time stepping scheme based on the invariant energy quadratization ieq method the novelty is that all nonlinear terms are treated semi explicitly and the resulted semi discrete equations form a linear system at each time step moreover the linear operator is symmetric positive definite and thus can be solved efficiently we then prove that all proposed schemes are unconditionally energy stable the semi discrete schemes are further discretized in space using finite difference methods and implemented on gpus for high performance computing various 2d and 3d numerical examples are presented to demonstrate stability and accuracy of the proposed schemes

self similarity in two phase curvature flow in this work we present a family of exact self similar solutions for the distribution of enclosed areas under mean curvature flow on the union of disjoint closed curves in the plane to our knowledge this is the first example of closed form exact self similar solutions for the mean curvature flow we perform numerical simulations for large sets of initial data obtained as the zero level set of gaussian random fields and observe numerical convergence to one particular member of this family

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

phase instability and coarsening in two dimensions instabilities and pattern formation is the rule in nonequilibrium systems selection of a persistent length scale or coarsening increase in the length scale with time are the two major alternatives when and under which conditions one dynamics prevails over the other is a long standing problem particularly beyond one dimension it is shown that the challenge can be defied in two dimensions using the concept of phase diffusion equation we find that coarsening is related to the ensuremath lambda dependence of a suitable phase diffusion coefficient d 11 ensuremath lambda depending on lattice symmetry and conservation laws these results are exemplified analytically on prototypical nonlinear equations

a variational method for geometric regularization of vascular segmentation in medical images in this paper a level set based geometric regularization method is proposed which has the ability to estimate the local orientation of the evolving front and utilize it as shape induced information for anisotropic propagation we show that preserving anisotropic fronts can improve elongations of the extracted structures while minimizing the risk of leakage to that end for an evolving front using its shape offset level set representation a novel energy functional is defined it is shown that constrained optimization of this functional results in an anisotropic expansion flow which is useful for vessel segmentation we have validated our method using synthetic data sets 2 d retinal angiogram images and magnetic resonance angiography volumetric data sets a comparison has been made with two state of the art vessel segmentation methods quantitative results as well as qualitative comparisons of segmentations indicate that our regularization method is a promising tool to improve the efficiency of both techniques

vessels as 4 d curves global minimal 4 d paths to extract 3 d tubular surfaces and centerlines in this paper we propose an innovative approach to the segmentation of tubular structures this approach combines all of the benefits of minimal path techniques such as global minimizers fast computation and powerful incorporation of user input while also having the capability to represent and detect vessel surfaces directly which so far has been a feature restricted to active contour and surface techniques the key is to represent the trajectory of a tubular structure not as a 3 d curve but to go up a dimension and represent the entire structure as a 4 d curve then we are able to fully exploit minimal path techniques to obtain global minimizing trajectories between two user supplied endpoints in order to reconstruct tubular structures from noisy or low contrast 3 d data without the sensitivity to local minima inherent in most active surface techniques in contrast to standard purely spatial 3 d minimal path techniques however we are able to represent a full tubular surface rather than just a curve which runs through its interior our representation also yields a natural notion of a tube s central curve we demonstrate and validate the utility of this approach on magnetic resonance mr angiography and computed tomography ct images of coronary arteries

a second order in time uniquely solvable unconditionally stable numerical scheme for cahn hilliard navier stokes equation we propose a novel second order in time numerical scheme for cahn hilliard navier stokes phase field model with matched density the scheme is based on second order convex splitting for the cahn hilliard equation and pressure projection for the navier stokes equation we show that the scheme is mass conservative satisfies a modified energy law and is therefore unconditionally stable moreover we prove that the scheme is unconditionally uniquely solvable at each time step by exploring the monotonicity associated with the scheme thanks to the simple coupling of the scheme we design an efficient picard iteration procedure to further decouple the computation of cahn hilliard equation and navier stokes equation we implement the scheme by the mixed finite element method ample numerical experiments are performed to validate the accuracy and efficiency of the numerical scheme

