fast sweeping algorithms for a class of hamilton jacobi equations we derive a godunov type numerical flux for the class of strictly convex homogeneous hamiltonians that includes h p q sqrt ap 2 bq 2 2cpq c 2 ab we combine our godunov numerical fluxes with simple gauss seidel type iterations for solving the corresponding hamilton jacobi hj equations the resulting algorithm is fast since it does not require a sorting strategy as found e g in the fast marching method in addition it providesa way to compute solutions to a class of hj equations for which the conventional fast marching method is not applicable our experiments indicate convergence after a few iterations even in rather difficult cases

redistancing by flow of time dependent eikonal equation construction of signed distance to a given interface is a topic of special interest to level set methods there are currently however few algorithms that can efficiently produce highly accurate solutions we introduce an algorithm for constructing an approximate signed distance function through manipulation of values calculated from flow of time dependent eikonal equations we provide operation counts and experimental results to show that this algorithm can efficiently generate solutions with a high order of accuracy comparison with the standard level set reinitialization algorithm shows ours is superior in terms of predictability and local construction which for example are critical in local level set methods we further apply the same ideas to extension of values off interfaces together our proposed approaches can be used to advance the level set method for fast and accurate computations of the latest scientific problems

shape description and matching using integral invariants on eccentricity transformed images matching occluded and noisy shapes is a problem frequently encountered in medical image analysis and more generally in computer vision to keep track of changes inside the breast for example it is important for a computer aided detection system to establish correspondences between regions of interest shape transformations computed both with integral invariants ii and with geodesic distance yield signatures that are invariant to isometric deformations such as bending and articulations integral invariants describe the boundaries of planar shapes however they provide no information about where a particular feature lies on the boundary with regard to the overall shape structure conversely eccentricity transforms ecc can match shapes by signatures of geodesic distance histograms based on information from inside the shape but they ignore the boundary information we describe a method that combines the boundary signature of a shape obtained from ii and structural information from the ecc to yield results that improve on them separately

an cal o n level set method for eikonal equations a propagating interface can develop corners and discontinuities as it advances level set algorithms have been extensively applied for the problems in which the solution has advancing fronts one of the most popular level set algorithms is the so called fast marching method fmm which requires total cal o n log 2n operations where n is the number of grid points the article is concerned with the development of an cal o n level set algorithm called the group marching method gmm the new method is based on the narrow band approach as in the fmm however it is incorporating a correction by iteration strategy to advance a group of grid points at a time rather than sorting the solution in the narrow band to march forward a single grid point after selecting a group of grid points appropriately the gmm advances the group in two iterations for the cost of slightly larger than one iteration numerical results are presented to show the efficiency of the method applied to the eikonal equation in two and three dimensions

high order time marching reinitialization for regional level set functions abstract in this work the time marching reinitialization method is extended to compute the unsigned distance function in multi region systems involving arbitrary number of regions high order and interface preservation are achieved by applying a simple mapping that transforms the regional level set function to the level set function and a high order two step reinitialization method which is a combination of the closest point finding procedure and the hj weno scheme the convergence failure of the closest point finding procedure in three dimensions is addressed by employing a proposed multiple junction treatment and a directional optimization algorithm simple test cases show that our method exhibits 4th order accuracy for reinitializing the regional level set functions and strictly satisfies the interface preserving property the reinitialization results for more complex cases with randomly generated diagrams show the capability our method for arbitrary number of regions n with a computational effort independent of n the proposed method has been applied to dynamic interfaces with different types of flows and the results demonstrate high accuracy and robustness

a parallel two scale method for eikonal equations numerous applications of eikonal equations prompted the development of many efficient numerical algorithms the heap cell method hcm is a recent serial two scale technique that has been shown to have advantages over other serial state of the art solvers for a wide range of problems a chacon and a vladimirsky siam j sci comput 34 2012 pp a547 a578 this paper presents a parallelization of hcm for a shared memory architecture the numerical experiments in r 3 show that the parallel hcm exhibits good algorithmic behavior and scales well resulting in a very fast and practical solver

a viscosity solutions approach to shape from shading the problem of recovering a lambertian surface from a single two dimensional image may be written as a first order nonlinear equation which presents the disadvantage of having several continuous and even smooth solutions a new approach based on hamilton jacobi bellman equations and viscosity solutions theories enables one to study non uniqueness phenomenon and thus to characterize the surface among the various solutions a consistent and monotone scheme approximating the surface is constructed thanks to the dynamic programming principle and numerical results are presented

a fast marching algorithm for the factored eikonal equation the eikonal equation is instrumental in many applications in several fields ranging from computer vision to geoscience this equation can be efficiently solved using the iterative fast sweeping fs methods and the direct fast marching fm methods however when used for a point source the original eikonal equation is known to yield inaccurate numerical solutions because of a singularity at the source in this case the factored eikonal equation is often preferred and is known to yield a more accurate numerical solution one application that requires the solution of the eikonal equation for point sources is travel time tomography this inverse problem may be formulated using the eikonal equation as a forward problem while this problem has been solved using fs in the past the more recent choice for applying it involves fm methods because of the efficiency in which sensitivities can be obtained using them however while several fs methods are available for solving the factored equation the fm method is available only for the original eikonal equation in this paper we develop a fast marching algorithm for the factored eikonal equation using both first and second order finite difference schemes our algorithm follows the same lines as the original fm algorithm and requires the same computational effort in addition we show how to obtain sensitivities using this fm method and apply travel time tomography formulated as an inverse factored eikonal equation numerical results in two and three dimensions show that our algorithm solves the factored eikonal equation efficiently and demonstrate the achieved accuracy for computing the travel time we also demonstrate a recovery of a 2d and 3d heterogeneous medium by travel time tomography using the eikonal equation for forward modeling and inversion by gauss newton

fast marching methods for stationary hamilton jacobi equations with axis aligned anisotropy the fast marching method fmm has proved to be a very efficient algorithm for solving the isotropic eikonal equation because it is a minor modification of dijkstra s algorithm for finding the shortest path through a discrete graph fmm is also easy to implement in this paper we describe a new class of hamilton jacobi hj pdes with axis aligned anisotropy which satisfy a causality condition for standard finite difference schemes on orthogonal grids and can hence be solved using the fmm the only modification required to the algorithm is in the local update equation for a node this class of hj pdes has applications in anelliptic wave propagation and robotic path planning and brief examples are included since our class of hj pdes and grids permit asymmetries we also examine some methods of improving the efficiency of the local update that do not require symmetric grids and pdes finally we include explicit update formulas for variations of the eikonal equation that use the manhattan euclidean and infinity norms on orthogonal grids of arbitrary dimension and with variable node spacing

