an adaptive spectral dg method for a reduced phase space based level set approach to geometrical optics on curved elements we present an adaptive spectral discontinuous galerkin dg method on curved elements to simulate high frequency wavefronts within a reduced phase space formulation of geometrical optics following recent work the approach is based on the use of level sets defined by functions satisfying the liouville equations in reduced phase space and in particular it relies on the smoothness of these functions to represent them by rapidly convergent spectral expansions in the phase variables the resulting hyperbolic system of equations for the coefficients in these expansions are then amenable to a high order accurate treatment via dg approximations in the present work we significantly expand on the applicability and efficiency of the approach by incorporating mechanisms that allow for its use in scattering simulations and for a reduced overall computational cost with regards to the former we demonstrate that the incorporation of curved elements is necessary to attain any kind of accuracy in calculations that involve scattering off non flat interfaces with regards to efficiency on the other hand we also show that the level set formulation allows for a space p adaptive scheme that under resolves the level set functions away from the wavefront without incurring in a loss of accuracy in the approximation of its location as we show these improvements enable simulations that are beyond the capabilities of previous implementations of these numerical procedures

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

numerical schemes for hamilton jacobi equations on unstructured meshes in this paper a numerical scheme is presented by applying the finite element method to the viscosity equations of the hamilton jacobi equations on unstructured meshes by improving the finite element scheme another numerical scheme is constructed under certain limitations the numerical solutions of the two schemes converge to the viscosity solutions of the hamilton jacobi equations the latter numerical scheme has weaker restrictions than the former scheme for convergence numerical examples are provided to test the stability convergence and sensitivity to different meshes

numerical homogenization of a second order discrete model for traffic flow the goal of this paper is to obtain a numerical approximation of the effective hamiltonian for a system of pdes deriving from a second order discrete model for traffic flow we will propose an explicit and an implicit discretization for this effective hamiltonian and we will provide the corresponding error estimates of crandall lions type finally we will also present some numerical simulations

on the maximum entropy negation of a probability distribution we suggest a transformation to obtain the negation of a probability distribution we investigate the properties of this negation using the dempster shafer theory of evidence we show of all the possible negations our proposed negation is one having a maximal type entropy

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

an eulerian method for capturing caustics a robust numerical method for the localization of caustics is proposed for general hamiltonians it is based on the direct resolution of a system of partial differential equations obtained through a local change of the time variable in the hamilton jacobi equation and complemented by a set of transport equations numerical results 1 to 3 d are presented

evolution implementation and application of level set and fast marching methods for advancing fronts abstract a variety of numerical techniques are available for tracking moving interfaces in this review we concentrate on techniques that result from the link between the partial differential equations that describe moving interfaces and numerical schemes designed for approximating the solutions to hyperbolic conservation laws this link gives rise to computational techniques for tracking moving interfaces in two and three space dimensions under complex speed laws we discuss the evolution of these techniques the fundamental numerical approximations involved implementation details and applications in particular we review some work on three aspects of materials sciences semiconductor process simulations seismic processing and optimal structural topology design

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

convex orderings in affine root systems ii abstract this paper contains several results concerning convex orderings in root systems these results complete the analysis developed in earlier papers by the author we obtain in the finite case a combinatorial characterization of reflections longest elements in standard parabolic subgroups of the weyl group of the root system and of coxeter elements in the affine case a refinement of the main construction done earlier by the author j algebra 172 1995 613 623 is described moreover infinite periodic compatible sets are classified

error estimates for a finite difference scheme associated with hamilton jacobi equations on a junction this paper is concerned with monotone time explicit finite difference schemes associated with first order hamilton jacobi equations posed on a junction they extend the schemes recently introduced by costeseque lebacque and monneau 2013 to general junction conditions on the one hand we prove the convergence of the numerical solution towards the viscosity solution of the hamilton jacobi equation as the mesh size tends to zero for general junction conditions on the other hand we derive optimal error estimates of order delta x frac 1 2 in l loc infty for junction conditions of optimal control type at least if the flux is strictly limited

an error estimate for a new scheme for mean curvature motion in this work we propose a new numerical scheme for the anisotropic mean curvature equation the solution of the scheme is not unique but for all numerical solutions we provide an error estimate between the continuous solution and the numerical approximation our scheme is also applicable to computing the solution to dislocation dynamics equations we also provide some numerical simulations

numerical approximation for a superreplication problem under gamma constraints we study a superreplication problem of european options with gamma constraints in mathematical finance the initially unbounded control problem is set back to a problem involving a viscosity pde solution with a set of bounded controls then a numerical approach is introduced unconditionally stable with respect to the mesh steps a generalized finite difference scheme is used since basic finite differences cannot work in our case numerical tests illustrate the validity of our approach

