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

a parallel two scale method for eikonal equations numerous applications of eikonal equations prompted the development of many efficient numerical algorithms the heap cell method hcm is a recent serial two scale technique that has been shown to have advantages over other serial state of the art solvers for a wide range of problems a chacon and a vladimirsky siam j sci comput 34 2012 pp a547 a578 this paper presents a parallelization of hcm for a shared memory architecture the numerical experiments in r 3 show that the parallel hcm exhibits good algorithmic behavior and scales well resulting in a very fast and practical solver

closed form solution of local shape from shading at critical points in the theory of shape from shading behaviours of the local solution around a critical point of the image play an important role this paper shows that the second derivatives of the object surface can be locally determined at these image critical points closed form expressions of the surface second derivatives in terms of the second derivatives of the image brightness and of the reflectance map are shown they are derived as follows by differentiating the image irradiance equation twice at an image critical point a set of polynomial equations is obtained that contains the second derivatives of the surface of the image brightness and of the reflectance map regarding these equations as simultaneous equations for unknown surface second derivatives they are algebraically solved and their explicit expressions are derived such a derivation is possible only at image critical points and is impossible at any other image point the applicability of the derived expressions to noisy images is tested using synthetic images

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

active contours without edges we propose a new model for active contours to detect objects in a given image based on techniques of curve evolution mumford shah 1989 functional for segmentation and level sets our model can detect objects whose boundaries are not necessarily defined by the gradient we minimize an energy which can be seen as a particular case of the minimal partition problem in the level set formulation the problem becomes a mean curvature flow like evolving the active contour which will stop on the desired boundary however the stopping term does not depend on the gradient of the image as in the classical active contour models but is instead related to a particular segmentation of the image we give a numerical algorithm using finite differences finally we present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable also the initial curve can be anywhere in the image and interior contours are automatically detected

nonconvex degenerate hamilton jacobi equations in this paper we prove a characterization of the maximal viscosity subsolution for a class of hamilton jacobi equations coupled with dirichlet boundary condition for which uniqueness of the viscosity solution does not necessarily hold

eikonal equation based front propagation for arbitrary complex configurations this paper presents a front propagation method using the eikonal equation 1 in which represents the smallest euclidean distance field to the front to be propagated the offset capturing approach consists in first calculating the field over a uniform cartesian grid fully covering the front to be propagated and then constructing the iso curves or surfaces as the propagated result the calculation of uses a 3d numerical scheme the fast sweeping scheme validation for accuracy of the method is presented using academic test cases a real 3d industry application draft tube with two piers is successfully propagated and demonstrated using special boundary conditions to cope with inlet and outlet planes during front propagtion this application involves the propagation of a front that exhibits both concave and convex shape regions sharp corners and local tangent plane surface discontinuities as well as a multi connected domain copyright 2007 john wiley sons ltd

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

sobolev active contours all previous geometric active contour models that have been formulated as gradient flows of various energies use the same l 2 type inner product to define the notion of gradient recent work has shown that this inner product induces a pathological riemannian metric on the space of smooth curves however there are also undesirable features associated with the gradient flows that this inner product induces in this paper we reformulate the generic geometric active contour model by redefining the notion of gradient in accordance with sobolev type inner products we call the resulting flows sobolev active contours sobolev metrics induce favorable regularity properties in their gradient flows in addition sobolev active contours favor global translations but are not restricted to such motions they are also less susceptible to certain types of local minima in contrast to traditional active contours these properties are particularly useful in tracking applications we demonstrate the general methodology by reformulating some standard edge based and region based active contour models as sobolev active contours and show the substantial improvements gained in segmentation

shape from shading viscosity solutions and edges this article deals with the so called shape from shading problem which arises when recovering a shape from a single image the general case of a distribution of light sources illuminating a lambertian surface is considered this involves original definitions of three types of edges mainly the apparent contours the grazing light edges and the shadow edges the elevation of the shape is expressed in terms of viscosity solution of a first order hamilton jacobi equation with various boundary conditions on these edges various existence and uniqueness results are presented

an hopf lax formula for a class of measurable hamilton jacobi equations abstract we consider the cauchy problem u t h x du 0 x t r n 0 u x 0 u 0 x x r n where h is measurable in x continuous convex and positive homogeneous in p we adapt the definition of viscosity solution to the measurable framework and we prove that the unique viscosity solution is given by a representation formula of hopf lax type

