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

convergence of the finite difference scheme for a general class of the spatial segregation of reaction diffusion systems abstract in this work we prove convergence of the finite difference scheme for equations of stationary states of a general class of the spatial segregation of reaction diffusion systems with m 2 components more precisely we show that the numerical solution u h l given by the difference scheme converges to the l t h component u l when the mesh size h tends to zero provided u l c 2 for every l 1 2 m in particular our proof provides convergence of a difference scheme for the multi phase obstacle problem

convergent finite difference solvers for viscosity solutions of the elliptic monge ampere equation in dimensions two and higher the elliptic monge ampere equation is a fully nonlinear partial differential equation that originated in geometric surface theory and has been applied in dynamic meteorology elasticity geometric optics image processing and image registration solutions can be singular in which case standard numerical approaches fail novel solution methods are required for stability and convergence to the weak viscosity solution in this article we build a wide stencil finite difference discretization for the monge ampere equation the scheme is monotone so the barles souganidis theory allows us to prove that the solution of the scheme converges to the unique viscosity solution of the equation solutions of the scheme are found using a damped newton s method we prove convergence of newton s method and provide a systematic method to determine a starting point for the newton iteration computational results are presented in two and three dimensions which demonstrates the speed and accuracy of the method on a number of exact solutions which range in regularity from smooth to nondifferentiable

on the topology and index of minimal surfaces we show that for an immersed two sided minimal surface in r 3 there is a lower bound on the index depending on the genus and number of ends using this we show the nonexistence of an embedded minimal surface in r 3 of index 2 as conjectured by choe cho90 moreover we show that the index of an immersed two sided minimal surface with embedded ends is bounded from above and below by a linear function of the total curvature of the surface

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

a partial differential equation for the rank one convex envelope a partial differential equation pde for the rank one convex envelope is introduced the existence and uniqueness of viscosity solutions to the pde is established elliptic finite difference schemes are constructed and convergence of finite difference solutions to the viscosity solution of the pde is proven computational results are presented and laminates are computed from the envelopes results include the kohn strang example the classical four gradient example and an example with eight gradients which produces nontrivial laminates

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

approximation schemes for viscosity solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of applications including the calculus of variations control theory and differential games recently m g crandall and p l lions established the correct notion of generalized solutions for these equations this article discusses the convergence of general approximation schemes to this solution and gives under certain hypotheses explicit error estimates these results are then applied to obtain various representations as limits of solutions of general explicit and implicit finite difference schemes with error estimates

a counterpart to nagata idealization abstract idealization of a module k over a commutative ring s produces a ring having k as an ideal all of whose elements are nilpotent we develop a method that under suitable field theoretic conditions produces from an s module k a subring r of s that behaves like the idealization of k but is such that when s is a domain so is r the ring s is contained in the normalization of r but is finite over r only when r s we determine conditions under which r is noetherian cohen macaulay gorenstein a complete intersection or a hypersurface when r is local then its m adic completion is the idealization of the m adic completions of s and k

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

regime switching constrained viscosity solutions approach for controlling dam reservoir systems abstract a new stochastic control problem of a dam reservoir system installed in a river is analyzed both mathematically and numerically water balance dynamics of the reservoir are piece wise deterministic and are driven by a stochastic regime switching inflow process the system is controlled to balance among the operation purpose and the internal and downstream environmental conditions finding the optimal operation policy of the system reduces to solving an optimality equation with a discontinuous hamiltonian which is a system of nonlinear degenerate parabolic or hyperbolic equations we show that the optimality equation has at most one constrained viscosity solution and find the solution explicitly under certain conditions the model is applied to numerical computation of the operation policy of an existing dam reservoir system using a high order finite difference scheme the computational results can suggest how the operation policy should be adapted according to the environmental concerns of the river

