an efficient threshold dynamics method for wetting on rough surfaces the threshold dynamics method developed by merriman bence and osher mbo is an efficient method for simulating the motion by mean curvature flow when the interface is away from the solid boundary direct generalization of mbo type methods to the wetting problem with interfaces intersecting the solid boundary is not easy because solving the heat equation in a general domain with a wetting boundary condition is not as efficient as it is with the original mbo method the dynamics of the contact point also follows a different law compared with the dynamics of the interface away from the boundary in this paper we develop an efficient volume preserving threshold dynamics method for simulating wetting on rough surfaces this method is based on minimization of the weighted surface area functional over an extended domain that includes the solid phase the method is simple stable with o n log n complexity per time step and is not sensitive to the inhomogeneity or roughness of the solid boundary

fast fourier transforms of piecewise constant functions we present an algorithm for the evaluation of the fourier transform of piecewise constant functions of two variables the algorithm overcomes the accuracy problems associated with computing the fourier transform of discontinuous functions in fact its time complexity is o n2 logn np log2 1 v log3 1 where is the accuracy n is the size of the problem p is the perimeter of the set of discontinuities and v is its number of vertices the algorithm is based on the lagrange interpolation formula and the green s theorem which are used to preprocess the data before applying the fast fourier transform it readily generalizes to higher dimensions and to piecewise smooth functions

approximation and comparison for motion by mean curvature with intersection points abstract consider the motion of a curve in the plane under its mean curvature it is a very interesting problem to investigate what happens when there are intersection points on the curve at which the mean curvature is singular in this paper we study this issue numerically by solving the allen cahn equation and the nonlocal evolution equation with kac potential the allen cahn equation is discretized by a monotone scheme and the nonlocal evolution equation with kac potential is discretized by the spectral method several curves with intersection points under motion by mean curvature are studied from a simple analysis and our numerical results we find that which direction to split of the curve at the intersection point depends on the angle of the curve at the point i e it splits in horizontal direction when the angle 2 in vertical direction when

efficient algorithms for diffusion generated motion by mean curvature the problem of simulating the motion of evolving surfaces with junctions according to some curvature dependent speed arises in a number of applications by alternately diffusing and sharpening characteristic functions for each region a variety of motions have been obtained which allow for topological mergings and breakings and produce no overlapping regions or vacuums however the usual finite difference discretization of these methods is often excessively slow when accurate solutions are sought even in two dimensions we propose a new spectral discretization of these diffusion generated methods which obtains greatly improved efficiency over the usual finite difference approach these efficiency gains are obtained in part through the use of a quadrature based refinement technique by integrating fourier modes exactly and by neglecting the contributions of rapidly decaying solution transients indeed numerical studies demonstrate that the new algorithm is often more than 1000 times faster than the usual finite difference discretization our findings are demonstrated on several examples

threshold dynamics for the piecewise constant mumford shah functional we propose an efficient algorithm for minimizing the piecewise constant mumford shah functional of image segmentation it is based on the threshold dynamics of merriman bence and osher for evolving an interface by its mean curvature we show that a very fast minimization can be achieved by alternating the solution of a linear parabolic partial differential equation and simple thresholding

algorithms for area preserving flows we propose efficient and accurate algorithms for computing certain area preserving geometric motions of curves in the plane such as area preserving motion by curvature these schemes are based on a new class of diffusion generated motion algorithms using signed distance functions in particular they alternate two very simple and fast operations namely convolution with the gaussian kernel and construction of the distance function to generate the desired geometric flow in an unconditionally stable manner we present applications of these area preserving flows to large scale simulations of coarsening

kernels with prescribed surface tension mobility for threshold dynamics schemes abstract we show how to construct a convolution kernel that has a desired anisotropic surface tension and desired anisotropic mobility to be used in threshold dynamics schemes for simulating weighted motion by mean curvature of interfaces including networks of them in both two and three dimensions moreover we discuss necessary and sufficient conditions for the positivity of the kernel which in the case of two phase flow ensures that the resulting scheme respects a comparison principle and implies convergence to the viscosity solution of the level set formulation of the flow in particular we show in a barrier type statement that the kernel cannot possibly be positive unless both the mobility and the surface tension satisfy necessary conditions in three dimensions and give a complete characterization among other results is a threshold dynamics scheme that is guaranteed to dissipate a non local approximation to the interfacial energy in the fully anisotropic multiphase setting using the new kernel construction