eikonal equation based front propagation for arbitrary complex configurations this paper presents a front propagation method using the eikonal equation 1 in which represents the smallest euclidean distance field to the front to be propagated the offset capturing approach consists in first calculating the field over a uniform cartesian grid fully covering the front to be propagated and then constructing the iso curves or surfaces as the propagated result the calculation of uses a 3d numerical scheme the fast sweeping scheme validation for accuracy of the method is presented using academic test cases a real 3d industry application draft tube with two piers is successfully propagated and demonstrated using special boundary conditions to cope with inlet and outlet planes during front propagtion this application involves the propagation of a front that exhibits both concave and convex shape regions sharp corners and local tangent plane surface discontinuities as well as a multi connected domain copyright 2007 john wiley sons ltd

semi lagrangian schemes for hamilton jacobi equations discrete representation formulae and godunov methods we study a class of semi lagrangian schemes which can be interpreted as a discrete version of the hopf lax oleinik representation formula for the exact viscosity solution of first order evolutive hamilton jacobi equations that interpretation shows that the scheme is potentially accurate to any prescribed order we discuss how the method can be implemented for convex and coercive hamiltonians with a particular structure and how this method can be coupled with a discrete legendre trasform we also show that in one dimension the first order semi lagrangian scheme coincides with the integration of the godunov scheme for the corresponding conservation laws several test illustrate the main features of semi lagrangian schemes for evolutive hamilton jacobi equations

the use of a legendre pseudospectral viscosity technique to solve a class of nonlinear dynamic hamilton jacobi equations in this research we study the problem of finding the approximate solution of a class of hamilton jacobi equations namely the eikonal equation we employ the legendre pseudospectral viscosity method to solve this problem this method basically consists of adding a spectral viscosity to the equation this spectral viscosity which is sufficiently small to retain the formal spectral accuracy is large enough to stabilize the numerical scheme several test problems are considered and the numerical results are given to show the efficiency of the proposed method

an efficient discontinuous galerkin method on triangular meshes for a pedestrian flow model in this paper we develop a discontinuous galerkin method on triangular meshes to solve the reactive dynamic user equilibrium model for pedestrian flows the pedestrian density in this model is governed by the conservation law in which the flow flux is implicitly dependent on the density through the eikonal equation to solve the eikonal equation efficiently at each time level we use the fast sweeping method two numerical examples are then used to demonstrate the effectiveness of the algorithm copyright 2008 john wiley sons ltd

weibull type limiting distribution for replicative systems the weibull function is widely used to describe skew distributions observed in nature however the origin of this ubiquity is not always obvious to explain in the present paper we consider the well known galton watson branching process describing simple replicative systems the shape of the resulting distribution about which little has been known is found essentially indistinguishable from the weibull form in a wide range of the branching parameter this can be seen from the exact series expansion for the cumulative distribution which takes a universal form we also find that the branching process can be mapped into a process of aggregation of clusters in the branching and aggregation process the number of events considered for branching and aggregation grows cumulatively in time whereas for the binomial distribution an independent event occurs at each time with a given success probability

shape optimization using the cut finite element method we present a cut finite element method for shape optimization in the case of linear elasticity the elastic domain is defined by a level set function and the evolution of the domain is obtained by

rapid and accurate computation of the distance function using grids we present two fast and simple algorithms for approximating the distance function for given isolated points on uniform grids the algorithms are then generalized to compute the distance to piecewise linear objects by incorporating the geometry of huygens principle in the reverse order with the classical viscosity solution theory for the eikonal equation u 1 the algorithms become almost purely algebraic and yield very accurate approximations the generalized closest point formulation used in the second algorithm provides a framework for further extension to compute the distance accurately to smooth geometric objects on different grid geometries without the construction of the voronoi diagrams this method provides a fast and simple translator of data commonly given in computational geometry to the volumetric representation used in level set methods

symmetric fast marching schemes for better numerical isotropy existing fast marching methods solve the eikonal equation using a continuous first order model to estimate the accumulated cost but a discontinuous zero order model for the traveling cost at each grid point as a result the estimate of the accumulated cost calculated numerically at a given point will vary based on the direction of the arriving front introducing an anisotropy into the discrete algorithm even though the continuous partial differential equation pde is itself isotropic to remove this anisotropy we propose two very different schemes in the first model we utilize a continuous interpolation of the traveling cost which is not biased by the direction of the propagating front in the second model we upsample the traveling cost on a higher resolution grid to overcome the directional bias we show the significance of removing the directional bias in the computation of the cost in some applications of the fast marching method demonstrating that both methods make the discrete implementation more isotropic in accordance with the underlying continuous pde

fast sweeping methods for static hamilton jacobi equations we propose a new sweeping algorithm which discretizes the legendre transform of the numerical hamiltonian using an explicit formula this formula yields the numerical solution at a grid point using only its immediate neighboring grid values and is easy to implement numerically the minimization that is related to the legendre transform in our sweeping scheme can either be solved analytically or numerically we illustrate the efficiency and accuracy approach with several numerical examples in two and three dimensions