coarsening dynamics of the one dimensional cahn hilliard model the dynamics of one dimensional cahn hilliard model is studied the stationary and particle type solutions the bubbles are perused as a function of initial conditions boundary conditions and system size we characterize the bubble solutions which are involved in the coarsening dynamics and establish the bifurcation scenarios of the system a set of ordinary differential equation permits us to describe the coarsening dynamics in very good agreement with numerical simulations we also compare these dynamics with the bubble dynamics deduced from the classical kink interaction computation where our model seems to be more appropriated in the case of two bubbles we deduce analytical expressions for the bubble s position and the bubble s width besides a simple description of the ulterior dynamics is presented

domain branching in uniaxial ferromagnets a scaling law for the minimum energy we address the branching of magnetic domains in a uniaxial ferromagnet our thesis is that branching is required by energy minimization to show this we consider the nonlocal nonconvex variational problem of micromagnetics we identify the scaling law of the minimum energy by proving a rigorous lower bound which matches the already known upper bound we further show that any domain pattern achieving this scaling law must have average width of order l 2 3 where l is the length of the magnet in the easy direction finally we argue that branching is required by considering the constrained variational problem in which branching is prohibited and the domain structure is invariant in the easy direction its scaling law is different

the gradient theory of phase transitions for systems with two potential wells on generalise la theorie du gradient des transitions de phase au cas a valeur vectoriel on considere la famille des perturbations e e u w u dx e 2 u 2 dx de la fonctionnelle non convexe e 0 u w u dx ou w r n r supporte deux phases et n 1

kinetics of phase ordering in the o n model with a conserved order parameter we study the phase ordering dynamics of the o n model with a conserved order parameter for systems with topological defects we present results from both cell dynamical simulations and predictions of a gaussian auxiliary field gaf approximation for the xy n 2 model in two and three dimensions and the heisenberg n 3 model in three dimensions we describe the results for the structure factor s q and growth law l t from simulations the growth laws obtained are consistent with theoretical predictions based on energy scaling arguments the structure factor shows good dynamical scaling using a length extracted from its first moment the simulations are compared with the theoretical predictions of the gaf for the scaling functions our results show that the gaf gives a good qualitative description of most features of the structure factor however it overestimates the amplitude of the porod tail in the large q limit moreover for small q the structure factor exhibits a q2 behaviour instead of the expected generalized yeung result of q4

non self similar behavior in the lsw theory of ostwald ripening the classical lifshitz slyozov wagner theory of domain coarsening predicts asymptotically self similar behavior for the size distribution of a dilute system of particles that evolve by diffusional mass transfer with a common mean field here we consider the long time behavior of measure valued solutions for systems in which particle size is uniformly bounded i e for initial measures of compact support we prove that the long time behavior of the size distribution depends sensitively on the initial distribution of the largest particles in the system convergence to the classically predicted smooth similarity solution is impossible if the initial distribution function is comparable to any finite power of distance to the end of the support we give a necessary criterion for convergence to other self similar solutions and conditional stability theorems for some such solutions for a dense set of initial data convergence to any self similar solution is impossible

front migration in the nonlinear cahn hilliard equation the method of matched asymptotic expansions is used to describe solutions of the nonlinear cahn hilliard equation for phase separation in n 1 space dimensions the expansion is formally valid when the thickness of internal transition layers is small compared with the distance separating layers and with their radii of curvature on the dominant slowest timescale the interface velocity is determined by the mean curvature of the interface by a non local relation which is identical to that in a well known quasi static model of solidification which exhibits a shape instability discovered by mullins sekerka j appl phys 34 323 329 1963 on a faster timescale the cahn hilliard equation regularizes a classic two phase stefan problem similarity solutions of the two phase stefan problem should describe boundary layers existence and uniqueness of such similarity solutions which admit metastable states is proved rigorously in an appendix