high order essentially nonsocillatory schemes for hamilton jacobi equations hamilton jacobi h j equations are frequently encountered in applications e g in control theory and differential games h j equations are closely related to hyperbolic conservation laws in one space dimension the former is simply the integrated version of the latter similarity also exists for the multidimensional case and this is helpful in the design of difference approximations in this paper high order essentially nonoscillatory eno schemes for h j equations are investigated which yield uniform high order accuracy in smooth regions and sharply resolve discontinuities in the derivatives the eno scheme construction procedure is adapted from that for hyperbolic conservation laws the schemes are numerically tested on a variety of one dimensional and two dimensional problems including a problem related to control optimization and high order accuracy in smooth regions good resolution of discontinuities in the derivatives and convergence to viscosity solutions are observed

large scale simulation of normal grain growth via diffusion generated motion diffusion generated motion is used to perform a very large scale simulation of normal grain growth in three dimensions with high accuracy the method is based on the diffusion of signed distance functions and shares similarities with level set methods the herring angle condition at junctions and topological transitions are naturally captured with this formulation this approach offers significant advantages over existing numerical methods and allows for accurate computations on scales not previously possible a fully resolved simulation of normal grain growth initially containing over 130 000 grains in three dimensions is presented and analysed it is shown that the average grain radius grows as the square root of time and the grain size distribution is self similar good agreement with other theoretical predictions experimental results and simulation results via other techniques is also demonstrated

motion of multiple junctions a level set approach abstract a coupled level set method for the motion of multiple junctions is proposed the new method extends the hamilton jacobi level set formulation of osher and sethian it retains the feature of tracking fronts by following level sets and allows the specification of arbitrary velocities on each front the diffusion equation is shown to generate curvature dependent motion and this is used to develop an algorithm to move multiple junctions with curvature dependent speed systems of reaction diffusion equations are shown to possess inherent properties which prohibit efficient numerical solutions when applied to curvature dependent motion

diffusion generated motion using signed distance functions we describe a new class of algorithms for generating a variety of geometric interfacial motions by alternating two steps construction of the signed distance function i e redistancing to the interface and convolution with a suitable kernel these algorithms can be seen as variants of merriman bence and osher s threshold dynamics 25 the new algorithms proposed here preserve the computational efficiency of the original threshold dynamics algorithm however unlike threshold dynamics the new algorithms also allow attaining high accuracy on uniform grids without adaptive refinement

numerical simulation and linear well posedness analysis for a class of three phase boundary motion problems we investigate the linear well posedness for a class of three phase boundary motion problems and perform some numerical simulations in a typical model three phase boundaries evolve under certain evolution laws with specified normal velocities the boundaries meet at a triple junction with appropriate conditions applied a system of partial differential equations and algebraic equations pdae is proposed to describe the problems with reasonable assumptions all problems are shown to be well posed if all three boundaries evolve under the same evolution law for problems involving two or more evolution laws we show the well posedness case by case for some examples numerical simulations are performed for some examples

diffusion generated motion for grain growth in two and three dimensions an efficient algorithm for accurately simulating curvature flow for large networks of curves in two dimensions and surfaces in three dimensions on uniform grids is proposed this motion arises in the technologically important problem of simulating grain boundary motion in polycrystalline materials in this formulation grain boundaries are zero level sets of signed distance functions curvature motion is achieved by first diffusing locally maintained signed distance functions followed by a reinitialization step a technique is devised to allow a single signed distance function to represent a large subset of spatially separated grains hundreds of thousands of grains can be simulated using a small number of signed distance functions in this work 32 in two dimensions and 64 in three dimensions are more than sufficient using modest computational hardware

a variational method for multiphase volume preserving interface motions we develop a numerical method for realizing mean curvature motion of interfaces separating multiple phases whose volumes are preserved throughout time the foundation of the method is a thresholding algorithm of the bence merriman osher type the original algorithm is reformulated in a vector setting which allows for a natural inclusion of constraints even in the multiphase case moreover a new method for overcoming the inaccuracy of thresholding methods on non adaptive grids is designed since this inaccuracy becomes especially prominent in volume preserving motions formal analysis of the method and numerical tests are presented

threshold growth dynamics we study the asymptotic shape of the occupied region for monotone deterministic dynamics in d dimensional euclidean space parametrized by a threshold 0 and a borel set n r d with positive and finite lebesgue measure if a n denotes the oocupied set of the dynamics at integer time n then a n 1 is obtained by adjoining any point x for which the volume of overlap between x n and a n exceeds except in some degenerate cases we prove that n 1 a n converges to a unique limiting shape l starting from any bounded initial region a 0 that is suitably large moreover l is computed as the polar transform for 1 w where w is an explicit width function that depends on n and