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

the lax friedrichs sweeping method for optimal control problems in continuous and hybrid dynamics abstract we implement the lax friedrichs sweeping lfs method to approximate the solution to the hamilton jacobi hj equation arising in an infinite horizon optimal control problem the fast sweeping method approximates the viscosity solution of a boundary value problem for hj equations the static hj equation is discretized by using the lax friedrichs numerical hamiltonian and the solution is approximated iteratively by using a gauss seidel like new updating process assuming that the hamiltonian is strictly convex in the control variable we derive a pair of hj equations in continuous time through the dynamic programming formulation the lfs method is applied to solve the coupled pdes by successive iteration of the optimal cost and control the lfs scheme also applies to optimal control problems in hybrid systems in which it admits the viscosity solution for the hj equation we demonstrate the efficiency of the method through some numerical examples in both continuous and hybrid dynamics

lax friedrichs sweeping scheme for static hamilton jacobi equations we propose a simple fast sweeping method based on the lax friedrichs monotone numerical hamiltonian to approximate viscosity solutions of arbitrary static hamilton jacobi equations in any number of spatial dimensions by using the lax friedrichs numerical hamiltonian we can easily obtain the solution at a specific grid point in terms of its neighbors so that a gauss seidel type nonlinear iterative method can be utilized furthermore by incorporating a group wise causality principle into the gauss seidel iteration by following a finite group of characteristics we have an easy to implement sweeping type and fast convergent numerical method however unlike other methods based on the godunov numerical hamiltonian some computational boundary conditions are needed in the implementation we give a simple recipe which enforces a version of discrete min max principle some convergence analysis is done for the one dimensional eikonal equation extensive 2 d and 3 d numerical examples illustrate the efficiency and accuracy of the new approach to our knowledge this is the first fast numerical method based on discretizing the hamilton jacobi equation directly without assuming convexity and or homogeneity of the hamiltonian

a parallel fast sweeping method for the eikonal equation we present an algorithm for solving in parallel the eikonal equation the efficiency of our approach is rooted in the ordering and distribution of the grid points on the available processors we utilize a cuthill mckee ordering the advantages of our approach is that 1 the efficiency does not plateau for a large number of threads we compare our approach to the current state of the art parallel implementation of zhao 2007 14 and 2 the total number of iterations needed for convergence is the same as that of a sequential implementation i e our parallel implementation does not increase the complexity of the underlying sequential algorithm numerical examples are used to illustrate the efficiency of our approach

revisiting the redistancing problem using the hopf lax formula this article presents a fast new numerical method for redistancing objective functions based on the hopf lax formula 1 the algorithm suggested here is a special case of the previous work in 2 and an extension that applies the hopf lax formula for computing the signed distance to the front we propose the split bregman approach to solve the minimization problem as a solution of the eikonal equation obtained from hopf lax formula our redistancing procedure is expected to be generalized and widely applied to many fields such as computational fluid dynamics the minimal surface problem and elsewhere

two approximations of solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games the associated initial value problems almost never have global time classical solutions and one must deal with suitable generalized solutions the correct class of generalized solutions has only recently been established by the authors this article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates analogous results are proved by similar means for the method of vanishing viscosity author

semi implicit second order schemes for numerical solution of level set advection equation on cartesian grids abstract a new parametric class of semi implicit numerical schemes for a level set advection equation on cartesian grids is derived and analyzed an accuracy and a stability study is provided for a linear advection equation with a variable velocity using partial lax wendroff procedure and numerical von neumann stability analysis the obtained semi implicit scheme is 2nd order accurate in space and time in any dimensional case when using a dimension by dimension extension of the one dimensional scheme that is not the case for analogous fully explicit or fully implicit schemes a further improvement is obtained by using so called corner transport upwind extension in two dimensional case the extended semi implicit scheme with a specific velocity dependent value of is 3rd order accurate in space and time for a constant advection velocity and it is unconditional stable according to the numerical von neumann stability analysis for the linear advection equation in general

a multiresolution stochastic level set method for mumford shah image segmentation the mumford shah model is one of the most successful image segmentation models however existing algorithms for the model are often very sensitive to the choice of the initial guess to make use of the model effectively it is essential to develop an algorithm which can compute a global or near global optimal solution efficiently while gradient descent based methods are well known to find a local minimum only even many stochastic methods do not provide a practical solution to this problem either in this paper we consider the computation of a global minimum of the multiphase piecewise constant mumford shah model we propose a hybrid approach which combines gradient based and stochastic optimization methods to resolve the problem of sensitivity to the initial guess at the heart of our algorithm is a well designed basin hopping scheme which uses global updates to escape from local traps in a way that is much more effective than standard stochastic methods in our experiments a very high quality solution is obtained within a few stochastic hops whereas the solutions obtained with simulated annealing are incomparable even after thousands of steps we also propose a multiresolution approach to reduce the computational cost and enhance the search for a global minimum furthermore we derived a simple but useful theoretical result relating solutions at different spatial resolutions

variational curve skeletons using gradient vector flow representing a 3d shape by a set of 1d curves that are locally symmetric with respect to its boundary i e curve skeletons is of importance in several machine intelligence tasks this paper presents a fast automatic and robust variational framework for computing continuous subvoxel accurate curve skeletons from volumetric objects a reference point inside the object is considered a point source that transmits two wave fronts of different energies the first front beta front converts the object into a graph from which the object salient topological nodes are determined curve skeletons are tracked from these nodes along the cost field constructed by the second front alpha front until the point source is reached the accuracy and robustness of the proposed work are validated against competing techniques as well as a database of 3d objects unlike other state of the art techniques the proposed framework is highly robust because it avoids locating and classifying skeletal junction nodes employs a new energy that does not form medial surfaces and finally extracts curve skeletons that correspond to the most prominent parts of the shape and hence are less sensitive to noise

approximations for viscosity solutions of hamilton jacobi equations with locally varying time and space grids a new monotone finite difference scheme is introduced that approximates viscosity solutions of first order nonlinear hamilton jacobi equations the main feature of the scheme is that it allows for locally varying time and space grids making it ideal for use with adaptive algorithms explicit a priori error estimates are given to establish convergence numerical examples including adaptive mesh refinement examples demonstrate the effectiveness of the proposed scheme

nonlocal pdes based morphology on weighted graphs for image and data processing mathematical morphology mm offers a wide range of operators to address various image processing problems these operators can be defined in terms of algebraic discrete sets or as partial differential equations pdes in this paper we introduce a nonlocal pdes based morphological framework defined on weighted graphs we present and analyze a set of operators that leads to a family of discretized morphological pdes on weighted graphs our formulation introduces nonlocal patch based configurations for image processing and extends pdes based approach to the processing of arbitrary data such as nonuniform high dimensional data finally we show the potentialities of our methodology in order to process segment and classify images and arbitrary data

