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

an approximation scheme for the minimum time function this paper presents an approximation scheme for the nonlinear minimum time problem with compact target the scheme is derived from a discrete dynamic programming principle and the main convergence result is obtained by applying techniques related to discontinuous viscosity solutions for hamilton jacobi equations the convergence is proved under general controllability assumptions on both the continuous time and the discrete time systems an explicit sufficient condition on the system and the target ensuring the desired controllability is given this condition is shown to be necessary and sufficient for the lipschitz continuity of the minimum time function if the target is smooth an extension to the case of a point shaped target is given

fast computation of weighted distance functions and geodesics on implicit hyper surfaces 730 an algorithm for the computationally optimal construction of intrinsic weighted distance functions on implicit hyper surfaces is introduced in this paper the basic idea is to approximate the intrinsic weighted distance by the euclidean weighted distance computed in a band surrounding the implicit hyper surface in the embedding space thereby performing all the computations in a cartesian grid with classical and efficient numerics based on work on geodesics on riemannian manifolds with boundaries we bound the error between the two distance functions we show that this error is of the same order as the theoretical numerical error in computationally optimal hamilton jacobi based algorithms for computing distance functions in cartesian grids therefore we can use these algorithms modified to deal with spaces with boundaries and obtain also for the case of intrinsic distance functions on implicit hyper surfaces a computationally efficient technique the approach can be extended to solve a more general class of hamilton jacobi equations defined on the implicit surface following the same idea of approximating their solutions by the solutions in the embedding euclidean space the framework here introduced thereby allows for the computations to be performed on a cartesian grid with computationally optimal algorithms in spite of the fact that the distance and hamilton jacobi equations are intrinsic to the implicit hyper surface c 2001 academic press

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

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

fast data extrapolating data extrapolating extension technique has important applications in image processing on implicit surfaces and in level set methods the existing data extrapolating techniques are inefficient because they are designed without concerning the specialities of the extrapolating equations besides there exists little work on locating the narrow band after data extrapolating a very important problem in narrow band level set methods in this paper we put forward the general huygens principle and based on the principle we present two efficient data extrapolating algorithms the algorithms can easily locate the narrow band in data extrapolating furthermore we propose a prediction correction version for the data extrapolating algorithms and the corresponding band locating method for a special case where the direct band locating method is hard to apply experiments demonstrate the efficiency of our algorithms and the convenience of the band locating method

ordered upwind methods for static hamilton jacobi equations abstract we introduce a family of fast ordered upwind methods for approximating solutions to a wide class of static hamilton jacobi equations with dirichlet boundary conditions standard techniques often rely on iteration to converge to the solution of a discretized version of the partial differential equation our fast methods avoid iteration through a careful use of information about the characteristic directions of the underlying partial differential equation these techniques are of complexity o m log m where m is the total number of points in the domain we consider anisotropic test problems in optimal control seismology and paths on surfaces

short note on reinitializing level set functions in this paper we consider reinitializing level functions through equation f t sgn f 0 f 1 0 16 the method of russo and smereka 11 is taken in the spatial discretization of the equation the spatial discretization is simply speaking the second order eno finite difference with subcell resolution near the interface our main interest is on the temporal discretization of the equation we compare the three temporal discretizations the second order runge kutta method the forward euler method and a gauss seidel iteration of the forward euler method the fact that the time in the equation is fictitious makes a hypothesis that all the temporal discretizations result in the same result in their stationary states the fact that the absolute stability region of the forward euler method is not wide enough to include all the eigenvalues of the linearized semi discrete system of the second order eno spatial discretization makes another hypothesis that the forward euler temporal discretization should invoke numerical instability our results in this paper contradict both the hypotheses the runge kutta and gauss seidel methods obtain the second order accuracy and the forward euler method converges with order between one and two examining all their properties we conclude that the gauss seidel method is the best among the three compared to the runge kutta it is twice faster and requires memory two times less with the same accuracy