numerical methods for the fractional laplacian a finite difference quadrature approach the fractional laplacian delta alpha 2 is a nonlocal operator which depends on the parameter alpha and recovers the usual laplacian as alpha to 2 a numerical method for the fractional laplacian is proposed based on the singular integral representation for the operator the method combines finite differences with numerical quadrature to obtain a discrete convolution operator with positive weights the accuracy of the method is shown to be o h 3 alpha convergence of the method is proven the treatment of far field boundary conditions using an asymptotic approximation to the integral is used to obtain an accurate method numerical experiments on known exact solutions validate the predicted convergence rates computational examples include exponentially and algebraically decaying solutions with varying regularity the generalization to nonlinear equations involving the operator is discussed the obstacle problem for the fractional laplacian is computed

a numerical approach for a general class of the spatial segregation of reaction diffusion systems arising in population dynamics abstract in the current work we consider the numerical solutions of equations of stationary states for a general class of the spatial segregation of reaction diffusion systems with m 2 population densities we introduce a discrete multi phase minimization problem related to the segregation problem which allows to prove the existence and uniqueness of the corresponding finite difference scheme based on that scheme we suggest an iterative algorithm and show its consistency and stability for the special case m 2 we show that the problem gives rise to the generalized version of the so called two phase obstacle problem in this particular case we introduce the notion of viscosity solutions and prove convergence of the difference scheme to the unique viscosity solution at the end of the paper we present computational tests for different internal dynamics and discuss numerical results

a geometrical optics based numerical method for high frequency electromagnetic fields computations near fold caustics part i this paper presents an eulerian numerical method for the computation of a bi valued solution of hamilton jacobi type equation in a particular geometric setting more precisely we consider high frequency electromagnetic fields in the vicinity of fold caustics

a reformulation of the conservative level set reinitialization equation for accurate and robust simulation of complex multiphase flows abstract this paper presents an accurate and robust reinitialization equation for the conservative level set that does not significantly deform stationary surfaces the compression and diffusion term of the reinitialization equation are reformulated to use a distance level set directly mapped from the conservative level set the normals are calculated using a distance level set reconstructed from the interface using a fast marching method increasing robustness and allowing the use of high order non tvd transport schemes using this new reinitialization equation we present results for canonical test cases such as zalesak s disk and spurious currents which show significant improvement a simulation of a liquid gas jet with r e 5000 is also presented to demonstrate the volume conservation properties of the method in more complex flows

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

a convergent monotone difference scheme for motion of level sets by mean curvature an explicit convergent finite difference scheme for motion of level sets by mean curvature is presented the scheme is defined on a cartesian grid using neighbors arranged approximately in a circle the accuracy of the scheme which depends on the radius of the circle dx and on the angular resolution d is formally o dx2 d the scheme is explicit and nonlinear the update involves computing the median of the values at the neighboring grid points numerical results suggest that despite the low accuracy acceptable results are achieved for small stencil sizes a numerical example is presented which shows that the centered difference scheme is non convergent

discontinuous galerkin finite element approximation of hamilton jacobi bellman equations with cordes coefficients we propose an hp version discontinuous galerkin finite element method for fully nonlinear second order elliptic hamilton jacobi bellman equations with cordes coefficients the method is proved to be consistent and stable with convergence rates that are optimal with respect to mesh size and suboptimal in the polynomial degree by only half an order numerical experiments on problems with nonsmooth solutions and strongly anisotropic diffusion coefficients illustrate the accuracy and computational efficiency of the scheme an existence and uniqueness result for strong solutions of the fully nonlinear problem and a semismoothness result for the nonlinear operator are also provided

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

complete embedded minimal surfaces of finite total curvature we will survey what is known about minimal surfaces s r 3 which are complete embedded and have finite total curvature int s k da infty the only classically known examples of such surfaces were the plane and the catenoid the discovery by costa 14 15 early in the last decade of a new example that proved to be embedded sparked a great deal of research in this area many new examples have been found even families of them as will be described below the central question has been transformed from whether or not there are any examples except surfaces of rotation to one of understanding the structure of the space of examples