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

approximation of viscosity solutions of elliptic partial differential equations on minimal grids we obtain estimates on the minimal size of an equally spaced grid which can be used to construct monotone and consistent approximations to elliptic problems the estimates are given in terms of the ellipticity of the equation this problem is related to diophantine approximations see the appendix written by w m schmidt

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

error bounds for monotone approximation schemes for parabolic hamilton jacobi bellman equations we obtain nonsymmetric upper and lower bounds on the rate of convergence of general monotone approximation numerical schemes for parabolic hamilton jacobi bellman equations by introducing a new notion of consistency our results are robust and general they improve and extend earlier results by krylov barles and jakobsen we apply our general results to various schemes including crank nicholson type finite difference schemes splitting methods and the classical approximation by piecewise constant controls in the first two cases our results are new and in the last two cases the results are obtained by a new method which we develop here

twisted filtrations of soergel bimodules and linear rouquier complexes abstract we consider twisted standard filtrations of soergel bimodules associated to arbitrary coxeter groups and show that the graded multiplicities in these filtrations can be interpreted as structure constants in the hecke algebra this corresponds to the positivity of the polynomials occurring when expressing an element of the canonical basis in a generalized standard basis twisted by a biclosed set of roots in the sense of dyer and comes as a corollary of soergel s conjecture we then show the positivity of the corresponding inverse polynomials in the case where the biclosed set is an inversion set of an element or its complement by generalizing a result of elias and williamson on the linearity of the rouquier complexes associated to lifts of these basis elements in the artin tits group these lifts turn out to be generalizations of mikado braids as introduced in a joint work with digne this second positivity property generalizes a result of dyer and lehrer from finite to arbitrary coxeter groups

approximations for viscosity solutions of hamilton jacobi equations with locally varying time and space grids a new monotone finite difference scheme is introduced that approximates viscosity solutions of first order nonlinear hamilton jacobi equations the main feature of the scheme is that it allows for locally varying time and space grids making it ideal for use with adaptive algorithms explicit a priori error estimates are given to establish convergence numerical examples including adaptive mesh refinement examples demonstrate the effectiveness of the proposed scheme

properties of measures of information in evidence and possibility theories abstract an overview of information measures recently introduced by several authors in the settting of shafer s theory of evidence is proposed new results pertaining to additivity and monotonicity properties of these measures of information are presented the interpretation of each measure of information as opposed to others is discussed the potential usefulness of measures of specificity or imprecision is suggested and a principle of minimal specificity is stated for the purpose of reconstructing a body of evidence from incomplete knowledge

two views of belief belief as generalized probability and belief as evidence abstract belief functions are mathematical objects defined to satisfy three axioms that look somewhat similar to the kolmogorov axioms defining probability functions we argue that there are at least two useful and quite different ways of understanding belief functions the first is as a generalized probability function which technically corresponds to the inner measure induced by a probability function the second is as a way of representing evidence evidence in turn can be understood as a mapping from probability functions to probability functions it makes sense to think of updating a belief if we think of it as a generalized probability on the other hand it makes sense to combine two beliefs using say dempster s rule of combination only if we think of the belief functions as representing evidence many previous papers have pointed out problems with the belief function approach the claim of this paper is that these problems can be explained as a consequence of confounding these two views of belief functions