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 monolithic conservative level set method with built in redistancing abstract we introduce a new level set method for representing evolving interfaces in the case of divergence free velocity fields the new method satisfies a conservation principle conservation is important for many applications such as modeling two phase incompressible flow in the present implementation the conserved quantity is defined as the integral of a smoothed characteristic function the new approach embeds level sets into a volume of fluid formulation the evolution of an approximate signed distance function is governed by a conservation law for its smoothed sign the non linear level set transport equation is regularized by adding a flux correction term that assures a non singular jacobian and penalizes deviations from a distance function the result is a locally conservative level set method with built in elliptic redistancing the continuous model is monolithic in the sense that the level set transport model the volume of fluid law of mass conservation and the minimization problem that preserves the approximate distance function property are incorporated into a single equation there is no need for any extra stabilization artificial compression flux limiting redistancing mass correction and other numerical fixes which are commonly used in level set or volume of fluid methods in addition there is just one free parameter that controls the strength of regularization and penalization in the model the accuracy and conservation properties of the monolithic finite element level set method are illustrated by the results of numerical studies for passive advection of free interfaces

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

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 boundary piecewise constant level set method for boundary control of eigenvalue optimization problems this paper investigates optimization of the least eigenvalue of d with the constraint of one dimension hausdorff measure of dirichlet boundary we propose the boundary piecewise constant level set bpcls method based on the regularity technique to combine two types of boundary conditions into a single robin boundary condition we derive the first variation of the least eigenvalue w r t the bpcls function and propose a penalty bpcls algorithm and an augmented lagrangian bpcls algorithm numerical results are reported for experiments on ellipse and l shape domains

a gpu accelerated level set reinitialization for an adaptive discontinuous galerkin method gpu accelerated high order reconstruction of signed distance function of the level set method is studied the flow based reinitialization equation is discretized in space by using a nodal discontinuous galerkin method on adaptive unstructured grids artificial diffusion with a modal decay rate based regularity estimator is used to damp out high frequency solution components near kinks where mesh adaptivity is applied a two rate adams bashforth time integrator is developed to avoid time step restrictions resulting from artificial diffusion stabilization and local mesh refinement platform independence of the solver is achieved by using an extensible multi threading programming api that allows runtime selection of different computing devices gpu and cpu and threading interfaces cuda opencl and openmp overall a highly scalable numerical scheme that preserves the simplicity of the original level set method is obtained performance and accuracy of the method to construct signed distance function on highly disturbed initial data with smooth and non smooth interfaces are tested through distinct two and three dimensional problems

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

control and stabilization of nonholonomic dynamic systems a class of inherently nonlinear control problems has been identified the nonlinear features arising directly from physical assumptions about constraints on the motion of a mechanical system models are presented for mechanical systems with nonholonomic constraints represented both by differential algebraic equations and by reduced state equations control issues for this class of systems are studied and a number of fundamental results are derived although a single equilibrium solution cannot be asymptotically stabilized using continuous state feedback a general procedure for constructing a piecewise analytic state feedback which achieves the desired result is suggested

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

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

favre averaged nonlinear harmonic method for compressible periodic flows the treatment of compressible flows subject to periodic disturbances of finite amplitude with nonlinear harmonic nlh methods requires a careful choice of averaging procedures and variable sets to

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

local minimization algorithms for dynamic programming equations the numerical realization of the dynamic programming principle for continuous time optimal control leads to nonlinear hamilton jacobi bellman equations which require the minimization of a nonlinear mapping over the set of admissible controls this minimization is often performed by comparison over a finite number of elements of the control set in this paper we demonstrate the importance of an accurate realization of these minimization problems and propose algorithms by which this can be achieved effectively the considered class of equations includes nonsmooth control problems with ell 1 penalization which lead to sparse controls

an efficient policy iteration algorithm for dynamic programming equations we present an accelerated algorithm for the solution of static hamilton jacobi bellman equations related to optimal control problems our scheme is based on a classic policy iteration procedure which is known to have superlinear convergence in many relevant cases provided the initial guess is sufficiently close to the solution this limitation often degenerates into a behavior similar to a value iteration method with an increased computation time the new scheme circumvents this problem by combining the advantages of both algorithms with an efficient coupling the method starts with a coarse mesh value iteration phase and then switches to a fine mesh policy iteration procedure when a certain error threshold is reached a delicate point is to determine this threshold in order to avoid cumbersome computations with the value iteration and at the same time to ensure the convergence of the policy iteration method to the optimal solution we analyze the methods and efficient coupling in a number of examples

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 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