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

solving partial differential equations on manifolds from incomplete interpoint distance solutions of partial differential equations pdes on manifolds have provided important applications in different fields in science and engineering existing methods are mainly based on discretization of manifolds as implicit functions triangle meshes or point clouds where the manifold structure is approximated by either zero level set of an implicit function or a set of points in many applications manifolds might be only provided as an interpoint distance matrix with possible missing values this paper discusses a framework to discretize pdes on manifolds represented as incomplete interpoint distance information without conducting a time consuming global coordinates reconstruction we propose a more efficient strategy by discretizing differential operators only based on pointwisely local reconstruction our local reconstruction model is based on the recent advances of low rank matrix completion theory where only a very small random portion of distance information is required this method enables u

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

wavelet frame based surface reconstruction from unorganized points applications of wavelet frames to image restoration problems e g image deblurring and inpainting have been successful due to their redundancy and capability of sparsely approximating piecewise smooth functions like images see e g 13 14 8 however wavelet frames have not yet been used for surface reconstruction problems recently in 6 connections between one of the wavelet frame based image restoration model 8 28 48 and variational models e g the rof model 45 were rigorously established such connections not only grant new insights to wavelet frame based image restorations it also case a geometric explanation to wavelet frame based approaches this leads us to a wavelet frame based model as well as a fast algorithm to reconstruct implicit surfaces from unorganized point sets in r 3 we will demonstrate the effectiveness of the proposed model using several commonly used examples

a self adaptive oriented particles level set method for tracking interfaces a new method for tracking evolving interfaces by lagrangian particles in conjunction with a level set approach is introduced this numerical technique is based on the use of time evolution equations for fundamental vector and tensor quantities defined on the front and represents a new and convenient way to couple the advantages of the eulerian description given by a level set function f to the use of lagrangian massless particles the term oriented points out that the information advected by the particles not only concern the spatial location but also the local outward normal vector n to the interface c and the second fundamental tensor the shape operator n the particles are exactly located upon c and provide all the requested information for tracking the interface on their own in addition a self adaptive mechanism suitably modifies at each time step the markers distribution in the numerical domain each particle behaves both as a potential seeder of new markers on c so as to guarantee an accurate reconstruction of the interface and a de seeder to avoid any useless gathering of markers and to limit the computational effort the algorithm is conceived to avoid any transport equation for f and to confine the level set function to the role of a mere post processing tool thus all the numerical diffusion problems usually affecting the level set methodology are removed the method has been tested both on 2d and 3d configurations it carries out a fast reconstruction of the interface and its accuracy is only limited by the spatial resolution of the mesh

local or global minima flexible dual front active contours most variational active contour models are designed to find local minima of data dependent energy functionals with the hope that reasonable initial placement of the active contour will drive it toward a desirable local minimum as opposed to an undesirable configuration due to noise or complex image structure as such there has been much research into the design of complex region based energy functionals that are less likely to yield undesirable local minima when compared to simpler edge based energy functionals whose sensitivity to noise and texture is significantly worse unfortunately most of these more robust region based energy functionals are applicable to a much narrower class of imagery compared to typical edge based energies due to stronger global assumptions about the underlying image data devising new implementation algorithms for active contours that attempt to capture more global minimizers of already proposed image based energies would allow us to choose an energy that makes sense for a particular class of energy without concern over its sensitivity to local minima such implementations have been proposed for capturing global minima however sometimes the completely global minimum is just as undesirable as a minimum that is too local in this paper we propose a novel fast and flexible dual front implementation of active contours motivated by minimal path techniques and utilizing fast sweeping algorithms which is easily manipulated to yield minima with variable degrees of localness and globalness by simply adjusting the size of active regions the ability to gracefully move from capturing minima that are more local according to the initial placement of the active contour surface to minima that are more global allows this model to more easily obtain desirable minimizers which often are neither the most local nor the most global experiments on various 2d and 3d images and comparisons with some active contour models and region growing methods are also given to illustrate the properties of this model and its performance in a variety of segmentation applications

a dynamic programming approach to first arrival traveltime computation in media with arbitrarily distributed velocities curved ray tomographic traveltime inversion reverse time migration and various other seismic modeling applications require the calculation of traveltime and raypath information throughout a two or three dimensional medium when arbitrary velocity distributions and curved rays are involved traditional ray shooting or bending procedures can be time consuming and error prone a two dimensional dynamic programming traveltime computation technique based upon fermat s principle uses simple calculus techniques and a systematic mapping scheme to determine first arrival times on a uniform grid given an arbitrary discrete velocity distribution it accurately handles large contrast discontinuous velocity distributions including those that generate caustics first arrival seismic energy can travel either as transmitted waves diffracted waves or headwaves and this technique models all types the traveltime computations begin with starting values computed near the source location then the mapping systemat

stabilized finite element methods for solving the level set equation without reinitialization new stabilized finite element methods are proposed for solving moving interface flow problems using the level set approach the formulations enhance the interface resolution without the need to resort to the reinitialization process these are established by adding a perturbation term that depends on the local residual of the eikonal equation to the supg variational formulation of the level set equation these methods are numerically evaluated for well known benchmark flow problems and compared with a modified variant of the penalty method of li et al 2005 the proposed stabilized finite element methods employing second order time and space approximations are promising simple and accurate techniques for solving complex moving interface flows

a fast sweeping method for eikonal equations in this paper a fast sweeping method for computing the numerical solution of eikonal equations on a rectangular grid is presented the method is an iterative method which uses upwind difference for discretization and uses gauss seidel iterations with alternating sweeping ordering to solve the discretized system the crucial idea is that each sweeping ordering follows a family of characteristics of the corresponding eikonal equation in a certain direction simultaneously the method has an optimal complexity of o n for n grid points and is extremely simple to implement in any number of dimensions monotonicity and stability properties of the fast sweeping algorithm are proven convergence and error estimates of the algorithm for computing the distance function is studied in detail it is shown that 2 n gauss seidel iterations is enough for the distance function in n dimensions an estimation of the number of iterations for general eikonal equations is also studied numerical examples are used to verify the analysis

weighted distance maps computation on parametric three dimensional manifolds we propose an efficient computational solver for eikonal equations on parametric three dimensional manifolds our approach is based on the fast marching method for solving the eikonal equation in o nlogn steps on n grid points by numerically simulating wavefront propagation the obtuse angle splitting problem is reformulated as a set of small integer linear programs that can be solved in o n numerical simulations demonstrate the accuracy of the proposed algorithm