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

nodal solutions for second order m point boundary value problems with nonlinearities across several eigenvalues abstract we study the existence of nodal solutions of the m point boundary value problem u f u 0 0 t 1 u 0 0 u 1 i 1 m 2 i u i where i q i 1 2 m 2 with 0 1 2 m 2 1 and i r i 1 2 m 2 with i 0 and 0 i 1 m 2 i 1 we give conditions on the ratio f s s at infinity and zero that guarantee the existence of nodal solutions the proofs of our main results are based upon bifurcation techniques

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

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

hamilton jacobi equations on a manifold and applications to grid generation or refinement in this paper we consider hamilton jacobi equations on a manifold typically on the graph of some previously computed function z x y and we show how the corresponding level set method allows us to generate and or to refine a mesh in regions where this function z has large derivatives such as it is the method needs to be strongly improved and accelerated but the principle is awfully natural and in principle the method is fully automatic similar ideas are also useful in image processing in particular for the active contours method

implicit brain imaging we describe how implicit surface representations can be used to solve fundamental problems in brain imaging this kind of representation is not only natural following the state of the art segmentation algorithms reported in the literature to extract the different brain tissues but it is also as shown in this paper the most appropriate one from the computational point of view examples are provided for finding constrained special curves on the cortex such as sulcal beds regularizing surface based measures such as cortical thickness and for computing warping fields between surfaces such as the brain cortex all these result from efficiently solving partial differential equations pdes and variational problems on surfaces represented in implicit form the implicit framework avoids the need to construct intermediate mappings between 3 d anatomical surfaces and parametric objects such planes or spheres a complex step that introduces errors and is required by many other cortical processing approaches

an adaptive grid scheme for the discrete hamilton jacobi bellman equation in this paper an adaptive finite difference scheme for the solution of the discrete first order hamilton jacobi bellman equation is presented local a posteriori error estimates are established and certain properties of these estimates are proved based on these estimates an adapting iteration for the discretization of the state space is developed an implementation of the scheme for two dimensional grids is given and numerical examples are discussed

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

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

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

local or global minima flexible dual front active contours most variational active contour models are designed to find local minima of data dependent energy functionals with the hope that reasonable initial placement of the active contour will drive it toward a desirable local minimum as opposed to an undesirable configuration due to noise or complex image structure as such there has been much research into the design of complex region based energy functionals that are less likely to yield undesirable local minima when compared to simpler edge based energy functionals whose sensitivity to noise and texture is significantly worse unfortunately most of these more robust region based energy functionals are applicable to a much narrower class of imagery compared to typical edge based energies due to stronger global assumptions about the underlying image data devising new implementation algorithms for active contours that attempt to capture more global minimizers of already proposed image based energies would allow us to choose an energy that makes sense for a particular class of energy without concern over its sensitivity to local minima such implementations have been proposed for capturing global minima however sometimes the completely global minimum is just as undesirable as a minimum that is too local in this paper we propose a novel fast and flexible dual front implementation of active contours motivated by minimal path techniques and utilizing fast sweeping algorithms which is easily manipulated to yield minima with variable degrees of localness and globalness by simply adjusting the size of active regions the ability to gracefully move from capturing minima that are more local according to the initial placement of the active contour surface to minima that are more global allows this model to more easily obtain desirable minimizers which often are neither the most local nor the most global experiments on various 2d and 3d images and comparisons with some active contour models and region growing methods are also given to illustrate the properties of this model and its performance in a variety of segmentation applications

spectral properties of p laplacian problems with neumann and mixed type multi point boundary conditions abstract we consider the boundary value problem consisting of the p laplacian equation 1 p u p u on 1 1 where p 1 p s s p 1 sgn s for s r r together with the multi point boundary conditions 2 p u 1 i 1 m i p u i or 3 u 1 i 1 m i u i or a mixed pair of these conditions with one condition holding at each of x 1 and x 1 in 2 3 m 1 are integers i 1 1 1 i m and the coefficients i satisfy i 1 m i 1 we term the conditions 2 and 3 respectively neumann type and dirichlet type boundary conditions since they reduce to the standard neumann and dirichlet boundary conditions when 0 given a suitable pair of boundary conditions a number is an eigenvalue of the corresponding boundary value problem if there exists a non trivial solution u an eigenfunction the spectrum of the problem is the set of eigenvalues in this paper we obtain various spectral properties of these eigenvalue problems we then use these properties to prove rabinowitz type global bifurcation theorems for related bifurcation problems and to obtain nonresonance conditions in terms of the eigenvalues for the solvability of related inhomogeneous problems