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

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

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

a p adic algorithm for computing the inverse of integer matrices a method for computing the inverse of an nxn integer matrix a using p adic approximation is given the method is similar to dixon s algorithm but ours has a quadratic convergence rate the complexity of this algorithm without using fft or fast matrix multiplication is o n 4 logn 2 the same as that of dixon s algorithm however experiments show that our method is faster this is because our methods decrease the number of matrix multiplications but increase the digits of the components of the matrix which suits modern cpus with fast integer multiplication instructions

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 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

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

a binary level set model and some applications to mumford shah image segmentation in this paper we propose a pde based level set method traditionally interfaces are represented by the zero level set of continuous level set functions instead we let the interfaces be represented by discontinuities of piecewise constant level set functions each level set function can at convergence only take two values i e it can only be 1 or 1 thus our method is related to phase field methods some of the properties of standard level set methods are preserved in the proposed method while others are not using this new method for interface problems we need to minimize a smooth convex functional under a quadratic constraint the level set functions are discontinuous at convergence but the minimization functional is smooth we show numerical results using the method for segmentation of digital images

compact implementation strategy for a harmonic balance method within implicit flow solvers a two step approximate factorization technique for implementing a computationally stable nonlinear unsteady frequency domain harmonic balance solution method within existing implicit computational fluid dynamic flow solver codes is presented the approach uses an explicit discretization of the harmonic balance source term and no new implicit code development is required both of these features enable the harmonic balance method to be implemented within existing implicit flow solver codes with minimal modification necessary to the underlying flow solver code the resulting harmonic balance solver can then be used for modeling nonlinear periodic unsteady flows the methodology is applied to the nasa overflow flow solver code and results are presented for transonic viscous flow past an unsteady pitching airfoil section

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

dynamic cluster formation using level set methods density based clustering has the advantages for 1 allowing arbitrary shape of cluster and 2 not requiring the number of clusters as input however when clusters touch each other both the cluster centers and cluster boundaries as the peaks and valleys of the density distribution become fuzzy and difficult to determine we introduce the notion of cluster intensity function cif which captures the important characteristics of clusters when clusters are well separated cifs are similar to density functions but when clusters become closed to each other cifs still clearly reveal cluster centers cluster boundaries and degree of membership of each data point to the cluster that it belongs clustering through bump hunting and valley seeking based on these functions are more robust than that based on density functions obtained by kernel density estimation which are often oscillatory or oversmoothed these problems of kernel density estimation are resolved using level set methods and related techniques comparisons with two existing density based methods valley seeking and dbscan are presented which illustrate the advantages of our approach

short note o n implementation of the fast marching algorithm in this note we present an implementation of the fast marching algorithm for solving eikonal equations that in practice reduces the original run time from o nlogn to linear this lower run time cost is obtained while keeping an error bound of the same order of magnitude as the original algorithm this improvement is achieved introducing the straight forward untidy priority queue obtained via a quantization of the priorities in the marching computation we present the underlying framework estimations on the error and examples showing the usefulness of the proposed approach

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

visibility and its dynamics in a pde based implicit framework we investigate the problem of determining visible regions given a set of moving obstacles and a moving vantage point our approach to this problem is through an implicit framework where the obstacles are represented by a level set function the visibility problem is formally formulated as a boundary value problem bvp of a first order partial differential equation it is based on the continuation of values along the given ray field we propose a one pass multi level algorithm for the construction of the solution on a grid furthermore we study the dynamics of shadow boundaries on the surfaces of the obstacles when the vantage point moves along a given trajectory in all of these situations topological changes such as merging and breaking occur in the regions of interest these are automatically handled by the level set framework proposed here finally we obtain additional useful information through simple operations in the level set framework

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

convergence of a generalized fast marching method for an eikonal equation with a velocity changing sign we present a new fast marching algorithm for an eikonal equation with a velocity changing sign this first order equation models a front propagation in the normal direction the algorithm is an extension of the fast marching method in two respects the first is that the new scheme can deal with a time dependent velocity and the second is that there is no restriction on its change in sign we analyze the properties of the algorithm and we prove its convergence in the class of discontinuous viscosity solutions finally we present some numerical simulations of fronts propagating in mathbb r 2