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

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

deformations of complete minimal surfaces a notion of deformation is defined and studied for complete minimal surfaces in r3 and r3 c c a group of translations the catenoid enneper s surface and the surface of meeks jorge modelled on a 3 punctured sphere are shown to be isolated minimal surfaces of total curvature 4sr in r3 z and r3 z2 are studied it is proved that the helicoid and scherk s surface are isolated under periodic perturbations let m be a submanifold of a riemannian manifold n and let te m be a tubular neighborhood of m in n of radius e a c1 e variation of m in n is a submanifold m1 c te m which is a graph over m and is e c1 close to m this means m1 is pointwise e close to m in each fibre of te m and the tangent planes as well we are interested in complete minimal submanifolds m of n abbreviated c m s and c1 e variations which are also c m s s and henceforth we always assume m and m1 are c m s s we say m is isolated if for some e 0 the only e c1 variations of m differ from m by an ambient isometry of n in this paper we shall study this question when n is r3 or a translation space r3 modulo a group of translations a flat plane in r3is isolated this follows immediately from bernstein s theorem a function of two variables on r2 whose graph is a c m s is linear we shall prove enneper s surface the catenoid and a certain 3 punctured sphere discovered by meeks and jorge are isolated in r3 we prove the helicoid and scherk s surface are isolated under periodic deformations i e the helicoid of total curvature 4sr in r3 modulo one translation is isolated in this translation space and scherk s surface of total curvature 4sr is isolated in r3 modulo two translations this scherk surface is conformally a 4 punctured sphere and the helicoid a 2 punctured sphere we shall study minimal submanifolds of translation spaces many of the tech niques developed by r osserman can be adapted to this context and yield information about periodic minimal surfaces in r3 for example we classify c m s s vf total curvature 4sr in translation space in r3 osserman has proved the catenoid and enneper s surface are the only c m s s of total curvature 4sr we obtain an analogous classification in r3 g in vi we describe these surfaces for g z we study c m s s m of finite total curvature in translation spaces we prove a deformation i e an e c1 variation m1 of a finite total curvature c m s in a translation space is also of finite total curvature and is conformally a compact

meshfree finite difference approximations for functions of the eigenvalues of the hessian we introduce meshfree finite difference methods for approximating nonlinear elliptic operators that depend on second directional derivatives or the eigenvalues of the hessian approximations are defined on unstructured point clouds which allows for very complicated domains and a non uniform distribution of discretisation points the schemes are monotone which ensures that they converge to the viscosity solution of the underlying pde as long as the equation has a comparison principle numerical experiments demonstrate convergence for a variety of equations including problems posed on random point clouds complex domains degenerate equations and singular solutions

high resolution nonoscillatory central schemes for hamilton jacobi equations in this paper we construct second order central schemes for multidimensional hamilton jacobi equations and we show that they are nonoscillatory in the sense of satisfying the maximum principle thus these schemes provide the first examples of nonoscillatory second order godunov type schemes based on global projection operators numerical experiments are performed l 1 l infty errors and convergence rates are calculated for convex hamiltonians numerical evidence confirms that our central schemes converge with second order rates when measured in the l1 norm advocated in our recent paper numer math to appear the standard l infty norm however fails to detect this second order rate

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

a convergent difference scheme for the infinity laplacian construction of absolutely minimizing lipschitz extensions this article considers the problem of building absolutely minimizing lipschitz extensions to a given function these extensions can be characterized as being the solution of a degenerate elliptic partial differential equation the infinity laplacian for which there exist unique viscosity solutions a convergent difference scheme for the infinity laplacian equation is introduced which arises by minimizing the discrete lipschitz constant of the solution at every grid point existence and uniqueness of solutions to the scheme is shown directly solutions are also shown to satisfy a discrete comparison principle solutions are computed using an explicit iterative scheme which is equivalent to solving the parabolic version of the equation

an upwind finite difference method for a nonlinear black scholes equation governing european option valuation under transaction costs in this paper we develop a numerical method for a nonlinear parabolic partial differential equation arising from pricing european options under transaction costs the method is based on an upwind finite difference scheme for the spatial discretization and a fully implicit time stepping scheme we prove that the system matrix from this scheme is an m matrix and that the approximate solution converges unconditionally to the viscosity solution to the equation by showing that the scheme is consistent monotone and unconditionally stable a newton iterative algorithm is proposed for solving the discretized nonlinear system of which the jacobian matrix is shown to be also an m matrix numerical experiments are performed to demonstrate the accuracy and robustness of the method