multiphase thermomechanics with interfacial structure 2 evolution of an isothermal interface paper 1 1988 g 1 of this series began an investigation whose goal is a thermomechanics of two phase continua based on gibbs s notion of a sharp phase interface endowed with thermomechanical structure in that paper a balance law balance of capillary forces was introduced and then applied in conjunction with suitable statements of the first two laws of thermodynamics the chief results are thermodynamic restrictions on constitutive equations exact and approximate free boundary conditions at the interface and a hierarchy of free boundary problems the simplest versions of these problems the mullins sekerka problems are essentially the classical stefan problem with the free boundary condition u 0 for the temperature replaced by the condition u h k where k is the curvature of the free boundary and h 0 is a material constant this dependence on curvature renders the problem difficult and apart from numerical studies involving linearization stability there are almost no supporting theoretical results

reinitialization free level set evolution via reaction diffusion this paper presents a novel reaction diffusion rd method for implicit active contours that is completely free of the costly reinitialization procedure in level set evolution lse a diffusion term is introduced into lse resulting in an rd lse equation from which a piecewise constant solution can be derived in order to obtain a stable numerical solution from the rd based lse we propose a two step splitting method to iteratively solve the rd lse equation where we first iterate the lse equation then solve the diffusion equation the second step regularizes the level set function obtained in the first step to ensure stability and thus the complex and costly reinitialization procedure is completely eliminated from lse by successfully applying diffusion to lse the rd lse model is stable by means of the simple finite difference method which is very easy to implement the proposed rd method can be generalized to solve the lse for both variational level set method and partial differential equation based level set method the rd lse method shows very good performance on boundary antileakage the extensive and promising experimental results on synthetic and real images validate the effectiveness of the proposed rd lse approach

second order threshold dynamics schemes for two phase motion by mean curvature abstract the threshold dynamics algorithm of merriman bence and osher is only first order accurate in the two phase setting its accuracy degrades further to half order in the multi phase setting a shortcoming it has in common with other related more recent algorithms such as the equal surface tension version of the voronoi implicit interface method as a first rigorous step in addressing this shortcoming we present two different second order accurate versions of two phase threshold dynamics unlike in previous efforts in this direction we present careful consistency calculations for both of our algorithms the first algorithm is consistent with its limit motion by mean curvature up to second order in any space dimension the second achieves second order accuracy only in dimension two but comes with a rigorous stability guarantee unconditional energy stability in any dimension a first for high order schemes of its type

auction dynamics a volume constrained mbo scheme abstract we show how auction algorithms originally developed for the assignment problem can be utilized in merriman bence and osher s threshold dynamics scheme to simulate multi phase motion by mean curvature in the presence of equality and inequality volume constraints on the individual phases the resulting algorithms are highly efficient and robust and can be used in simulations ranging from minimal partition problems in euclidean space to semi supervised machine learning via clustering on graphs in the case of the latter application numerous experimental results on benchmark machine learning datasets show that our approach exceeds the performance of current state of the art methods while requiring a fraction of the computation time

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

diffusion generated motion of curves on surfaces we present a new method for computing the curvature driven motion of a curve constrained to move on a given surface it is based on the diffusion generated motion algorithm and retains both the novel simplicity of that method as well as the natural extension to curves with junctions to general geometric motion laws and to higher dimensions the result is an extremely simple algorithm for curvature dependent motion on surfaces wherein the only evolution operation is linear diffusion in three dimensional euclidean space

a simple proof of convergence for an approximation scheme for computing motions by mean curvature we prove the convergence of an approximation scheme recently proposed by bence merriman and osher for computing motions of hypersurfaces by mean curvature our proof is based on viscosity solutions methods

a numerical method for tracking curve networks moving with curvature motion abstract a finite difference method is proposed to track curves whose normal velocity is given by their curvature and which meet at different types of junctions the prototypical example is that of phase interfaces that meet at prescribed angles although eutectic junctions and interactions through nonlocal effects are also considered the method is based on a direct discretization of the underlying parabolic problem and boundary conditions a linear stability analysis is presented for our scheme as well as computational studies that confirm the second order convergence to smooth solutions after a singularity in the curve network where the solution is no longer smooth we demonstrate almost second order convergence a numerical study of singularity types is done for the case of networks that meet at prescribed angles at triple junctions finally different discretizations and methods for implicit time stepping are presented and compared

a convolution thresholding approximation of generalized curvature flows we construct a convolution thresholding approximation scheme for the geometric surface evolution in the case when the velocity of the surface at each point is a given function of the mean curvature conditions for the monotonicity of the scheme are found and the convergence of the approximations to the corresponding viscosity solution is proved we also discuss some aspects of the numerical implementation of such schemes and present several numerical results