algorithm for overcoming the curse of dimensionality for state dependent hamilton jacobi equations abstract in this paper we develop algorithms to overcome the curse of dimensionality in non convex state dependent hamilton jacobi partial differential equations hj pdes arising from optimal control and differential game problems the subproblems are independent and they can be implemented in an embarrassingly parallel fashion this is ideal for perfect scaling in parallel computing the algorithm is proposed to overcome the curse of dimensionality 1 2 when solving hj pde the major contribution of the paper is to change either the solving of a pde problem or an optimization problem over a space of curves to an optimization problem of a single vector which goes beyond the work of 40 we extend the method in 7 9 15 and conjecture a lax type minimization principle to solve state dependent hj pde when the hamiltonian is convex as well as a hopf type maximization principle to solve state dependent hj pde when the hamiltonian is non convex as a generalization of the well known hopf formula in 18 25 50 we showed the validity of the formula under restricted assumption for the sake of completeness and would like to bring our readers to 62 which validates our conjectures in a more general setting we conjectured the weakest assumption of our formula to hold is a pseudoconvexity assumption similar to one stated in 50 our method is expected to have application in control theory differential game problems and elsewhere

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

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

acceleration of euclidean algorithm and rational number reconstruction we accelerate the known algorithms for computing a selected entry of the extended euclidean algorithm for integers and consequently for the modular and numerical rational number reconstruction problems the acceleration is from quadratic to nearly linear time matching the known complexity bound for the integer gcd which our algorithm computes as a special case

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

the computational efficiency of non linear frequency domain methods this paper demonstrates efficient solver technologies applied to the non linear frequency domain nlfd method the basis of the nlfd method is to assume the time period of the solution s oscillation and to transform both the solution and residual using a discrete fourier transform an unsteady residual is formed in the frequency domain and iteratively driven to a negligible value this method is amenable to many of the convergence acceleration techniques used for steady state flows including pseudo local time stepping implicit residual averaging coarse grid viscosity and multigrid this paper will address the implementation of these techniques such that convergence rates of the modified unsteady solver are equivalent to those of the original steady state techniques

some improvements of the fast marching method the fast marching method published by sethian proc natl acad sci usa 93 1996 pp 1591 1595 is an optimally efficient algorithm for solving problems of front evolution where the front speed is monotonic it has been used in a wide variety of applications such as robotic path planning r kimmel and j sethian fast marching methods for computing distance maps and shortest paths tech report 669 cpam university of california berkeley 1996 crack propagation m stolarska et al modelling crack growth by level sets in the extended finite element method comput methods appl mech engrg to appear internat j numer methods engrg 51 2001 pp 943 960 n sukumar d l chopp and b moran extended finite element method and fast marching method for three dimensional fatigue crack propagation j comput phys submitted seismology j sethian and a popovici geophysics 64 1999 pp 516 523 photolithography j sethian fast marching level set methods for three dimensional photolithography development in proceedings of the spie 1996 international symposium on microlithography santa clara ca 1996 and medical imaging r malladi and j sethian proc natl acad sci usa 93 1996 pp 9389 9392 it has also been a valuable tool for the implementation of modern level set methods where it is used to efficiently compute the distance to the front and or an extended velocity function in this paper we improve upon the second order fast marching method of sethian siam rev 41 1999 pp 199 235 by constructing a second order approximation of the interface generated from local data on the mesh the data is interpolated on a single box of the mesh using a bicubic approximation the distance to the front is then calculated by using a variant of newton s method to solve both the level curve equation and the orthogonality condition for the nearest point to a given node the result is a second order approximation of the distance to the interface which can then be used to produce second order accurate initial conditions for the fast marching method and a third order fast marching method

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

an adaptive high order discontinuous galerkin method with error control for the hamilton jacobi equations part i the one dimensional steady state case we propose and study an adaptive version of the discontinuous galerkin method for hamilton jacobi equations it works as follows given the tolerance and the degree of the polynomial of the approximate solution the adaptive algorithm finds a mesh on which the approximate solution has an l 1 distance to the viscosity solution no bigger than the prescribed tolerance the algorithm uses three main tools the first is an iterative solver combining the explicit runge kutta discontinuous galerkin method and the implicit newton s method that enables us to solve the hamilton jacobi equations efficiently the second is a new a posteriori error estimate based on the approximate resolution of an approximate problem for the actual error the third is a method that allows us to find a new mesh as a function of the old mesh and the ratio of the a posteriori error estimate to the tolerance we display extensive numerical evidence that indicates that for any given polynomial degree the method achieves its goal with optimal complexity independently of the tolerance this is done in the framework of one dimensional steady state model problems with periodic boundary conditions

