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

an 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

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

the principle of equivalence and cosmological metrics this paper is concerned with the extent to which the geodesics of space time that is the experimental consequences of the principle of equivalence determine the metric in general relativity theory and in particular in friedmann robertson walker lemaitre frwl space times thus it discusses projective structure in these space times the approach will be from a geometrical point of view and it is shown that if two space time metrics share the same unparametrized geodesics and one is a generic frwl metric then so is the other and that each is a member of a well defined family of projectively related frwl metrics similar techniques are then applied to study the existence and properties of symmetries of the weyl projective tensor and projective symmetries in frwl space times

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

picard vessiot theory of differentially simple rings abstract in picard vessiot theory the galois theory for linear differential equations the picard vessiot ring plays an important role since it is the picard vessiot ring which is a torsor principal homogeneous space for the galois group scheme like fields are simple rings having only 0 and 1 as ideals the picard vessiot ring is a differentially simple ring i e a differential ring having only 0 and 1 as differential ideals having in mind that the classical galois theory is a theory of extensions of fields i e of simple rings it is quite natural to ask whether one can also set up a picard vessiot theory where the base is not a differential field but more generally a differentially simple ring it is the aim of this article to give a positive answer to this question i e to set up such a differential galois theory

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

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

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

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

polynomials with psl 2 monodromy let k be a field of characteristic p 0 let q be a power of p and let u be transcendental over k we determine all polynomials f k x k x p of degree q q 1 2 for which the galois group of f x u over k u has a transitive normal subgroup isomorphic to psl 2 q subject to a certain ramification hypothesis as a consequence we describe all polynomials f k x such that deg f is not a power of p and f is functionally indecomposable over k but f decomposes over an extension of k moreover except for one ramification configuration which is handled in a companion paper with rosenberg we describe all indecomposable polynomials f k x such that deg f is not a power of p and f is exceptional in the sense that x y is the only absolutely irreducible factor of f x f y which lies in k x y it is known that when k is finite a polynomial f is exceptional if and only if it induces a bijection on infinitely many finite extensions of k

cyclic quotients of transitive groups abstract let a be a transitive subgroup of s n we show that the largest cyclic quotient of a has order at most n this can be interpreted as an equivalent result about extensions of constants in the galois closure of a covering of curves over a finite field we also prove that the point stabilizer in a finite primitive permutation group always has a faithful orbit

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

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

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

iterative differential embedding problems in positive characteristic abstract in this paper we prove that every iterative differential embedding problem over an algebraic function field in positive characteristic with an algebraically closed field of constants has a proper solution

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

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

galois theory for iterative connections and nonreduced galois groups this article presents a theory of modules with iterative connection this theory is a generalisation of the theory of modules with connection in characteristic zero as given by n katz to modules over rings of arbitrary characteristic we show that these modules with iterative connection and also the modules with integrable iterative connection form a tannakian category assuming some nice properties for the underlying ring and we show how this generalises to modules over schemes stratifications on modules as introduced by a grothendieck generalise the notion of integrable ordinary connections in another way however over smooth rings we obtain an equivalence of stratifications and integrable iterative connections furthermore over a regular ring in positive characteristic we show that the category of modules with integrable iterative connection is also equivalent to the category of flat bundles as defined by d gieseker in the second part of this article we set up a picard vessiot theory for fields of solutions for such a picard vessiot extension we obtain a galois correspondence which takes into account even nonreduced closed subgroup schemes of the galois group scheme on one hand and inseparable intermediate extensions of the picard vessiot extension on the other hand finally we compare our galois theory with the galois theory for purely inseparable field extensions given by s chase

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

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

some invariants of holomorphically projective mappings of generalized kahlerian spaces abstract in the present paper a generalized kahlerian space is considered as a generalized riemannian space gr n with almost complex structure f i h that is covariantly constant with respect to the first and the second kind of covariant derivative in the general case of a holomorphically projective mapping f of two non symmetric generalized kahlerian spaces gk n and g k n it is impossible to obtain a generalization of the holomorphically projective curvature tensor in the present paper we study the case when gk n and g k n have the same torsion in corresponding points such a mapping we call equitorsion mapping we obtain quantities h p w 1 5 that are generalizations of the holomorphically projective tensor i e they are invariants based on f among h p w only h p w 5 is a tensor using another five linearly independent curvature tensors we can prove that there exist three holomorphically projective tensors

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

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

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

estimating the order of vanishing at infinity of drinfeld quasi modular forms we introduce and study certain deformations of drinfeld quasi modular forms by using rigid analytic trivialisations of corresponding anderson s t motives we show that a sub algebra of these deformations has a natural graduation by the group z 2 x z q 1 z and an homogeneous automorphism and we deduce from this and other properties multiplicity estimates

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

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

necessary and sufficient conditions for equitorsion geodesic mapping abstract in the present paper we study equitorsion geodesic mappings between two generalized riemannian spaces f gr n g r n in this case these spaces have the same torsions at corresponding points we prove that a generalized riemannian space gr n admits an equitorsion geodesic mapping onto a generalized riemannian space g r n if and only if in gr n certain differential equations of cauchy type in covariant derivatives of the 1 4 kinds have a solution with respect to the unknown tensor a i j the gradient vector i 0 and the differentiable function 1 4 in fact we find four systems of pde all equivalent to the existence of an equitorsion geodesic mapping and discuss the number of linearly independent solutions of this system of pde we establish an upper bound for the number of solutions for the geometrical problem under consideration