well posedness and spectral properties of heat and wave equations with non local conditions abstract we consider the one dimensional heat and wave equations but instead of boundary conditions we impose on the solution certain non local integral constraints an appropriate hilbert setting leads to an integration by parts formula in sobolev spaces of negative order and eventually allows us to use semigroup theory leading to analytic well posedness hence sharpening regularity results previously obtained by other authors in doing so we introduce a parametrization of such integral conditions that includes known cases but also shows the connection with more usual boundary conditions like periodic ones in the self adjoint case we even obtain eigenvalue asymptotics of so called weyl s type

computational studies of coarsening rates for the cahn hilliard equation with phase dependent diffusion mobility we study computationally coarsening rates of the cahn hilliard equation with a smooth double well potential and with phase dependent diffusion mobilities the latter is a feature of many materials systems and makes accurate numerical simulations challenging our numerical simulations confirm earlier theoretical predictions on the coarsening dynamics based on asymptotic analysis we demonstrate that the numerical solutions are consistent with the physical gibbs thomson effect even if the mobility is degenerate in one or both phases for the two sided degenerate mobility we report computational results showing that the coarsening rate is on the order of l c t 1 4 independent of the volume fraction of each phase for the one sided degenerate mobility that is non degenerate in the positive phase but degenerate in the negative phase we illustrate that the coarsening rate depends on the volume fraction of the positive phase for large positive volume fractions the coarsening rate is on the order of l c t 1 3 and for small positive volume fractions the coarsening rate becomes l c t 1 4

isosurfaces as deformable models for magnetic resonance angiography vascular disease produces changes in lumenal shape evident in magnetic resonance angiography mra however quantification of vascular shape from mra is problematic due to image artifacts prior deformable models for vascular surface reconstruction primarily resolve problems of initialization of the surface mesh however initialization can be obtained in a trivial manner for mra using isosurfaces we propose a methodology for deforming the isosurface to conform to the boundaries of objects in the image with minimal a priori assumptions of object shape as in conventional methods external forces attract the surface toward edges in the image however smoothing is produced by a moment that aligns the normals of adjacent surface triangles notably the moment produces no translational motion of surface triangles the deformable isosurface was applied to a digital phantom of a stenotic artery to mra of three renal arteries with atherosclerotic disease and mra of one carotid artery with atherosclerotic disease results of the surface reconstruction from the deformable model were compared with conventional x ray angiography for the renal arteries measurement of the degree of stenosis of the renal arteries was within 12 spl plusmn 6 the deformable model provided improvements over the isosurface in all cases in terms of measurement of the degree of stenosis or improving the surface smoothness

on steady motions of an elastic rod with application to contact problems based on the directed theory of an elastic rod developed by green naghdi and several of their co workers this paper presents a formulation which is convenient to use when considering steady motions of elastic rods a properly invariant linear theory of this formulation is also developed and used to analyze several systems these systems involve surface constraints and find application in various technologies where elastic rod like bodies are present

compactness in ginzburg landau energy by kinetic averaging we consider a ginzburg landau energy for two dimensional divergence free fields appearing in the gradient theory of phase transition for instance we prove that as the relaxation parameter vanishes families of such fields with finite energy are compact in l p our proof is based on a kinetic interpretation of the entropies which were introduced by desimone kohn muller and otto the so called kinetic averaging lemmas allow to generalize their compactness results also the method yields a kinetic equation for the limit where the righthand side is an unknown kinetic defect bounded measurefrom which we deduce some sobolev regularity this measure also satisfies some cancellation properties depending on its local regularity which seem to indicate several level of singularities in the limit c 2001 john wiley sons inc

a mixed finite element method for thin film epitaxy we present a mixed finite element method for the thin film epitaxy problem comparing to the primal formulation which requires c 2 elements in the discretization the mixed formulation only needs to use c 1 elements by introducing proper dual variables the dual variable in our method is defined naturally from the nonlinear term in the equation and its accurate approximation will be essential for understanding the long time effect of the nonlinear term for time discretization we use a backward euler semi implicit scheme which involves a convex concave decomposition of the nonlinear term the scheme is proved to be unconditionally stable and its convergence rate is analyzed