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

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

ordered upwind methods for static hamilton jacobi equations theory and algorithms we develop a family of fast methods for approximating the solutions to a wide class of static hamilton jacobi pdes these fast methods include both semi lagrangian and fully eulerian versions numerical solutions to these problems are typically obtained by solving large systems of coupled nonlinear discretized equations our techniques which we refer to as ordered upwind methods oums use partial information about the characteristic directions to decouple these nonlinear systems greatly reducing the computational labor our techniques are considered in the context of control theoretic and front propagation problems we begin by discussing existing oums focusing on those designed for isotropic problems we then introduce a new class of oums which decouple systems for general anisotropic problems we prove convergence of one such scheme to the viscosity solution of the corresponding hamilton jacobi pde next we introduce a set of finite differences methods based on an analysis of the role played by anisotropy in the context of front propagation and optimal trajectory problems the performance of the methods is analyzed and computational experiments are performed using test problems from computational geometry and seismology

flow stability of patchy vector fields and robust feedback stabilization the paper is concerned with patchy vector fields a class of discontinuous piecewise smooth vector fields that were introduced by the authors to study feedback stabilization problems we prove the stability of the corresponding solution set w r t a wide class of impulsive perturbations these results yield the robustness of patchy feedback controls in the presence of measurement errors and external disturbances

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

distance functions and geodesics on submanifolds of r d and point clouds a theoretical and computational framework for computing intrinsic distance functions and geodesics on submanifolds of bbb r d given by point clouds is introduced and developed in this paper

block jacobi implicit algorithms for the time spectral method many industrial applications involve flows periodic in time such flows are not simulated with enough efficiency when using classical unsteady techniques as a transient regime must be bypassed new techniques dedicated to time periodic flows and based on fourier analysis have been developed recently among these the time spectral method casts a time periodic flow computation in several coupled steady computations corresponding to a uniform sampling of the period up until now the steady states were reached using an explicit pseudotime algorithm thus very small time steps were needed and the convergence rate slowed down when increasing the number of harmonics in this paper a block jacobi approach is presented to solve the stationary problems with an implicit algorithm numerical simulations show on one hand the good quality of the results and on the other hand the interest of the proposed method to reduce the sensitivity of computations to a large number of harmonics

level set methods an overview and some recent results the level set method was devised by s osher and j a sethian 1988 j comput phys 79 12 49 as a simple and versatile method for computing and analyzing the motion of an interface in two or three dimensions bounds a possibly multiply connected region the goal is to compute and analyze the subsequent motion of under a velocity field v this velocity can depend on position time the geometry of the interface and the external physics the interface is captured for later time as the zero level set of a smooth at least lipschitz continuous function x t i e t x x t 0 is positive inside negative outside and is zero on t topological merging and breaking are well defined and easily performed in this review article we discuss recent variants and extensions including the motion of curves in three dimensions the dynamic surface extension method fast methods for steady state problems diffusion generated motion and the variational level set approach we also give a user s guide to the level set dictionary and technology and couple the method to a wide variety of problems involving external physics such as compressible and incompressible possibly reacting flow stefan problems kinetic crystal growth epitaxial growth of thin films vortex dominated flows and extensions to multiphase motion we conclude with a discussion of applications to computer vision and image processing

motion of curves constrained on surfaces using a level set approach the level set method has been successfully applied to a variety of problems that deal with curves in r2 or surfaces in r3 we present here a combination of these two cases creating a level set representation for curves constrained to lie on surfaces we study primarily geometrically based motions of these curves on stationary surfaces while allowing topological changes in the curves i e merging and breaking to occur applications include finding geodesic curves and shortest paths and curve shortening on surfaces further applications can be arrived at by extending those for curves moving in r2 to surfaces the problem of moving curves on surfaces can also be viewed as a simple constraint problem and may be useful in studying more difficult versions results show that our representation can accurately handle many geometrically based motions of curves on a wide variety of surfaces while automatically enforcing topological changes in the curves when they occur and automatically fixing the curves to lie on the surfaces the method can also be easily extended to higher dimensions