linear and nonlinear second order problems with sturm liouville type multi point boundary conditions abstract we consider the nonlinear boundary value problem consisting of the equation u f u on 1 1 where f r r is continuous together with general sturm liouville type multi point boundary conditions at 1 we obtain rabinowitz type global bifurcation results and then use these to obtain nodal solutions of the problem we conclude with a nonresonance result for an inhomogeneous form of the problem these results rely on the spectral properties of the eigenvalue problem consisting of the equation u u on 1 1 together with the multi point boundary conditions in a previous paper it was shown that under certain optimal conditions the basic spectral properties of this eigenvalue problem are similar to those of the standard sturm liouville problem with single point boundary conditions in particular for each integer k 0 there exists a unique simple eigenvalue k whose eigenfunctions have oscillation count equal to k where the oscillation count was defined in terms of a complicated prufer angle construction unfortunately it seems to be difficult to apply the prufer angle construction to the nonlinear problem accordingly in this paper we use alternative non optimal oscillation counting methods to obtain the required spectral properties of the linear problem and these are then applied to the nonlinear problem to yield the results mentioned above

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

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

p laplacian problems with jumping nonlinearities we consider the p laplacian boundary value problem 1 p u x f x u x u x a e x 0 1 2 c00u 0 c01u 0 c10u 1 c11u 1 where p 1 is a fixed number p s s p 2s s r and for each j 0 1 cj0 cj1 0 the function f 0 1 r2 r is a caratheodory function satisfying for x s t 0 1 r2 x p s e x s t f x s t x p s e x s t s 0 where l1 0 1 and e has the form e x s t x e s t with l1 0 1 0 e 0 and limr e r r1 p 0 this condition allows the nonlinearity in 1 to behave differently as u such a nonlinearity is often termed jumping related to 1 2 is the problem 3 p u a p u b p u p u in 0 1 together with 2 where a b l1 0 1 r and u x max u x 0 for x 0 1 this problem is positively homogeneous and jumping values of for which 2 3 has a nontrivial solution u will be called half eigenvalues while the corresponding solutions u will be called half eigenfunctions we show that a sequence of half eigenvalues exists the corresponding half eigenfunctions having certain nodal properties and we obtain certain spectral and degree theoretic properties of the set of half eigenvalues these properties lead to existence and nonexistence results for the problem 1 2 we also consider a related bifurcation problem and obtain a global bifurcation result similar to the well known rabinowitz global bifurcation theorem this then leads to a multiplicity result for 1 2 when the functions a and b are constant the set of half eigenvalues is closely related to the fucik spectrum of the problem and equivalent solvability results are obtained using the two approaches however when a and b are not constant the half eigenvalue approach yields stronger results

reinitialization free level set evolution via reaction diffusion this paper presents a novel reaction diffusion rd method for implicit active contours that is completely free of the costly reinitialization procedure in level set evolution lse a diffusion term is introduced into lse resulting in an rd lse equation from which a piecewise constant solution can be derived in order to obtain a stable numerical solution from the rd based lse we propose a two step splitting method to iteratively solve the rd lse equation where we first iterate the lse equation then solve the diffusion equation the second step regularizes the level set function obtained in the first step to ensure stability and thus the complex and costly reinitialization procedure is completely eliminated from lse by successfully applying diffusion to lse the rd lse model is stable by means of the simple finite difference method which is very easy to implement the proposed rd method can be generalized to solve the lse for both variational level set method and partial differential equation based level set method the rd lse method shows very good performance on boundary antileakage the extensive and promising experimental results on synthetic and real images validate the effectiveness of the proposed rd lse approach

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

error estimation and adaptive discretization for the discrete stochastic hamilton jacobi bellman equation generalizing an idea from deterministic optimal control we construct a posteriori error estimates for the spatial discretization error of the stochastic dynamic programming method based on a discrete hamilton jacobi bellman equation these error estimates are shown to be efficient and reliable furthermore a priori bounds on the estimates depending on the regularity of the approximate solution are derived based on these error estimates we propose an adaptive space discretization scheme whose performance is illustrated by two numerical examples