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

error estimates for approximate solutions to bellman equations associated with controlled jump diffusions we derive error estimates for approximate viscosity solutions of bellman equations associated to controlled jump diffusion processes which are fully nonlinear integro partial differential equations two main results are obtained i error bounds for a class of monotone approximation schemes which under some assumptions includes finite difference schemes and ii bounds on the error induced when the original levy measure is replaced by a finite measure with compact support an approximation process that is commonly used when designing numerical schemes for integro partial differential equations our proofs use and extend techniques introduced by krylov and barles jakobsen

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

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 convergent scheme for a non local hamilton jacobi equation modelling dislocation dynamics we study dislocation dynamics with a level set point of view the model we present here looks at the zero level set of the solution of a non local hamilton jacobi equation as a dislocation in a plane of a crystal the front has a normal speed depending on the solution itself we prove existence and uniqueness for short time in the set of continuous viscosity solutions we also present a first order finite difference scheme for the corresponding level set formulation of the model the scheme is based on monotone numerical hamiltonian proposed by osher and sethian the non local character of the problem makes it not monotone we obtain an explicit convergence rate of the approximate solution to the viscosity solution we finally provide numerical simulations

on the combinatorics of b b orbits on group compactifications abstract it is shown that there is an order isomorphism from the poset v of b b orbits on the wonderful compactification of a semi simple adjoint group g with weyl group w to an interval in reverse chevalley bruhat order on a non canonically associated coxeter group w in general neither finite nor affine moreover preserves the corresponding kazhdan lusztig polynomials springer s partly conjectural construction of kazhdan lusztig polynomials for the analogues of v for general coxeter groups w is completed by reducing it by a similar order isomorphism to known results involving a twisted chevalley bruhat order on w

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

spectral viscosity approximations to hamilton jacobi solutions the spectral viscosity approximate solution of convex hamilton jacobi equations with periodic boundary conditions is studied it is proved in this paper that the approximation and its gradient remain uniformly bounded formally spectral accurate and converge to the unique viscosity solution the l1 convergence rate of the order 1 varepsilon forall varepsilon 0 is obtained

hybrid control synthesis for eventuality specifications using level set methods this paper is concerned with the extraction of controllers for hybrid systems with respect to eventuality specifications given a hybrid system modelled by a hybrid automaton and a target set of states the objective is to compute the maximal set of initial states together with the hybrid control policy such that all the trajectories of the controlled system reach the target in finite time due to the existence of set valued disturbance inputs the problem is studied in a game theoretic framework having shown that a least restrictive solution does not exist we propose a dynamic programming algorithm that computes the maximal initial set and a controller with the desired property to implement the algorithm reachable sets of pursuit evasion differential games need to be computed for that reason level set methods are employed where the boundary of the reachable set is characterized as the zero level set of a hamilton jacobi equation the procedure for the numerical extraction of the controller is presented in detail and examples illustrate the methodology finally to demonstrate the practical character of our results a control design problem in the benchmark system of the batch evaporator is considered as an eventuality synthesis problem and solved using the proposed methodology

a matrix method of basic belief assignment s negation in dempster shafer theory negation is a new perspective to represent knowledge the negation of probability distribution has been proposed and it has a lot of interesting properties which can reach a maximum entropy because of the defects of the classical probability theory in the expression of uncertainty the basic belief assignment bba in the dempster shafer theory d s theory are widely used in decision theory thus negation provides a new perspective for d s theory to measure fuzziness in this paper a new definition of negation of bba is presented in the proposed negation bbas are represented as vectors and negation is realized by matrix operators this method has a good interpretation of the matrix operators and has the merit of simplifying the problem with several different definitions of entropy to determinate the uncertainty of bba the proposed negation of bba can reach a maximum belief entropy when the entropies satisfy a certain property