operator splitting based fast sweeping methods for isotropic wave propagation in a moving fluid wave propagation in an isotropic acoustic medium occupied by a moving fluid is governed by an anisotropic eikonal equation since this anisotropic eikonal equation is associated with an inhomogeneous hamiltonian most of existing anisotropic eikonal solvers are either inapplicable or of unpredictable behavior in convergence realizing that this anisotropic eikonal equation is defined by a sum of two well understood first order differential operators we propose novel operator splitting based fast sweeping methods to solve this generalized eikonal equation we develop various operator splitting methods relying on the peaceman rachford scheme the douglas rachford scheme the theta scheme and the regularized theta scheme after applying the operator splitting strategy each splitting step corresponds to a much simpler hamilton jacobi equation so that we can apply the lax friedrichs sweeping method to solve these splitted equations efficiently and easily two and three dimensional examples demons

fast methods for the eikonal and related hamilton jacobi equations on unstructured meshes the fast marching method is a numerical algorithm for solving the eikonal equation on a rectangular orthogonal mesh in o m log m steps where m is the total number of grid points the scheme relies on an upwind finite difference approximation to the gradient and a resulting causality relationship that lends itself to a dijkstra like programming approach in this paper we discuss several extensions to this technique including higher order versions on unstructured meshes in rn and on manifolds and connections to more general static hamilton jacobi equations

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

convergent difference schemes for degenerate elliptic and parabolic equations hamilton jacobi equations and free boundary problems convergent numerical schemes for degenerate elliptic partial differential equations are constructed and implemented simple conditions are identified which ensure that nonlinear finite difference schemes are monotone and nonexpansive in the maximum norm explicit schemes endowed with an explicit cfl condition are built for time dependent equations and are used to solve stationary equations iteratively explicit and implicit formulations of monotonicity for first and second order equations are unified bounds on orders of accuracy are established an example of a scheme which is stable but nonmonotone and nonconvergent is presented schemes for hamilton jacobi equations obstacle problems one phase free boundary problems and stochastic games are built and computational results are presented

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

a moving grid framework for geometric deformable models geometric deformable models based on the level set method have become very popular in the last decade to overcome an inherent limitation in accuracy while maintaining computational efficiency adaptive grid techniques using local grid refinement have been developed for use with these models this strategy however requires a very complex data structure yields large numbers of contour points and is inconsistent with the implementation of topology preserving geometric deformable models tgdms in this paper we investigate the use of an alternative adaptive grid technique called the moving grid method with geometric deformable models in addition to the development of a consistent moving grid geometric deformable model framework our main contributions include the introduction of a new grid nondegeneracy constraint the design of a new grid adaptation criterion and the development of novel numerical methods and an efficient implementation scheme the overall method is simpler to implement than using grid refinement requiring no large complex hierarchical data structures it also offers an extra benefit of automatically reducing the number of contour vertices in the final results after presenting the algorithm we demonstrate its performance using both simulated and real images

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

2 5d face recognition using patch geodesic moments in this paper we propose a novel patch geodesic distance pgd to transform the texture map of an object through its shape data for robust 2 5d object recognition local geodesic paths within patches and global geodesic paths for patches are combined in a coarse to fine hierarchical computation of pgd for each surface point to tackle the missing data problem in 2 5d images shape adjusted texture patches are encoded into local patterns for similarity measurement between two 2 5d images with different viewing angles and or shape deformations an extensive experimental investigation is conducted on 2 5 face images using the publicly available bu 3dfe and bosphorus databases covering face recognition under expression and pose changes the performance of the proposed method is compared with that of three benchmark approaches the experimental results demonstrate that the proposed method provides a very encouraging new solution for 2 5d object recognition

geometric optics in a phase space based level set and eulerian framework we introduce a level set approach for ray tracing and the construction of wavefronts in geometric optics this is important in a wide variety of applications in wave propagation our approach automatically handles the multivalued solutions that appear and automatically resolves the wavefronts this is achieved through solving for the bicharacteristic strips whose projection to spatial space gives the wavefronts in a reduced phase space under a eulerian and partial differential equation based framework the bicharacteristic strips are represented using a level set approach for handling higher codimensional objects and the partial differential equations responsible for the evolution are reduced forms of the liouville equations results for the two dimensional case for constant and variable indices of refraction are shown and compared to those of other current methods in the field results are also introduced to show the ability to handle reflection and to extend the method to the three dimensional case