a patchy dynamic programming scheme for a class of hamilton jacobi bellman equations in this paper we present a new algorithm for the solution of hamilton jacobi bellman equations related to optimal control problems the key idea is to divide the domain of computation into subdomains which are shaped by the optimal dynamics of the underlying control problem this can result in a rather complex geometrical subdivision but it has the advantage that every subdomain is invariant with respect to the optimal dynamics and then the solution can be computed independently in each subdomain the features of this dynamics dependent domain decomposition can be exploited to speed up the computation and for an efficient parallelization since the classical transmission conditions at the boundaries of the subdomains can be avoided for their properties the subdomains are patches in the sense introduced by ancona and bressan esaim control optim calc var 4 1999 pp 445 471 several examples in two and three dimensions illustrate the properties of the new method

optimal control with budget constraints and resets we consider both discrete and continuous control problems constrained by a fixed budget of some resource which may be renewed upon entering a preferred subset of the state space in the discrete case we consider deterministic shortest path problems on graphs with a full budget reset in all preferred nodes in the continuous case we derive augmented pdes of optimal control which are then solved numerically on the extended state space with a full instantaneous budget reset on the preferred subset we introduce an iterative algorithm for solving these problems efficiently the method s performance is demonstrated on a range of computational examples including optimal path planning with constraints on prolonged visibility by a static enemy observer

a fast marching like algorithm for geometrical shock dynamics we develop a new algorithm for the computation of the geometrical shock dynamics gsd model the method relies on the fast marching paradigm and enables the discrete evaluation of the first arrival time of a shock wave and its local velocity on a cartesian grid the proposed algorithm is based on a first order upwind finite difference scheme and reduces to a local nonlinear system of two equations solved by an iterative procedure reference solutions are built for a smooth radial configuration and for the 2d riemann problem the link between the gsd model and p systems is given numerical experiments demonstrate the efficiency of the scheme and its ability to handle singularities

fast and accurate redistancing by directional optimization a fast and accurate algorithm for the reinitialization of the signed distance function in two and three spatial dimensions is presented the algorithm has computational complexity o n log n for the reinitialization of n grid points the order of accuracy of the reinitialization is demonstrated to depend primarily on the interpolation algorithm used bicubic interpolation is demonstrated to result in fourth order accuracy for smooth interfaces simple numerical examples demonstrating the convergence and computational complexity of the reinitialization algorithm in two and three dimensions are presented as verification of the algorithm

computational study of fast methods for the eikonal equation a computational study of the fast marching and the fast sweeping methods for the eikonal equation is given it is stressed that both algorithms should be considered as direct as opposed to iterative methods on realistic grids fast sweeping is faster than fast marching for problems with simple geometry for strongly nonuniform problems and or complex geometry the situation may be reversed finally fully second order generalizations of methods of this type for problems with obstacles are proposed and implemented

fast sweeping methods for eikonal equations on triangular meshes the original fast sweeping method which is an efficient iterative method for stationary hamilton jacobi equations relies on natural ordering provided by a rectangular mesh we propose novel ordering strategies so that the fast sweeping method can be extended efficiently and easily to any unstructured mesh to that end we introduce multiple reference points and order all the nodes according to their l p metrics to those reference points we show that these orderings satisfy the two most important properties underlying the fast sweeping method 1 these orderings can cover all directions of information propagating efficiently 2 any characteristic can be decomposed into a finite number of pieces and each piece can be covered by one of the orderings we prove the convergence of the new algorithm the computational complexity of the algorithm is nearly optimal in the sense that the total computational cost consists of o m flops for iteration steps and o m rm log m flops for sorting at the predetermined initialization step which can be efficiently optimized by adopting a linear time sorting method where m is the total number of mesh points extensive numerical examples demonstrate that the new algorithm converges in a finite number of iterations independent of mesh size

a second order discontinuous galerkin fast sweeping method for eikonal equations in this paper we construct a second order fast sweeping method with a discontinuous galerkin dg local solver for computing viscosity solutions of a class of static hamilton jacobi equations namely the eikonal equations our piecewise linear dg local solver is built on a dg method developed recently y cheng c w shu a discontinuous galerkin finite element method for directly solving the hamilton jacobi equations journal of computational physics 223 2007 398 415 for the time dependent hamilton jacobi equations the causality property of eikonal equations is incorporated into the design of this solver the resulting local nonlinear system in the gauss seidel iterations is a simple quadratic system and can be solved explicitly the compactness of the dg method and the fast sweeping strategy lead to fast convergence of the new scheme for eikonal equations extensive numerical examples verify efficiency convergence and second order accuracy of the proposed method

discrete time high order schemes for viscosity solutions of hamilton jacobi bellman equations a general method for constructing high order approximation schemes for hamilton jacobi bellman equations is given the method is based on a discrete version of the dynamic programming principle we prove a general convergence result for this class of approximation schemes also obtaining under more restrictive assumptions an estimate in l infty of the order of convergence and of the local truncation error the schemes can be applied in particular to the stationary linear first order equation in bbb r n we present several examples of schemes belonging to this class and with fast convergence to the solution

numerical schemes and rates of convergence for the hamilton jacobi equation continuum limit of nondominated sorting non dominated sorting arranges a set of points in n dimensional euclidean space into layers by repeatedly removing the coordinatewise minimal elements it was recently shown that nondominated sorting of random points has a hamilton jacobi equation continuum limit the obvious numerical scheme for this pde has a slow convergence rate of o h frac 1 n o h1n in this paper we introduce two new numerical schemes that have formal rates of o h and we prove the usual o sqrt h o h theoretical rates we also present the results of numerical simulations illustrating the difference between the formal and theoretical rates

parallel re initialization of level set functions on distributed unstructured tetrahedral grids level set functions are employed to track interfaces in various application areas including simulation of two phase flows and image segmentation often a re initializing algorithm is incorporated to transform a numerically instable level set function to a signed distance function in this note we present a parallel algorithm for re initializing level set functions on unstructured three dimensional tetrahedral grids the main idea behind this new domain decomposition approach is to combine a parallel brute force re initializing algorithm with an efficient way to compute distances between the interface and grid points time complexity and error analysis of the algorithm are investigated detailed numerical experiments demonstrate the accuracy and scalability on up to 128 processes