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

derivatives with respect to metrics and applications subgradient marching algorithm this paper introduces a subgradient descent algorithm to compute a riemannian metric that minimizes an energy involving geodesic distances the heart of the method is the subgradient marching algorithm to compute the derivative of the geodesic distance with respect to the metric the geodesic distance being a concave function of the metric this algorithm computes an element of the subgradient in o n 2 log n operations on a discrete grid of n points it performs a front propagation that computes a subgradient of a discrete geodesic distance we show applications to landscape modeling and to traffic congestion both applications require the maximization of geodesic distances under convex constraints and are solved by subgradient descent computed with our subgradient marching we also show application to the inversion of travel time tomography where the recovered metric is the local minimum of a non convex variational problem involving geodesic distances

a first order asymptotic preserving scheme for front propagation in a one dimensional kinetic reaction transport equation abstract in this work we propose an asymptotic preserving scheme for a nonlinear kinetic reaction transport equation in the regime of sharp interface with a nonlinear reaction term of kpp type a phenomenon of front propagation was proven in 9 this behaviour can be highlighted by considering a suitable hyperbolic limit of the kinetic equation using a hopf cole transform it was proven in 6 8 11 that the logarithm of the distribution function then converges to the viscosity solution of a constrained hamilton jacobi equation the hyperbolic scaling and the hopf cole transform make the kinetic equation stiff thus the numerical resolution of the problem is challenging since the standard numerical methods usually lead to high computational costs in these regimes asymptotic preserving ap schemes have typically been introduced to deal with this difficulty since they are designed to be stable along the transition to the macroscopic regime the scheme we propose is adapted to the non linearity of the problem enjoys a discrete maximum principle and solves the limit equation in the sense of viscosity it is based on a dedicated micro macro decomposition attached to the hopf cole transform as it is well adapted to the singular limit our scheme is able to cope with singular behaviours in space sharp interface and possibly in velocity concentration in the velocity distribution various numerical tests are proposed to illustrate the properties and the efficiency of our scheme

an evidential fusion approach for gender profiling cctv closed circuit television systems are broadly deployed in the present world to ensure in time reaction for intelligent surveillance it is a fundamental task for real world applications to determine the gender of people of interest however normal video algorithms for gender profiling usually face profiling have three drawbacks first the profiling result is always uncertain second the profiling result is not stable the degree of certainty usually varies over time sometimes even to the extent that a male is classified as a female and vice versa third for a robust profiling result in cases that a person s face is not visible other features such as body shape are required these algorithms may provide different recognition results at the very least they will provide different degrees of certainties to overcome these problems in this paper we introduce an dempster shafer ds evidential approach that makes use of profiling results from multiple algorithms over a period of time in particular denoeux s cautious rule is applied for fusing mass functions through time lines experiments show that this approach does provide better results than single profiling results and classic fusion results furthermore it is found that if severe mis classification has occurred at the beginning of the time line the combination can yield undesirable results to remedy this weakness we further propose three extensions to the evidential approach proposed above incorporating notions of time window time attenuation and time discounting respectively these extensions also applies denoeux s rule along with time lines and take the ds approach as a special case experiments show that these three extensions do provide better results than their predecessor when mis classifications occur

3d level set methods for evolving fronts on tetrahedral meshes with adaptive mesh refinement the level set method is commonly used to model dynamically evolving fronts and interfaces in this work we present new methods for evolving fronts with a specified velocity field or in the surface normal direction on 3d unstructured tetrahedral meshes with adaptive mesh refinement amr the level set field is located at the nodes of the tetrahedral cells and is evolved using new upwind discretizations of hamiltonjacobi equations combined with a rungekutta method for temporal integration the level set field is periodically reinitialized to a signed distance function using an iterative approach with a new upwind gradient the details of these level set and reinitialization methods are discussed results from a range of numerical test problems are presented

continuous dependence results for non linear neumann type boundary value problems abstract we obtain estimates on the continuous dependence on the coefficient for second order non linear degenerate neumann type boundary value problems our results extend previous work of cockburn et al jakobsen and karlsen and gripenberg to problems with more general boundary conditions and domains a new feature here is that we account for the dependence on the boundary conditions as one application of our continuous dependence results we derive for the first time the rate of convergence for the vanishing viscosity method for such problems we also derive new explicit continuous dependence on the coefficients results for problems involving bellman isaacs equations and certain quasilinear equation

a finite difference scheme for option pricing in jump diffusion and exponential levy models we present a finite difference method for solving parabolic partial integro differential equations with possibly singular kernels which arise in option pricing theory when the random evolution of the underlying asset is driven by a levy process or more generally a time inhomogeneous jump diffusion process we discuss localization to a finite domain and provide an estimate for the localization error under an integrability condition on the levy measure we propose an explicit implicit finite difference scheme which can be used to price european and barrier options in such models we study stability and convergence of the scheme proposed and under additional conditions provide estimates on the rate of convergence numerical tests are performed with smooth and nonsmooth initial conditions