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

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

finite difference methods for the infinity laplace and p laplace equations abstract we build convergent discretizations and semi implicit solvers for the infinity laplacian and the game theoretical p laplacian the discretizations simplify and generalize earlier ones we prove convergence of the solution of the wide stencil finite difference schemes to the unique viscosity solution of the underlying equation we build a semi implicit solver which solves the laplace equation as each step it is fast in the sense that the number of iterations is independent of the problem size this is an improvement over previous explicit solvers which are slow due to the cfl condition

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

exact solution of linear equations usingp adic expansions a method is described for computing the exact rational solution to a regular systemax b of linear equations with integer coefficients the method involves i computing the inverse modp ofa for some primep ii using successive refinements to compute an integer vector bar x such that a bar x equiv b modp m for a suitably large integerm and iii deducing the rational solutionx from thep adic approximation bar x for matricesa andb with entries of bounded size and dimensionsn n andn 1 this method can be implemented in timeo n 3 logn 2 which is better than methods previously used

a discontinuous galerkin finite element method for directly solving the hamilton jacobi equations in this paper we propose a new discontinuous galerkin finite element method to solve the hamilton jacobi equations unlike the discontinuous galerkin method of c hu c w shu a discontinuous galerkin finite element method for hamilton jacobi equations siam journal on scientific computing 21 1999 666 690 which applies the discontinuous galerkin framework on the conservation law system satisfied by the derivatives of the solution the method in this paper applies directly to the solution of the hamilton jacobi equations for the linear case this method is equivalent to the traditional discontinuous galerkin method for conservation laws with source terms thus stability and error estimates are straightforward for the nonlinear convex hamiltonians numerical experiments demonstrate that the method is stable and provides the optimal k 1 th order of accuracy for smooth solutions when using piecewise kth degree polynomials singularities in derivatives can also be resolved sharply if the entropy condition is not violated special treatment is needed for the entropy violating cases both one and two dimensional numerical results are provided to demonstrate the good qualities of the scheme

numerical simulation of a continuum model for bi directional pedestrian flow an algorithm for an extended reactive dynamic user equilibrium model of pedestrian counterflow as a continuum is developed it is based on a cell centered high resolution finite volume scheme with a fast sweeping method for an eikonal type equation on an orthogonal grid a high order total variation diminishing runge kutta method is adopted for the time integration of semi discrete equations the numerical results demonstrate the rationality of the model and efficiency of the algorithm some crowd pedestrian flow phenomena such as dynamic lane formation in bi directional flow are observed which are helpful for a global comprehension of pedestrian dynamics also the model can be utilized with different potential applications

fast semi lagrangian schemes for the eikonal equation and applications we introduce and analyze a fast version of the semi lagrangian algorithm for front propagation originally proposed in m falcone the minimum time problem and its applications to front propagation in motion by mean curvature and related topics a visintin and g buttazzo eds de gruyter berlin 1994 pp 70 88 the new algorithm is obtained using the local definition of the approximate solution typical of semi lagrangian schemes and redefining the set of neighboring nodes necessary for fast marching schemes a new proof of convergence is needed since that definition produces a new narrow band centered at the interphase which is larger than the one used in fast marching methods based on finite differences we show that the new algorithm converges to the viscosity solution of the problem and that its complexity is o n log n nb as it is for the fast marching method based on finite difference n and n nb being respectively the total number of nodes and the number of nodes in the narrow band a new sufficient condition for the convergence of the standard finite difference fast marching method is also given we present several tests comparing the two algorithms and other fast methods e g fast sweeping on a series of benchmarks which include the minimum time problem and the shape from shading problem

an implicit interface boundary integral method for poisson s equation on arbitrary domains abstract we propose a simple formulation for constructing boundary integral methods to solve poisson s equation on domains with smooth boundaries defined through their signed distance function our formulation is based on averaging a family of parameterizations of an integral equation defined on the boundary of the domain where the integrations are carried out in the level set framework using an appropriate jacobian by the coarea formula the algorithm operates in the euclidean space and does not require any explicit parameterization of the boundaries we present numerical results in two and three dimensions