an eulerian approach for computing the finite time lyapunov exponent we propose efficient eulerian methods for approximating the finite time lyapunov exponent ftle the idea is to compute the related flow map using the level set method and the liouville equation there are several advantages of the proposed approach unlike the usual lagrangian type computations the resulting method requires the velocity field defined only at discrete locations no interpolation of the velocity field is needed also the method automatically stops a particle trajectory in the case when the ray hits the boundary of the computational domain the computational complexity of the algorithm is o dx d 1 with d the dimension of the physical space since there are the same number of mesh points in the x t space the computational complexity of the proposed eulerian approach is optimal in the sense that each grid point is visited for only o 1 time we also extend the algorithm to compute the ftle on a co dimension one manifold the resulting algorithm does not require computation on any local coordinate system and is simple to implement even for an evolving manifold

semi lagrangian schemes for linear and fully non linear diffusion equations for linear and fully non linear diffusion equations of bellmanisaacs type we introduce a class of monotone approximation schemes relying on monotone interpolation as opposed to classical numerical methods these schemes converge for degenerate diffusion equations having general nondiagonal dominant coefficient matrices such schemes have to have a wide stencil in general besides providing a unifying framework for several known first order accurate schemes our class of schemes also includes more efficient versions and a new second order scheme that converges only for essentially monotone solutions the methods are easy to implement and analyze and they are more efficient than some other known schemes we prove stability and convergence of the schemes in the general case and provide error estimates in the convex case which are robust in the sense that they apply to degenerate equations and non smooth solutions the methods are extensively tested

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

some remarks on deformations of minimal surfaces we consider complete minimal surfaces c m s s in r3 and their deformations m1 is an s deformation of mo if m1 is a graph over mo in an e tubular neighborhood of mo and m1 is e c1 close to mo a minimal surface m is isolated if all c m s s which are sufficiently small deformations of m are congruent to m in this paper we construct an example of a nonisolated c m s it is modelled on a 4 punctured sphere and is of finite total curvature on the other hand we prove that a c m s discovered by meeks and jorge modelled on the sphere punctured at the fourth roots of unity is isolated introduction we consider complete minimal surfaces c m s s in r3 and their deformations m1 in an s deformation of mo if m1 is a graph over mo is an e tubular neighborhood of m1 and m1 is e c1 close to mo a c m s mo is isolated if all minimal surfaces m1 which are sufficiently small deformations of mo are congruent to mo many of the classical minimal surfaces in r3 are known to be isolated 2 however no example was known of a nonisolated minimal surface in this paper we construct such an example it is modelled on a 4 punctured sphere and is of finite total curvature on the other hand we prove that a c m s discovered by meeks and jorge modelled on the sphere punctured at the four roots of unity 1 1 i i is isolated the analogous surface modelled on the sphere punctured at the cube roots of unity was shown to be isolated in 2 the question is raised there of whether deformations of the four puncture case can be realized by deforming the conformal structure i e changing the cross ratio of the four points thus for the meeks jorge example in question the answer is no as we shall see the conformal structure of our example that admits deformations does not change either perhaps the conformal structure never changes by small deformations we wish to thank w meeks and b morin for helpful conversations and greatly simplifying suggestions i a deformable surface let m be a c m s of finite total curvature so that m is conformally equivalent to a compact riemann surface m punctured at a finite number of points an end e of m is said to be bounded if e is a bounded distance from a plane if e corresponds to the puncture p and p is the plane orthogonal to g p g the gauss map then e bounded means e is a bounded distance from p if e is embedded then it can be expressed as the graph of a log r o rl where r is the distance from the origin in p and this holds for r large then e is bounded if and only if a o let 9 be a weierstrass representation of m and 1 x2 x3 the associated analytic differentials on m we know the real periods of each xk are zero on m received by the editors december 2 1983 and in revised form may 1 1984 1980 mathematics subject classification 1985 revision primary 53a10 1986 american mathematical society 0002 9947 86 1 00 25 per page