a variational approach to path planning in three dimensions using level set methods in this paper we extend the two dimensional methods set forth in t cecil d marthaler a variational approach to search and path planning using level set methods ucla cam report 04 61 2004 proposing a variational approach to a path planning problem in three dimensions using a level set framework after defining an energy integral over the path we use gradient flow on the defined energy and evolve the entire path until a locally optimal steady state is reached we follow the framework for motion of curves in three dimensions set forth in p burchard l t cheng b merriman s osher motion of curves in three spatial dimensions using a level set approach j comput phys 170 2 2001 720 741 modified appropriately to take into account that we allow for paths with positive varying widths applications of this method extend to robotic motion and visibility problems for example numerical methods and algorithms are given and examples are presented

simplex free adaptive tree fast sweeping and evolution methods for solving level set equations in arbitrary dimension we introduce simplex free adaptive tree numerical methods for solving static and time dependent hamilton jacobi equations arising in level set problems in arbitrary dimension the data structure upon which our method is built in a generalized n dimensional binary tree but it does not require the complicated splitting of cubes into simplices aka generalized n dimensional triangles or hypertetrahedrons that current tree based methods require it has enough simplicity that minor variants of standard numerical hamiltonians developed for uniform grids can be applied yielding consistent monotone convergent schemes combined with the fast sweeping strategy the resulting tree based methods are highly efficient and accurate thus without changing more than a few lines of code when changing dimension we have obtained results for calculations in up to n 7 dimensions

a high order and interface preserving discontinuous galerkin method for level set reinitialization abstract a high order numerical method for interface preserving level set reinitialization is presented in this paper in the interface cells the gradient of the level set function is determined by a weighted local projection scheme and the missing additive constant is determined such that the position of the zero level set is preserved in the non interface cells we compute the gradient of the level set function by solving a hamilton jacobi equation as a conservation law system using the discontinuous galerkin method following the work by hu and shu siam j sci comput 21 1999 660 690 the missing constant is then recovered by the continuity of the level set function while taking into account the characteristics to handle highly distorted initial conditions we develop a hybrid numerical flux that combines the lax friedrichs flux and the penalty flux our method is stable for non trivial test cases and handles singularities away from the interface very well when derivative singularities are present on the interface a second derivative limiter is designed to suppress the oscillations at least n 1 th order accuracy in the interface cells and nth order in the whole domain are observed for smooth solutions when nth degree polynomials are used two dimensional test cases are presented to demonstrate superior properties such as accuracy long term stability interface preserving capability and easy treatment of contact lines we also show some preliminary results on the pinch off process of a pendant drop where topological changes of the fluid interface are involved our method is readily extendable to three dimensions and adaptive meshes

vials an eulerian tool based on total variation and the level set method for studying dynamical systems abstract we propose a new eulerian tool to study complicated dynamical systems based on the average growth in the surface area of a family of level surfaces represented implicitly by a level set function since this proposed quantity determines the temporal variation of the averaged surface area of all level surfaces we name the quantity the variation of the integral over area of level surfaces vials numerically all these infinitely many level surfaces are advected according to the given dynamics by solving one single linear advection equation to develop a computationally efficient approach we apply the coarea formula and rewrite the surface area integral as a simple integral relating the total variation tv of the level set function the proposed method can be easily incorporated with a recent eulerian algorithm for efficient computation of flow maps to speed up our approach we will also prove that the proposed vials is closely related to the computation of the finite time lyapunov exponent ftle in the lagrangian coherent structure lcs extraction this connects our proposed eulerian approach to widely used lagrangian techniques for understanding complicated dynamical systems

a mean field game of evacuation in multilevel building this paper puts forward a simple mean field game that captures some of the key dynamic features of crowd and pedestrian flows in multilevel building evacuations it considers both microscopic and macroscopic route choice by strategic agents to achieve this we use mean field differential game with local congestion measure based on the location of the agent in the building including the local mean field term and its evolution along the path causes a sort of dispersion of the flow the agents will try to avoid high density areas in order to reduce their overall walking costs and queuing costs at the stairs and exits each agent state is represented by a center of a box that follows a simple first order dynamical system in an euclidean space each agent will move to one of the closest exits that is safer and with less congested path first we formulate the problem and derive optimality equations using maximum principle and dynamic programming with boundary conditions second well posedness and existence results are provided numerics and simulations are carried out to illustrate mean field equilibria of a safer evacuation process finally the methodology is shown to be flexible enough to include movement noises and stochastic structural component of the building

an adaptive domain decomposition technique for parallelization of the fast marching method abstract the fast marching method fmm is an efficient technique to solve numerically the eikonal equation the parallelization of the fmm is not easy because of its intrinsic sequential nature in this paper we propose a novel approach to parallelize the fmm it leads to an equation dependent domain decomposition and it turns out to be particularly suitable for machines with two or four cores that are in common use today compared to other techniques in the field the proposed method is much simpler to implement and it gives a slightly better computational speed up in order to test the new method on a real world application we solve the shape from shading problem based on a hamilton jacobi equation on a standard four core machine the method confirms the good properties it shows a reasonable speedup factor of about 2 5 and it reveals its potential to good performance if the arithmetic density of the problem is high

efficient algorithms for globally optimal trajectories we present serial and parallel algorithms for solving a system of equations that arises from the discretization of the hamilton jacobi equation associated to a trajectory optimization problem of the following type a vehicle starts at a prespecified point x sub o and follows a unit speed trajectory x t inside a region in spl rscr sup m until an unspecified time t that the region is exited a trajectory minimizing a cost function of the form spl int sub 0 sup t r x t dt q x t is sought the discretized hamilton jacobi equation corresponding to this problem is usually solved using iterative methods nevertheless assuming that the function r is positive we are able to exploit the problem structure and develop one pass algorithms for the discretized problem the first algorithm resembles dijkstra s shortest path algorithm and runs in time o n log n where n is the number of grid points the second algorithm uses a somewhat different discretization and borrows some ideas from a variation of dial s shortest path algorithm 1969 that we develop here it runs in time o n which is the best possible under some fairly mild assumptions finally we show that the latter algorithm can be efficiently parallelized for two dimensional problems and with p processors its running time becomes o n p provided that p o spl radic n log n