ecm an evidential version of the fuzzy c means algorithm a new clustering method for object data called ecm evidential c means is introduced in the theoretical framework of belief functions it is based on the concept of credal partition extending those of hard fuzzy and possibilistic ones to derive such a structure a suitable objective function is minimized using an fcm like algorithm a validity index allowing the determination of the proper number of clusters is also proposed experiments with synthetic and real data sets show that the proposed algorithm can be considered as a promising tool in the field of exploratory statistics

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

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 adaptive method with rigorous error control for the hamilton jacobi equations part ii the two dimensional steady state case in this paper we devise and study an adaptive method for finding approximations to the viscosity solution of hamilton jacobi equations the method which is an extension to two space dimensions of a similar method previously proposed for one space dimension is studied in the framework of steady state hamilton jacobi equations with periodic boundary conditions it seeks numerical approximations whose l distance to the viscosity solution is no bigger than a prescribed tolerance a thorough numerical study is carried out which shows that a strict error control is achieved and that the method exhibits an optimal computational complexity which does not depend on the value of the tolerance or on the type of hamiltonian

numerical schemes for the hamilton jacobi and level set equations on triangulated domains borrowing from techniques developed for conservation law equations numerical schemes which discretize the hamilton jacobi h j level set and eikonal equations on triangulated domains are presented the first scheme is a provably monotone discretization for the h j equations unfortunately the basic scheme lacks lipschitz continuity of the numerical hamiltonian by employing a virtual edge flipping technique local lipschitz continuity of the numerical flux is restored on acute triangulations next schemes are introduced and developed based on the weaker concept of positive coefficient approximations for homogeneous hamiltonians these schemes possess a discrete maximum principle on arbitrary triangulations and exhibit local lipschitz continuity of the numerical hamiltonian finally a class of petrov galerkin approximations is considered these schemes are stabilized via a least squares bilinear form the petrov galerkin schemes do not possess a discrete maximum principle but generalize to high order accuracy discretization of the level set equation also requires the numerical approximation of a mean curvature term a simple mass lumped galerkin approximation is presented in section 6 and analyzed using maximum principle analysis the use of unstructured meshes permits several forms of mesh adaptation which have been incorporated into numerical examples these numerical examples include discretizations of convex and nonconvex forms of the h j equation the eikonal equation and the level set equation

numerical schemes for conservation laws via hamilton jacobi equations we present some difference approximation schemes which converge to the entropy solution of a scalar conservation law having a convex flux the numerical methods described here take their origin from approximation schemes for hamilton jacobi bellman equations related to optimal control problems and exhibit several interesting features the convergence result still holds for quite arbitrary time steps the main assumption for convergence can be interpreted as a discrete analogue of oleinik s entropy condition numerical diffusion around the shocks is very limited some tests are included in order to compare the performances of these methods with other classical methods godunov tvd

adjoint and compensated compactness methods for hamilton jacobi pde we investigate the vanishing viscosity limit for hamilton jacobi pde with nonconvex hamiltonians and present a new method to augment the standard viscosity solution approach the main idea is to introduce a solution e of the adjoint of the formal linearization and then to integrate by parts with respect to the density e this procedure leads to a natural phase space kinetic formulation and also to a new compensated compactness technique

decision templates for multiple classifier fusion an experimental comparison abstract multiple classifier fusion may generate more accurate classification than each of the constituent classifiers fusion is often based on fixed combination rules like the product and average only under strict probabilistic conditions can these rules be justified we present here a simple rule for adapting the class combiner to the application c decision templates one per class are estimated with the same training set that is used for the set of classifiers these templates are then matched to the decision profile of new incoming objects by some similarity measure we compare 11 versions of our model with 14 other techniques for classifier fusion on the satimage and phoneme datasets from the database elena our results show that decision templates based on integral type measures of similarity are superior to the other schemes on both data sets