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

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 generalized fast marching method on unstructured triangular meshes in this paper we extend the generalized fast marching method gfmm presented in e carlini et al siam j numer anal 46 2008 pp 2920 2952 to unstructured meshes the gfmm generalizes the classical fast marching method in the sense that it can be applied to propagate interfaces with time dependent and changing sign velocity the main motivation for this extension is that in many areas such as in fluid dynamics the method for tracking the motion of an interface in time has to be coupled with other solvers based on finite elements or finite volumes typically constructed on triangular grids and on domains with complex geometries we prove an abstract convergence result for the two dimensional 2d case which requires only some properties of the mesh and of the local solvers used to compute the solution on a narrow band in particular this result applies to a class of monotone schemes for 2d problems on triangular grids with acute angles finally some numerical tests in two dimensions illu

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

a fast level set method for propagating interfaces a method is introduced to decrease the computational labor of the standard level set method for propagating interfaces the fast approach uses only points close to the curve at every time step we describe this new algorithm and compare its efficiency and accuracy with the standard level set approach

on the transport diffusion algorithm and its applications to the navier stokes equations this paper deals with an algorithm for the solution of diffusion and or convection equations where we mixed the method of characteristics and the finite element method globally it looks like one does one step of transport plus one step of diffusion or projection but the mathematics show that it is also an implicit time discretization of thepde in lagrangian form we give an error bound h t h h t in the interesting case that holds also for the navier stokes equations even when the reynolds number is infinite euler equation

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

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

piecewise linear recursive convolution for dispersive media using fdtd electromagnetic propagation through linear dispersive media can be analyzed using the finite difference time domain fdtd method by employing the recursive convolution rc approach to evaluate the discrete time convolution of the electric field and the dielectric susceptibility function the rc approach results in a fast and computationally efficient algorithm however the accuracy achieved is not generally as good as that obtained with other methods a new piecewise linear recursive convolution plrc method is described here that has greatly improved accuracy over the original rc approach but retains its speed and efficiency advantages

stochastic resonance and self tuning a simple threshold system to put it in the framework of information processing stochastic resonance sr is a phenomenon in which the transfer of information from input to output signals can be significantly increased by noise with appropriate nonzero intensity the minimum requirements for sr are generally input signals nonlinearity and noise in addition to a quantifier to measure the efficiency of information transfer we study a simple threshold system and propose an adaptation mechanism for system parameters such as a threshold and noise intensity i e temperature which are held fixed in a normal sr setting our emphasis is put on parameter dynamics induced by this adaptation which we call self tuning st and on how this dynamics modifies or changes the sr picture with respect to information processing efficiency as a measure for performance of the threshold system we take mutual information and the signal to noise ratio these quantities are calculated by an analytical method and by a simulational one st of temperature results in oscillatory time variation in temperature with its average located around the temperature at which performance is maximized in the sr setting st of a threshold turns out to improve performance in the weak noise region

some convergence results for howard s algorithm this paper deals with convergence results of howard s algorithm for the resolution of min a in mathcal a b a x c a 0 where b a is a matrix c a is a vector and mathcal a is a compact set we show a global superlinear convergence result under a monotonicity assumption on the matrices b a extensions of howard s algorithm for a max min problem of the form max b in mathcal b min a in mathcal a b a b x c a b 0 are also proposed in the particular case of an obstacle problem of the form min ax b x g 0 where a is an n times n matrix satisfying a monotonicity assumption we show the convergence of howard s algorithm in no more than n iterations instead of the usual 2 n bound still in the case of an obstacle problem we establish the equivalence between howard s algorithm and a primal dual active set algorithm m hintermuller k ito and k kunisch siam j optim 13 2002 pp 865 888 the algorithms are illustrated on the discretization of nonlinear pdes arising in the context of mathematical finance american option and merton s portfolio problem of front propagation problems and for the double obstacle problem

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

the mechanism of stochastic resonance it is shown that a dynamical system subject to both periodic forcing and random perturbation may show a resonance peak in the power spectrum which is absent when either the forcing or the perturbation is absent

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

a dynamic mesh algorithm for curvature dependent evolving interfaces a new finite element method is discussed for approximating evolving interfaces in2nwhose normal velocity equals mean curvature plus a forcing function the method is insensitive to singularity formation and retains the local structure of the limit problem and thus exhibits a computational complexity typical of3n 1without having the drawbacks of front tracking strategies a graded dynamic mesh around the propagating front is the sole partition present at any time step and is significantly smaller than a full mesh time stepping is explicit but stability constraints force small time steps only when singularities develop whereas relatively large time steps are allowed before or past singularities when the evolution is smooth the explicit marching scheme also guarantees that at most one layer of elements has to be added or deleted per time step thereby making mesh updating simple and thus practical performance and potentials are fully documented via a number of numerical simulations in 2d 3d 4d and 8d with axial symmetries they include tori and cones for the mean curvature flow minimal and prescribed mean curvature surfaces with given boundary fattening for smooth driving force and volume constraint

bits and brains information flow in the nervous system abstract until recently there have been no convincing quantitative measurements on the rates of information transmission in real neurons here we review the theoretical basis for making such measurements together with the data which demonstrate remarkably high information rates in a variety of systems in fact these rates are within a factor of two of the absolute physical limits set by the entropy of neural spike trains theses observations lead to sharp theoretical questions about the structure of the code and the strategy for adapting the code to different ensembles of input signals