some multipoint boundary value problems of neumann dirichlet type involving a multipoint p laplace like operator abstract let and be two increasing homeomorphisms from r onto r with 0 0 0 0 let f 0 1 r r r be a function satisfying caratheodory s conditions and for each i i 1 2 m 2 let a i r r be a continuous function with i 1 m 2 a i 0 1 i 0 1 0 1 2 m 2 1 in this paper we first prove a suitable continuation lemma of leray schauder type which we use to obtain several existence results for the m point boundary value problem u f t u u t 0 1 u 0 0 u 1 i 1 m 2 u i a i u i we note that this problem is at resonance in the sense that the associated m point boundary value problem u t 0 t 0 1 u 0 0 u 1 i 1 m 2 u i a i u i has the non trivial solution u t where r is an arbitrary constant vector in view of the assumption i 1 m 2 a i 0 1

numerical approximation of the maximal solutions for a class of degenerate hamilton jacobi equations in this paper we study an approximation scheme for a class of hamilton jacobi problems for which uniqueness of the viscosity solution does not hold this class includes the eikonal equation arising in the shape from shading problem we show that if an appropriate stability condition is satisfied the scheme converges to the maximal viscosity solution of the problem furthermore we give an estimate for the discretization error

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 dispersive effective medium for wave propagation in periodic composites the problem of wave propagation through a periodic medium is considered it is assumed that the ratio between the cell size and the shortest wavelength of the initial disturbance is small within this regime an effective medium model of the propagation phenomena is obtained using the bloch expansion the effective medium obtained is shown to be dispersive it is known that a dominant feature of the transient wave train in a periodic medium is that of dispersion it is shown that our model recovers homogenization formally when the cell size approaches zero explicit formula for the constants appearing in the effective medium equation is derived for the case of one dimension the accuracy of the resulting approximation is assessed in numerical simulations

a calculus for finely layered anisotropic media matrix algebra and group theory combine to offer a formalism for the simple calculation of the elastic anisotropic homogeneous medium which is equivalent in the long wavelength limit to a heterogeneous distribution of fine layers each layer itself an elastic anisotropic medium the properties of each anisotropic constituent in a set of fine layers map to an element of a commutative group a reverse mapping returns the material properties of the constituent adding group elements gives the group element for the homogeneous medium equivalent to a heterogeneous set of layers addition of an inverse element that is subtraction provides the means to remove a set of layers from an anisotropic medium then if the remaining layer is a stable anisotropic medium a valid decomposition of the original medium into anisotropic constituents is obtained within the group structure eight subgroups corresponding to eight types of elastic symmetry systems may be identified immediately yielding the symmetry of the e

viscoelastic stiffness tensor of anisotropic media from oscillatory numerical experiments a finely layered media behaves as an anisotropic medium when the dominat wavelengths are much larger than the layer thickness if the constituent are anelastic a generalization of backus averaging predicts that the medium is effectively a transversely isotropic viscoelastic tiv medium to test and validate the theory we present a novel procedure to determine the complex and frequency dependent stiffness components of a tiv medium the methodology consists in performing numerical compressibility and shear harmonic tests on a representative sample of the material these tests are described by a collection of non coercive elliptic boundary value problems formulated in the space frequency domain which are solved using a galerkin finite element procedure results on the existence and uniqueness of the continuous and discrete problems as well as optimal error estimates for the galerkin finite element method are derived numerical examples illustrates the implementation of the numerical oscillatory tests to determine the set of complex and frequency dependent effective tiv coefficients and the associated phase velocities and quality factors for a periodic sequence of epoxy and glass thin layers the results are compared to the analytical phase velocities and quality factors predicted by the backus carcione theory

non resonance conditions for semilinear sturm liouville problems with jumping non linearities abstract we consider the sturm liouville boundary value problem p x u x q x u x f x u x h x x 0 c00u 0 c01u 0 0 c10u c11u 0 where p c1 0 q c0 0 with p x 0 x 0 c2i0 c2i1 0 i 0 1 h l2 0 and f 0 r r is a caratheodory function we assume that the rate of growth of f x is at most linear as but the asymptotic behaviour may be different as so the non linearity is termed jumping conditions for existence of solutions of this problem are usually expressed in terms of non resonance with respect to the standard fucik spectrum in this paper we give conditions for both existence and non existence of solutions in terms of a slightly different idea of the spectrum these conditions extend the usual fucik spectrum conditions