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

a numerical method for the elliptic monge ampere equation with transport boundary conditions the problem of optimal mass transport arises in numerous applications including image registration mesh generation reflector design and astrophysics one approach to solving this problem is via the monge ampere equation while recent years have seen much work in the development of numerical methods for solving this equation very little has been done on the implementation of the transport boundary condition in this paper we propose a method for solving the transport problem by iteratively solving a monge ampere equation with neumann boundary conditions to enable mappings between variable densities we extend an earlier discretization of the equation to allow for right hand sides that depend on gradients of the solution b d froese and a m oberman siam j numer anal 49 2011 pp 1692 1714 this discretization provably converges to the viscosity solution the resulting system is solved efficiently with newton s method we provide several challenging computational examples that demonstra

uniqueness and existence results on the hele shaw and the stefan problems in this paper we introduce a notion of viscosity solutions for the one phase hele shaw and stefan problems when there is no surface tension we prove the uniqueness and existence of the solutions for both problems and the uniform convergence of solutions of porous medium equation to those of the hele shaw problem

the boundary method for semi discrete optimal transport partitions and wasserstein distance computation abstract we introduce a new technique which we call the boundary method for solving semi discrete optimal transport problems with a wide range of cost functions the boundary method reduces the effective dimension of the problem thus improving complexity for cost functions equal to a p norm with p 1 we provide mathematical justification convergence analysis and algorithmic development our testing supports the boundary method with these p norms as well as other more general cost functions

a study on moving mesh finite element solution of the porous medium equation an adaptive moving mesh finite element method is studied for the numerical solution of the porous medium equation with and without variable exponents and absorption the method is based on the moving mesh partial differential equation approach and employs its newly developed implementation the implementation has several improvements over the traditional one including its explicit compact form of the mesh velocities ease to program and less likelihood of producing singular meshes three types of metric tensor that correspond to uniform and arclength based and hessian based adaptive meshes are considered the method shows first order convergence for uniform and arclength based adaptive meshes and second order convergence for hessian based adaptive meshes it is also shown that the method can be used for situations with complex free boundaries emerging and splitting of free boundaries and the porous medium equation with variable exponents and absorption two dimensional numerical results are presented

convergent difference schemes for nonlinear parabolic equations and mean curvature motion explicit finite difference schemes are given for a collection of parabolic equations which may have all of the following complex features degeneracy quasilinearity full nonlinearity and singularities in particular the equation of motion by mean curvature is included the schemes are monotone and consistent so that convergence is guaranteed by the general theory of approximation of viscosity solutions of fully nonlinear problems in addition an intriguing new type of nonlocal problem is analyzed which is related to the schemes and another very different sort of approximation is presented as well

numerical solution of the optimal transportation problem using the monge ampere equation a numerical method for the solution of the elliptic monge ampere partial differential equation with boundary conditions corresponding to the optimal transportation ot problem is presented a local representation of the ot boundary conditions is combined with a finite difference scheme for the monge ampere equation email protected s method is implemented leading to a fast solver comparable to solving the laplace equation on the same grid several times theoretical justification for the method is given by a convergence proof in the companion paper 4 solutions are computed with densities supported on non convex and disconnected domains computational examples demonstrate robust performance on singular solutions and fast computational times

unconditionally stable methods for hamilton jacobi equations we present new numerical methods for constructing approximate solutions to the cauchy problem for hamilton jacobi equations of the form ut h dxu 0 the methods are based on dimensional splitting and front tracking for solving the associated non strictly hyperbolic system of conservation laws pt dx h p 0 where p dx u in particular our methods depend heavily on a front tracking method for one dimensional scalar conservation laws with discontinuous coefficients the proposed methods are unconditionally stable in the sense that the time step is not limited by the space discretization and they can be viewed as large time step godunov type or front tracking methods we present several numerical examples illustrating the main features of the proposed methods we also compare our methods with several methods from the literature