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

semi lagrangian methods for level set equations a new numerical method for solving geometric moving interface problems is presented the method combines a level set approach and a semi lagrangian time stepping scheme which is explicit yet unconditionally stable the combination decouples each mesh point from the others and the time step from the cfl stability condition permitting the construction of methods which are efficient adaptive and modular analysis of a linear one dimensional model problem suggests a surprising convergence criterion which is supported by heuristic arguments and confirmed by an extensive collection of two dimensional numerical results the new method computes correct viscosity solutions to problems involving geometry anisotropy curvature and complex topological events

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

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

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

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

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

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

fast marching methods fast marching methods are numerical schemes for computing solutions to the nonlinear eikonal equation and related static hamilton jacobi equations based on entropy satisfying upwind schemes and fast sorting techniques they yield consistent accurate and highly efficient algorithms they are optimal in the sense that the computational complexity of the algorithms is o n log n where n is the total number of points in the domain the schemes are of use in a variety of applications including problems in shape offsetting computing distances from complex curves and surfaces shape from shading photolithographic development computing first arrivals in seismic travel times construction of shortest geodesics on surfaces optimal path planning around obstacles and visibility and reflection calculations in this paper we review the development of these techniques including the theoretical and numerical underpinnings provide details of the computational schemes including higher order versions and demonstrate the techniques in a collection of different areas

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

primal dual extragradient methods for nonlinear nonsmooth pde constrained optimization we study the extension of the chambolle pock primal dual algorithm to nonsmooth optimization problems involving nonlinear operators between function spaces local convergence is shown under technical conditions including metric regularity of the corresponding primal dual optimality conditions we also show convergence for a nesterov type accelerated variant provided one part of the functional is strongly convex we show the applicability of the accelerated algorithm to examples of inverse problems with l 1 and l infty fitting terms as well as of state constrained optimal control problems where convergence can be guaranteed after introducing an arbitrarily small still nonsmooth moreau yosida regularization this is verified in numerical examples

minimum time control problems for non autonomous differential equations in this paper we investigate a minimum time problem for controlled non autonomous differential systems with dynamics depending on the final time the minimal time function associated to this problem does not satisfy the dynamic programming principle however we will prove that it is related to a standard front propagation problem via the reachability function two simple numerical examples are given to illustrate our approach

on the rate of convergence of finite difference approximations for bellmans equations with variable coefficients the estimates presented here for parabolic bellman s equations with variable coefficients extend the ones earlier obtained for constant coefficients

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

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

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

convergent network approximation for the continuous euclidean length constrained minimum cost path problem in many path planning situations we would like to find a path of constrained euclidean length in mathbb r 2 that minimizes a line integral we call this the continuous length constrained minimum cost path problem c lcmcpp generally this will be a nonconvex optimization problem for which continuous approaches ensure only locally optimal solutions however network discretizations yield weight constrained network shortest path problems wcspps which can in practice be solved to global optimality even for large networks we can readily find a globally optimal solution to an approximation of the c lcmcpp solutions to these wcspps yield feasible solutions and hence upper bounds we show how networks can be constructed and a wcspp in these networks formulated so that the solutions provide lower bounds on the global optimum of the continuous problem we give a general convergence scheme for our network discretizations and use it to prove that both the upper and lower bounds so generated converge to the global optimum of the c lcmcpp as the network discretization is refined our approach provides a computable lower bound formula of course the upper bounds are readily computable we give computational results showing the lower bound formula in practice and compare the effectiveness of our network construction technique with that of standard grid based approaches in generating good quality solutions we find that for the same computational effort we are able to find better quality solutions particularly when the length constraint is tighter

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

an fdtd formulation for dispersive media using a current density a novel finite difference time domain fdtd formulation for dispersive media called the je convolution jec method is derived using the convolution relationship between the current density j and the electric field e the high accuracy of the jec method is confirmed by computing the reflection and transmission coefficients for a nonmagnetized plasma slab in one dimension it is found that the new method has an accuracy comparable to the auxiliary differential equation ade while having the same computational efficiency as the less accurate recursive convolution rc method numerical simulations also show that the jec method exhibits significantly higher accuracy than the rc method in modeling wave attenuation inside the plasma