uniformly accurate discontinuous galerkin fast sweeping methods for eikonal equations in f li c w shu y t zhang h zhao j comput phys 227 2008 pp 8191 8208 we developed a fast sweeping method based on a hybrid local solver which is a combination of a discontinuous galerkin dg finite element solver and a first order finite difference solver for eikonal equations the method has second order accuracy in the l 1 norm and a very fast convergence speed but only first order accuracy in the l infty norm for the general cases this is an obstacle to the design of higher order dg fast sweeping methods in this paper we overcome this problem by developing uniformly accurate dg fast sweeping methods for solving eikonal equations we design novel causality indicators which guide the information flow directions for the dg local solver the values of these indicators are initially provided by the first order finite difference fast sweeping method and they are updated during iterations along with the solution we observe both a uniform second order accuracy in the l infty norm in smooth regions and the fast convergence speed linear computational complexity in the numerical examples

a level set based variational principal flow method for nonparametric dimension reduction on riemannian manifolds we propose a variational formulation for dimension reduction on riemannian manifolds the algorithm is developed based on the level set method together with a recently developed principal flow algorithm the original principal flow algorithm is a lagrangian technique which extends the principal component analysis pca to dimension reduction on riemannian manifolds we propose to incorporate the level set method to obtain a fully implicit formulation so that the overall algorithm can naturally handle various topological changes in the curve evolution the variational formulation consists of two terms which try to balance the contributions from both the dataset itself and the principal direction by the pca we will demonstrate that the method is insensitive to the initial guess and is robust enough for noisy data

properties of the single bit single user mmse receiver for ds cdma systems the single bit single user minimum mean squared error mmse receiver for direct sequence code division multiple access systems is studied in great detail it is shown that the mmse receiver is near far resistant and asymptotically optimum as the interference or noise disappear i e in interference or noise dominated scenarios it is experimentally shown that the popular gaussian approximation of the error probability is reasonable for the mmse receiver over a broad range of operating conditions i e signal to noise ratios and interference situations even though this may not be the case for the standard matched filter receiver the concept of dimension reduction is introduced to reduce the complexity of the adaptive part of the receiver and the optimum dimension reduction transform is derived unfortunately the optimum transform is dependent on unknown side information and is therefore of limited practical use however several practical schemes are proposed one which is asymptotically optimum the schemes are shown to outperform previously known methods

newton type gauss seidel lax friedrichs high order fast sweeping methods for solving generalized eikonal equations at large scale discretization abstract we propose a newton type gauss seidel lax friedrichs sweeping method to solve the generalized eikonal equation arising from wave propagation in a moving fluid the lax friedrichs numerical hamiltonian is used in discretization of the generalized eikonal equation different from traditional lax friedrichs sweeping algorithms we design a novel approach with a line wise sweeping strategy in the local solver the values of traveltime on an entire line are updated simultaneously by newton s method the global solution is then obtained by gauss seidel iterations with line wise sweepings we first develop the newton based first order scheme and on top of that we further develop high order schemes by applying weighted essentially non oscillatory weno approximations to derivatives extensive 2 d and 3 d numerical examples demonstrate the efficiency and accuracy of the new algorithm the combination of newton s method and gauss seidel iterations improves upon the convergence speed of the original lax friedrichs sweeping algorithm in addition the newton type sweeping method manipulates data in a vectorized manner so that it can be efficiently implemented in modern programming languages that feature array programming and the resulting advantages are extremely significant for large scale 3 d computations

factored singularities and high order lax friedrichs sweeping schemes for point source traveltimes and amplitudes in the high frequency regime the geometrical optics approximation for the helmholtz equation with a point source results in an eikonal equation for traveltime and a transport equation for amplitude because the point source traveltime field has an upwind singularity at the source point all formally high order finite difference eikonal solvers exhibit first order convergence and relatively large errors in this paper we propose to first factor out the singularities of traveltimes takeoff angles and amplitudes and then we design high order lax friedrichs sweeping schemes for point source traveltimes takeoff angles and amplitudes numerical examples are presented to demonstrate the performance of our new method

the wave front tracking algorithm for hughes model of pedestrian motion we code the wave front tracking algorithm for hughes model of pedestrian motion with generalized running cost this model displays a nonclassical dynamic at the splitting point between the two directions of motion the wave front tracking scheme provides us with reference solutions for testing numerically the convergence of classical finite volume schemes which do not treat explicitly the dynamics at the turning point

fast sweeping method for the factored eikonal equation we develop a fast sweeping method for the factored eikonal equation by decomposing the solution of a general eikonal equation as the product of two factors the first factor is the solution to a simple eikonal equation such as distance or a previously computed solution to an approximate eikonal equation the second factor is a necessary modification correction appropriate discretization and a fast sweeping strategy are designed for the equation of the correction part the key idea is to enforce the causality of the original eikonal equation during the gauss seidel iterations using extensive numerical examples we demonstrate that 1 the convergence behavior of the fast sweeping method for the factored eikonal equation is the same as for the original eikonal equation i e the number of iterations for the gauss seidel iterations is independent of the mesh size 2 the numerical solution from the factored eikonal equation is more accurate than the numerical solution directly computed from the original eikonal equation especially for point sources

can local single pass methods solve any stationary hamilton jacobi bellman equation the use of local single pass methods like e g the fast marching method has become popular in the solution of some hamilton jacobi equations the prototype of these equations is the eikonal equation for which the methods can be applied saving cpu time and possibly memory allocation then some questions naturally arise can local single pass methods solve any hamilton jacobi equation if not where should the limit be set this paper tries to answer these questions in order to give a complete picture we present an overview of some fast methods available in the literature and briefly analyze their main features we also introduce some numerical tools and provide several numerical tests which are intended to exhibit the limitations of the methods we show that the construction of a local single pass method for general hamilton jacobi equations is very hard if not impossible nevertheless some special classes of problems can actually be solved making local single pass methods very useful from a p