a multigrid finite element solver for the cahn hilliard equation a multigrid finite element solver for the cahn hilliard equation is presented that has mesh independent convergence rates for any time step size including in the important limit e 0 which is examined via numerical examples numerics are performed for a number of test problems which show that the features of the cahn hilliard equation minimising interface measure lyapunov energy functional etc are preserved we also explore the use of this solver in conjunction with adaptive time stepping and adaptive mesh strategies

on constitutive relations for elastic rods this paper explores the relationships among several existing procedures for specifying linear and nonlinear constitutive relations for hyperelastic cosserat or directed curves from three dimensional considerations these procedures are designed to ensure that exact solutions of the balance laws of the directed curve are in approximate agreement with the corresponding exact solutions of classical continuum mechanics the particular case of the linearized bernoulli euler beam theory is also examined in addition a new procedure is proposed in this paper this procedure is used to facilitate comparisons among various existing procedures although it is conceptually simple it illustrates the inherent arbitrariness in specifying constitutive relations for elastic rods

a microscopic time scale approximation to the behavior of the local slope on the faceted surface under a nonuniformity in supersaturation abstract the morphological stability of a growing faceted crystal is discussed we argue that the interplay between nonuniformity in supersaturation on a growing facet and anisotropy of surface kinetics derived from the lateral motion of steps leads to a faceted instability qualitatively speaking as long as the nonuniformity in supersaturation on the facet is not too large it can be compensated for by a variation of step density along the facet and the faceted crystal can grow in a stable manner the problem can be modeled as a hamilton jacobi equation for height of the crystal surface the notion of a maximal stable region of a growing facet is introduced for microscopic time scale approximation of the original hamilton jacobi equation it is shown that the maximal stable region keeps its shape determined by profile of the surface supersaturation with constant growth rate by studying large time behavior of solution of macroscopic time scale approximation a quantitative criterion for the facet stability is given

on lower semicontinuity of a defect energy obtained by a singular limit of the ginzburg landau type energy for gradient fields a defect energy j which measures jump discontinuities of a unit length gradient field is studied the number indicates the power of the jumps of the gradient fields that appear in the density of j it is shown that j for 3 is lower semicontinuous on the space of unit gradient fields belonging to bv in l 1 convergence of gradient fields a similar result holds for the modified energy which measures only a particular type of defect the result turns out to be very subtle since with 3 is not lower semicontinuous as is shown in this paper the key idea behind semicontinuity is a duality representation for j 3 and the duality representation is also important for obtaining a lower bound by using j 3 for the relaxation limit of the ginzburg landau type energy for gradient fields the lower bound obtained here agrees with the conjectured value of the relaxation limit

on the two frequency spectra of timoshenko beams abstract this paper is concerned with the question of whether there are indeed two distinct spectra of frequencies for the transverse vibrations of timoshenko beams as has been claimed by a number of prior authors for the case of the simply supported beam and more recently for beams supported in an arbitrary manner elementary analysis leads to the conclusion that there is only a single frequency spectrum in the particular case of the simply supported beam the two frequency spectra viewpoint may be expedient as a device to compute frequencies but does not serve otherwise to explain the complex dynamical behavior of timoshenko beams

higher order cahn hilliard equations with dynamic boundary conditions abstract our aim in this paper is to study the well posedness and the dissipativity of higher order cahn hilliard equations with dynamic boundary conditions more precisely we prove the existence and uniqueness of solutions and the existence of the global attractor

on coupled longitudinal and lateral vibrations of elastic rods abstract in this paper the coupled longitudinal and lateral vibrations of a rectangular parallelepiped composed of a uniform isotropic elastic body are examined the one dimensional model for the body was developed by green naghdi and several of their co workers it is based on a cosserat or directed rod theory the frequency spectrum is found to divide naturally into three regions which are separated by degenerate cases when the cross section of the body is square a natural splitting of the modes occurs one half of this split involves motions coupling the axial extension and symmetric lateral deformation of the body the other half involves asymmetric lateral deformations the cases of a free free rod and a cantilevered rod are discussed and the results compared to existing works in the literature which use the three dimensional theory of elasticity to model the vibrations