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

filtered schemes for hamilton jacobi equations we build a simple and general class of finite difference schemes for first order hamilton jacobi hj partial differential equations these filtered schemes are convergent to the unique viscosity solution of the equation the schemes are accurate we implement second third and fourth order accurate schemes in one dimension and second order accurate schemes in two dimensions indicating how to build higher order ones they are also explicit which means they can be solved using the fast sweeping method the accuracy of the method is validated with computational results for the eikonal equation and other hj equations in one and two dimensions using filtered schemes made from standard centered differences higher order upwinding and eno interpolation

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

memory optimized shift operator alternating direction implicit finite difference time domain method for plasma abstract through introducing the alternating direction implicit adi technique and the memory optimized algorithm to the shift operator so finite difference time domain fdtd method the memory optimized so adi fdtd for nonmagnetized collisional plasma is proposed and the corresponding formulae of the proposed method for programming are deduced in order to further the computational efficiency the iteration method rather than gauss elimination method is employed to solve the equation set in the derivation of the formulae complicated transformations and convolutions are avoided in the proposed method compared with the z transforms zt adi fdtd method and the piecewise linear je recursive convolution pljerc adi fdtd method the numerical dispersion of the so adi fdtd method with different plasma frequencies and electron collision frequencies is analyzed and the appropriate ratio of grid size to the minimum wavelength is given the accuracy of the proposed method is validated by the reflection coefficient test on a nonmagnetized collisional plasma sheet the testing results show that the proposed method is advantageous for improving computational efficiency and saving computer memory the reflection coefficient of a perfect electric conductor pec sheet covered by multilayer plasma and the rcs of the objects coated by plasma are calculated by the proposed method and the simulation results are analyzed

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

level sets of viscosity solutions some applications to fronts and rendez vous problems the authors treat some applications of hamilton jacobi equations to the study of a flame front propagation model and the rendez vows problem the solution of both problems requires the determination of the level sets of the viscosity solution for the corresponding equation in the flame front propagation model described here it is assumed that the evolution is driven by a vector field satisfying a transversality condition at time t 0 the evolution in the normal direction with variable velocity c x geq 0 is considered as a special case this approach is constructive permitting the numerical solution of such problems

transport and its enhancement caused by coupling using the weiss mean field approximation theory and the particles transport theory for the spatially periodic stochastic systems we derive an exact analytical expression for the stationary probability current of a coupled lattice system driven by dichotomous noise it is shown that for this coupled lattice system the spatial asymmetry of the system the asymmetry of the dichotomous noise and the coupling among nearest neighbors are the ingredients for the stationary probability current by applying our theory to two special models we find that 1 the coupling can lead to the directional transport of the particles even when the potential and the dichotomous noise are symmetric and 2 the coupling among nearest neighbors can enhance the transport of the particles in some circumstances our results are applied to a device of two dimensional josephson junction arrays and a large protein motors cluster

frequency dependent fdtd methods using z transforms the frequency dependent finite difference time domain method fd sup 2 td method has been shown to be capable of correctly calculating electromagnetic propagation through media whose dielectric properties are frequency dependent however as researchers search for more elaborate applications the formulation of the fd sup 2 td methods becomes more complex in this work the mathematics of the fd sup 2 td method is developed using z transform theory this has the advantages of presenting a clearer formulation and allowing researchers to draw on the literature of systems analysis and signal processing disciplines

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

optimal bilinear control of an abstract schro dinger equation well posedness of abstract quantum mechanical systems is considered and the existence of optimal control of such systems is proved first order optimality systems are derived convergence of the monotone scheme for the solution of the optimality system is proved

some estimates for finite difference approximations some estimates for the approximation of optimal stochastic control problems by discrete time problems are obtained in particular an estimate for the solutions of the continuous time versus the discrete time hamilton jacobi bellman equations is given the technique used is more analytic than probabilistic

stable numerical schemes for solving hamilton jacobi bellman isaacs equations stable approximation schemes for solving hamilton jacobi bellman isaacs equations in r n are proposed the efficiency of numerical procedures based on such approximations is demonstrated by solving several differential games in two three and four dimensions

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

convergence of a generalized fast marching method for a non convex eikonal equation we present a new fast marching algorithm for a non convex eikonal equation modeling front evolutions in the normal direction the algorithm is an extension of the fast marching method since the new scheme can deal with a emph time dependent velocity without emph any restriction on its 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 r 2

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

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

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

stochastic resonance as dithering a direct correspondence is demonstrated between the phenomenon of stochastic resonance in static nonlinear systems and the dithering effect well known in the theory of digital waveform coding it is argued that many static systems displaying stochastic resonance are forms of dithered quantizers and that the existence or absence of stochastic resonance in such systems can be predicted from the effects of dither averaging upon their transfer characteristics also results are introduced regarding stochastic resonance in certain nonlinear systems with memory e g hysteretic systems