threshold dynamics for networks with arbitrary surface tensions we present and study a new algorithm for simulating the n phase mean curvature motion for an arbitrary set of isotropic surface tensions the departure point is the threshold dynamics algorithm of merriman bence and osher for the two phase case a new energetic interpretation of this algorithm allows us to extend it in a natural way to the case of n phases for arbitrary surface tensions and arbitrary isotropic mobilities for a large class of surface tensions the algorithm is shown to be consistent in the sense that at every time step it decreases an energy functional that is an approximation in the sense of gamma convergence of the interfacial energy a broad range of numerical tests shows good convergence properties an important application is the motion of grain boundaries in polycrystalline materials it is also established that the above mentioned large class of surface tensions contains the read shockley surface tensions both in the two dimensional and three dimensional settings 2015 wiley periodicals inc

fast fourier transforms for nonequispaced data a group of algorithms is presented generalizing the fast fourier transform to the case of noninteger frequencies and nonequispaced nodes on the interval pi pi the schemes of this paper are based on a combination of certain analytical considerations with the classical fast fourier transform and generalize both the forward and backward ffts each of the algorithms requires o n cdot log n n cdot log 1 varepsilon arithmetic operations where varepsilon is the precision of computations and n is the number of nodes the efficiency of the approach is illustrated by several numerical examples

set valued quasi metrics and a general ekeland s variational principle in vector optimization we introduce the notion of compatibility between a set valued quasi metric and the original metric by using this notion we prove a general set valued ekeland s variational principle evp where the perturbation contains a set valued quasi metric which is compatible with the original metric here we need not assume that the order cone is w normal by using the unified approach for approximate solutions introduced by gutierrez jimenez and novo we deduce a general version of set valued evp based on c epsilon efficient solutions where c is a coradiant set contained in the order cone by choosing two specific versions of the coradiant set c in the general version of evp we obtain several particular set valued evps for epsilon efficient solutions in the sense of nemeth and of dentcheva and helbig respectively these set valued evps improve and generalize the related interesting results in c gutierrez b jimenez and v novo siam j control optim 47 2008 pp 883 903

on three phase boundary motion and the singular limit of a vector valued ginzburg landau equation we present a formal asymptotic analysis which suggests a model for three phase boundary motion as a singular limit of a vector valued ginzburg landau equation we prove short time existence and uniqueness of solutions for this model that is for a system of three phase boundaries undergoing curvature motion with assigned angle conditions at the meeting point such models pertain to grain boundary motion in alloys the method we use based on linearization about the initial conditions applies to a wide class of parabolic systems we illustrate this further by its application to an eutectic solidification problem

a variational level set approach to multiphase motion a coupled level set method for the motion of multiple junctions of e g solid liquid and grain boundaries which follows the gradient flow for an energy functional consisting of surface tension proportional to length and bulk energies proportional to area is developed the approach combines the level set method of s osher and j a sethian with a theoretical variational formulation of the motion by f reitich and h m soner the resulting method uses as many level set functions as there are regions and the energy functional is evaluated entirely in terms of level set functions the gradient projection method leads to a coupled system of perturbed by curvature terms hamilton jacobi equations the coupling is enforced using a single lagrange multiplier associated with a constraint which essentially prevents a regions from overlapping and b the development of a vacuum the numerical implementation is relatively simple and the results agree with and go beyond the theory as given in 12 other applications of this methodology including the decomposition of a domain into subregions with minimal interface length are discussed finally some new techniques and results in level set methodology are presented

acceleration of quasi monte carlo approximations with applications in mathematical finance a procedure to accelerate the convergence of quasi monte carlo approximations is described and investigated the procedure applies a weighted least squares smoothing method to improve the approximations that were obtained using low discrepancy sequences the acceleration procedure is tested computationally on several numerical integration problems including two problems of interest in mathematical finance the results indicate that when halton low discrepancy points are used in the quasi monte carlo method the accelerated sequence often converges a factor of five or more faster than the original quasi monte carlo approximations

a diffusion generated approach to multiphase motion in this article we present a diffusion generated approach for evolving multiple junctions this work generalizes an earlier method by merriman bence and osher which alternately diffuses and sharpens characteristic functions for each phase region to produce pure mean curvature flow specifically our new method produces a normal velocity which depends on a positive multiple of the curvature of the interface plus the difference in bulk energy densities for prescribed junction angles this simple method naturally treats topological mergings and breakings produces no overlapping regions or vacuums and can be made very fast numerical studies are provided which show that our method agrees with front tracking and a recent variational approach for a variety of examples asymptotic expansions are also carried out near junctions to justify our algorithms