fast finite difference solvers for singular solutions of the elliptic monge ampere equation the elliptic monge ampere equation is a fully nonlinear partial differential equation which originated in geometric surface theory and has been applied in dynamic meteorology elasticity geometric optics image processing and image registration solutions can be singular in which case standard numerical approaches fail in this article we build a finite difference solver for the monge ampere equation which converges even for singular solutions regularity results are used to select a priori between a stable provably convergent monotone discretization and an accurate finite difference discretization in different regions of the computational domain this allows singular solutions to be computed using a stable method and regular solutions to be computed more accurately the resulting nonlinear equations are then solved by newton s method computational results in two and three dimensions validate the claims of accuracy and solution speed a computational example is presented which demonstrates the necessity of the use of the monotone scheme near singularities

discrete methods for fully nonlinear elliptic equations this paper exhibits and proves the stability of discrete approximations for the solution of the dirichlet problem for fully nonlinear uniformly elliptic partial differential equations in situations where classical solutions are not known to exist the resulting solutions are characterized in the viscosity sense of crandall and lions traps amer math soc 177 1983 pp 1 42 and our stability analysis depends on estimates for nonlinear difference equations of positive type which are consequences of earlier work on linear difference equations with random coefficients

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

finite element methods for fully nonlinear second order pdes based on a discrete hessian with applications to the monge ampere equation the purpose of this paper is twofold first we modify a method due to lakkis and pryer where the notion of a discrete hessian is introduced to compute fully nonlinear second order pdes the discrete hessian used in our approach is entirely local making the resulting linear system within the newton iteration much easier to solve the second contribution of this paper is to analyze both lakkis and pryer s method and its modification in parallel applied to the two dimensional monge ampere equation in both cases we show the well posedness of the methods as well as derive optimal error estimates numerical experiments are presented which i back up the theoretical findings and ii indicate that the methods are able to capture weak viscosity solutions

hamilton jacobi bellman quasi variational inequality arising in an environmental problem and its numerical discretization abstract a hamilton jacobi bellman quasi variational inequality hjbqvi for a river environmental restoration problem with wise use of sediment is formulated and its mathematical properties are analyzed a finite difference scheme with a penalization technique is then established for solving the hjbqvi the scheme is free from any iterative solvers and is unconditionally stable and convergent in the viscosity sense under certain conditions a demonstrative application example of the hjbqvi is finally presented

computation of nonclassical solutions to hamilton jacobi problems this paper is devoted to the construction of numerical methods for the approximation of nonclassical solutions to multidimensional hamilton jacobi equations for both scalar and vectorial problems recent theoretical results have yielded existence of solutions in many cases for which the usual viscosity approach was ill suited or not applicable the selection criterion used here is based on a viscoelasticity capillarity approach common in solid mechanics numerical methods adapted to this framework are built consistency of the model equation with the given selection criterion is essential it is achieved here through the use of high order finite difference schemes by considering applications to potential well problems the convergence of the methods are investigated

on the convergence of finite element methods for hamilton jacobi bellman equations we study the convergence of monotone p1 finite element methods on unstructured meshes for fully nonlinear hamilton jacobi bellman equations arising from stochastic optimal control problems with possibly degenerate isotropic diffusions using elliptic projection operators we treat discretizations which violate the consistency conditions of the framework by barles and souganidis we obtain strong uniform convergence of the numerical solutions and under nondegeneracy assumptions strong l 2 convergence of the gradients

weakly chained matrices policy iteration and impulse control this work is motivated by numerical solutions to hamilton jacobi bellman quasi variational inequalities hjbqvis associated with combined stochastic and impulse control problems in particular we consider i direct control ii penalized and iii semi lagrangian discretization schemes applied to the hjbqvi problem scheme i takes the form of a bellman problem involving an operator which is not necessarily contractive we consider the well posedness of the bellman problem and give sufficient conditions for convergence of the corresponding policy iteration to do so we use weakly chained diagonally dominant matrices which give a graph theoretic characterization of weakly diagonally dominant m matrices we compare schemes i iii under the following examples a optimal control of the exchange rate b optimal consumption with fixed and proportional transaction costs and c pricing guaranteed minimum withdrawal benefits in variable annuities we find that one should abstain from using sche