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

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

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

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

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

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

the partition functions of classical systems in the gaussian equivalent representation of functional integrals abstract the gaussian equivalent representation method developed in quantum physics for approximate calculations of functional integrals is applied in the classical statistical physics of liquids to compute the partition and distribution functions for any densities and temperatures this method works in the case of systems of particles interacting via two body potentials with the positive fourier transform the partition functions for canonical and grand canonical ensembles of classical particles are presented in the form of the gaussian equivalent representation the equation of state and the phase transition for the yukawa potential and the debye screening for the coulomb potential are derived

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

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

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

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

an ordered line integral method for computing the quasi potential in the case of variable anisotropic diffusion abstract nongradient stochastic differential equations sdes with position dependent and anisotropic diffusion are often used in biological modeling the quasi potential is a crucial function in the large deviation theory that allows one to estimate transition rates between attractors of the corresponding ordinary differential equation and find the maximum likelihood transition paths unfortunately the quasi potential can rarely be found analytically it is defined as the solution to a certain action minimization problem in this work the recently introduced ordered line integral method olim is extended for computing the quasi potential for 2d sdes with anisotropic and position dependent diffusion scaled by a small parameter on a regular rectangular mesh the presented solver employs the dynamical programming principle at each step a local action minimization problem is solved using straight line path segments and the midpoint quadrature rule the solver is tested on two examples where analytic formulas for the quasi potential are available the dependence of the computational error on the mesh size the update factor k a key parameter of olims as well as the degree and the orientation of anisotropy is established the effect of anisotropy on the quasi potential and the maximum likelihood paths is demonstrated on the maier stein model the proposed solver is applied to find the quasi potential and the maximum likelihood transition paths in a model of the genetic switch in lambda phage between the lysogenic state where the phage reproduces inside the infected cell without killing it and the lytic state where the phage destroys the infected cell

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

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

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

clifford theory for g functors abstract when does a g functor admit a clifford theory in this paper we give a simple axiomatic property which characterizes the existence of such a theory satisfaction of this condition in the different contexts leads automatically to the satisfaction of the corresponding theorems of clifford we apply it to several examples some of them already known including the character and burnside rings

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

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

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

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

clifford theory for semisimple g groups abstract in this paper a clifford theory for semisimple g groups is developed as a particular case of an abstract clifford theory for g functors

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 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

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

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 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

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

clifford theory for mackey algebras we develop a clifford theory for mackey algebras for simple mackey functors using their classification we prove mackey algebra versions of clifford s theorem and the clifford correspondence let r g be the mackey algebra of a finite group g over a commutative unital ring r a nd let 1 n be the unity of r n where n is a normal subgroup of g observing that 1n r g 1n is a crossed product of g n over r n a number of results concerning group graded algebras are extended to the context of mackey algebras including fong s theorem green s indecomposibility theorem and some reduction and extension techniques for indecomposable mackey functors

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

on deterministic control problems an approximation procedure for the optimal cost i the stationary problem we study deterministic optimal control problems having stopping time continuous and impulse controls in each strategy we obtain the optimal cost considered as the maximum element of a suitable set of subsolutions of the associated hamilton jacobi equation using an approximation method a particular derivative discretization scheme is employed convergence of approximate solutions is shown taking advantage of a discrete maximum principle which is also proved for the numerical solutions of approximate problems we use a method of relaxation type the algorithm is very simple it can be run on computers with small central memory in part i we study the stationary case in part ii siam j control optim 23 1985 pp 267 285 we study the nonstationary case

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

fast two scale methods for eikonal equations fast marching and fast sweeping are the two most commonly used methods for solving the eikonal equation each of these methods performs best on a different set of problems fast sweeping for example will outperform fast marching on problems where the characteristics are largely straight lines fast marching on the other hand is usually more efficient than fast sweeping on problems where characteristics frequently change their directions and on domains with complicated geometry in this paper we explore the possibility of combining the best features of both approaches by using marching on a coarser scale and sweeping on a finer scale we present three new hybrid methods based on this idea and illustrate their properties in several numerical examples with continuous and piecewise constant speed functions in r 2

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

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

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

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

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

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

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

computation within the auxiliary field approach recently the classical auxiliary field methodology has been developed as a new simulation technique for performing calculations within the framework of classical statistical mechanics since the approach suffers from a sign problem a judicious choice of the sampling algorithm allowing a fast statistical convergence and an efficient generation of field configurations is of fundamental importance for a successful simulation in this paper we focus on the computational aspects of this simulation methodology we introduce two different types of algorithms the single move auxiliary field metropolis monte carlo algorithm and two new classes of force based algorithms which enable multiple move propagation in addition to further optimize the sampling we describe a preconditioning scheme which permits to treat each field degree of freedom individually with regard to the evolution through the auxiliary field configuration space finally we demonstrate the validity and assess the competitiveness of these algorithms on a representative practical example we believe that they may also provide an interesting possibility for enhancing the computational efficiency of other auxiliary field methodologies

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

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

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

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

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

representation of complex probabilities let a complex probability be a normalizable complex distribution p x defined on rd a real and positive probability distribution p z defined on the complex plane cd is said to be a positive representation of p x if q x p q z p where q x is any polynomial in rd and q z its analytical extension on cd in this paper it is shown that every complex probability admits a real representation and a constructive method is given among other results explicit positive representations in any number of dimensions are given for any complex distribution of the form gaussian times polynomial for any complex distributions with support at one point and for any periodic gaussian times polynomial

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

label setting methods for multimode stochastic shortest path problems on graphs stochastic shortest path ssp problems arise in a variety of discrete stochastic control contexts an optimal solution to such a problem is typically computed using the value function which can be found by solving the corresponding dynamic programming equations in the deterministic case these equations can be often solved by highly efficient label setting methods such as dijkstra s and dial s algorithms in this paper we define and study a class of multimode stochastic shortest path mssp problems and develop sufficient conditions for the applicability of label setting methods we illustrate our approach in a number of discrete stochastic control examples we also discuss the relationship of ssps with discretizations of static hamilton jacobi equations and provide an alternative derivation for several fast noniterative numerical methods for these partial differential equations pdes

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