an eikonal based formulation for traveltime perturbation with respect to the source location traveltime calculations amount to solving the nonlinear eikonal equation for a given source location the relationship between the eikonal solution and its perturbations is analyzed with respect to the source location and a partial differential equation is developed that relates the traveltime field for one source location to that for a nearby source this linear first order equation in one form depends on lateral changes in velocity and in another form is independent of the velocity field and relies on second order derivatives of the original traveltime field for stable finite difference calculations this requires the velocity field to be smooth in a sense similar to ray tracing requirements our formulation for traveltime perturbation has several potential applications such that as traveltime calculation by source location perturbation velocity independent interpolation including datuming and velocity estimation additionally higher order expansions provide parameters necessary for gaussian beam computations

modified gradient search for level set based image segmentation level set methods are a popular way to solve the image segmentation problem the solution contour is found by solving an optimization problem where a cost functional is minimized gradient descent methods are often used to solve this optimization problem since they are very easy to implement and applicable to general nonconvex functionals they are however sensitive to local minima and often display slow convergence traditionally cost functionals have been modified to avoid these problems in this paper we instead propose using two modified gradient descent methods one using a momentum term and one based on resilient propagation these methods are commonly used in the machine learning community in a series of 2 d 3 d experiments using real and synthetic data with ground truth the modifications are shown to reduce the sensitivity for local optima and to increase the convergence rate the parameter sensitivity is also investigated the proposed methods are very simple modifications of the basic method and are directly compatible with any type of level set implementation downloadable reference code with examples is available online

markov chain approximations for deterministic control problems with affine dynamics and quadratic cost in the control we consider the construction of markov chain approximations for an important class of deterministic control problems the emphasis is on the construction of schemes that can be easily implemented and which possess a number of highly desirable qualitative properties the class of problems covered is that for which the control is affine in the dynamics and with quadratic running cost this class covers a number of interesting areas of application including problems that arise in large deviations risk sensitive and robust control robust filtering and certain problems in computer vision examples are given as well as a proof of convergence

multistencils fast marching methods a highly accurate solution to the eikonal equation on cartesian domains a wide range of computer vision applications require an accurate solution of a particular hamilton jacobi hj equation known as the eikonal equation in this paper we propose an improved version of the fast marching method fmm that is highly accurate for both 2d and 3d cartesian domains the new method is called multistencils fast marching msfm which computes the solution at each grid point by solving the eikonal equation along several stencils and then picks the solution that satisfies the upwind condition the stencils are centered at each grid point and cover its entire nearest neighbors in 2d space two stencils cover 8 neighbors of the point whereas in 3d space six stencils cover its 26 neighbors for those stencils that are not aligned with the natural coordinate system the eikonal equation is derived using directional derivatives and then solved using higher order finite difference schemes the accuracy of the proposed method over the state of the art fmm based techniques has been demonstrated through comprehensive numerical experiments

a uniformly second order fast sweeping method for eikonal equations a uniformly second order method with a local solver based on the piecewise linear discontinuous galerkin formulation is introduced to solve the eikonal equation with dirichlet boundary conditions the method utilizes an interesting phenomenon referred as the superconvergence phenomenon that the numerical solution of monotone upwind schemes for the eikonal equation is first order accurate on both its value and gradient when the solution is smooth this phenomenon greatly simplifies the local solver based on the discontinuous galerkin formulation by reducing its local degrees of freedom from two 1 d or three 2 d or four 3 d to one with the information of the gradient frozen when considering the eikonal equation with point source conditions we further utilize a factorization approach to resolve the source singularities of the eikonal by decomposing it into two parts either multiplicatively or additively one part is known and captures the source singularities the other part serves as a correction term that is differentiable at the sources and satisfies the factored eikonal equations we extend the second order method to solve the factored eikonal equations to compute the correction term with second order accuracy then recover the eikonal with second order accuracy numerical examples are presented to demonstrate the performance of the method

fast huygens sweeping methods for multiarrival green s functions of helmholtz equations in the high frequency regime abstractmultiarrival green s functions are essential in seismic modeling migration and inversion huygens kirchhoff hk integrals provide a bridge to integrate locally valid first arrival green s functions into a globally valid multiarrival green s function we have designed robust and accurate finite difference methods to compute first arrival traveltimes and amplitudes so that first arrival green s functions can be constructed rapidly we adapted a fast butterfly algorithm to evaluate discretized hk integrals the resulting fast huygens sweeping method has the following unique features 1 it precomputes a set of local traveltime and amplitude tables 2 it automatically takes care of caustics 3 it constructs green s functions of the helmholtz equation for arbitrary frequencies and for many point sources and 4 for a fixed number of points per wavelength it constructs each green s function in nearly optimal complexity o n log n in terms of the total number of mesh points n where the prefa

kirchhoff migration using eikonal based computation of traveltime source derivatives abstractthe computational efficiency of kirchhoff type migration can be enhanced by using accurate traveltime interpolation algorithms we addressed the problem of interpolating between a sparse source sampling by using the derivative of traveltime with respect to the source location we adopted a first order partial differential equation that originates from differentiating the eikonal equation to compute the traveltime source derivatives efficiently and conveniently unlike methods that rely on finite difference estimations the accuracy of the eikonal based derivative did not depend on input source sampling for smooth velocity models the first order traveltime source derivatives enabled a cubic hermite traveltime interpolation that took into consideration the curvatures of local wavefronts and can be straightforwardly incorporated into kirchhoff antialiasing schemes we provided an implementation of the proposed method to first arrival traveltimes by modifying the fast marching eikonal solver severa

an oriented particle level set method based on surface coordinates in this work we present a new method for tracking evolving interfaces like the adaptive oriented particle level set method aopls ianniello and mascio 2010 it uses a combination of unconnected particles containing geometric information and a level set function to track the interface in the present method the geometric information solved for in each particle is a set of tangent vectors and the corresponding curvature tensor of the interface compared to the aopls method the amount of geometric data stored in each particle is roughly halved while still retaining a description of the interface that includes both orientation and curvature a mechanism for ensuring a sufficiently dense distribution of particles on the interface is also included from the information stored in each particle a continuous representation of the interface is obtained by transferring the information to a level set function stored on an eulerian grid unlike the aopls method the present method takes the curvature into account when the zero level set is constructed this results in significant improvements in regions of high curvature two different formulations of the new method have been explored in the first the tangent vectors deform purely according to the velocity field in the second the tangent vectors are forced to coincide with the principal directions of the surface both formulations are shown to be applicable