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

godunov type numerical methods for a model of granular flow we propose and analyze finite volume godunov type methods based on discontinuous flux for a 2 2 system of non linear partial differential equations proposed by hadeler and kuttler to model the dynamics of growing sandpiles generated by a vertical source on a flat bounded rectangular table the scheme is made well balanced by modifying the flux function locally by including source term as a part of the convection term its extension to multi dimensions is not straightforward for which an approach has been introduced here based on transport rays this approach is compared with another approach for inclusion of source term which uses the idea of inverting the divergence operator relying on the curl free component of the helmholtz decomposition of the source term numerical experiments are presented to illustrate the efficiency of the proposed scheme for both unsteady and steady state calculations and to make comparisons with the previously studied finite difference and semi lagrangian approaches by falcone and finzi vita

exit time problems in optimal control and vanishing viscosity method the authors study the connections between deterministic exit time control problems and possibly discontinuous viscosity solutions of a first order hamilton jacobi hj equation up to the boundary this equation admits a maximum and a minimum solution that are the value functions associated to stopping time problems on the boundary when these solutions are equal they can be obtained through the vanishing viscosity method finally when the hj equation has a continuous solution it is proved to be the value function for the first exit time of the domain it is also the vanishing viscosity limit arising in particular in some large deviations problems

fast legendre fenchel transform and applications to hamilton jacobi equations and conservation laws we are interested in the study of a fast algorithm introduced by brenier computing the discrete legendre fenchel transform of a real function we present convergence results and show how the order of convergence grows with the regularity of the function to be transformed applications to hamilton jacobi equations for front propagation problems and conservation laws are presented

an efficient interface preserving level set redistancing algorithm and its application to interfacial incompressible fluid flow in sussman smereka and osher j comp phys 94 1994 pp 146 159 a numerical scheme was presented for computing incompressible air water flows using the level set method crucial to the above method was a new iteration method for maintaining the level set function as the signed distance from the zero level set in this paper we implement a constraint along with higher order difference schemes in order to make the iteration method more accurate and efficient accuracy is measured in terms of the new computed signed distance function and the original level set function having the same zero level set we apply our redistancing scheme to incompressible flows with noticeably better resolved results at reduced cost we validate our results with experiment and theory we show that our distance level set scheme with the added constraint competes well with available interface tracking schemes for basic advection of an interface we perform basic accuracy checks and more stringent tests involving complicated interfacial structures as with all level set schemes our method is easy to implement

global minimum for active contour models a minimal path approach a new boundary detection approach for shape modeling is presented it detects the global minimum of an active contour model s energy between two end points initialization is made easier and the curve is not trapped at a local minimum by spurious edges we modify the snake energy by including the internal regularization term in the external potential term our method is based on finding a path of minimal length in a riemannian metric we then make use of a new efficient numerical method to find this shortest path it is shown that the proposed energy though based only on a potential integrated along the curve imposes a regularization effect like snakes we explore the relation between the maximum curvature along the resulting contour and the potential generated from the image the method is capable to close contours given only one point on the objects boundary by using a topology based saddle search routine we show examples of our method applied to real aerial and medical images

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

a numerical upscaling procedure to estimate effective plane wave and shear moduli in heterogeneous fluid saturated poroelastic media an important loss effect in heterogeneous poroelastic biot media is the dissipation mechanism due to wave induced fluid flow caused by mesoscopic scale heterogeneities which are larger than the pore size but much smaller than the predominant wavelengths of the fast compressional and shear waves these heterogeneities can be due to local variations in lithological properties or to patches of immiscible fluids for example a fast compressional wave traveling across a porous rock saturated with water and patches of gas induces a smaller fluid pressure in the gas patches than in the water saturated parts of the material this in turn generates fluid flow and slow biot waves which diffuse away from the gas water interfaces generating significant energy losses and velocity dispersion to perform numerical simulations using biot s equations of motion it would be necessary to employ extremely fine meshes to properly represent these mesoscopic heterogeneities and their attenuation effects on the fast waves an alternative approach to model wave propagation in these type of biot media is to employ a numerical upscaling procedure to determine effective complex p wave and shear moduli defining locally a viscoelastic medium having in the average the same properties than the original biot medium in this work the complex p wave and shear moduli in heterogeneous fluid saturated porous media are obtained using numerical gedanken experiments in a monte carlo fashion the experiments are defined as local boundary value problems on a reference representative volume of bulk material containing stochastic heterogeneities characterized by their statistical properties these boundary value problems represent compressibility and shear tests needed to determine these moduli for a given realization the average and variance of the phase velocities and quality factors associated with these moduli are obtained by averaging over realizations of the stochastic parameters the monte carlo realizations were stopped when the variance of the computed quantities stabilized at an almost constant value the approximate solution of the local boundary value problems was obtained using a galerkin finite element procedure and the method was validated by reproducing known solutions in the case of periodic layered media for the spatial discretization standard bilinear finite element spaces are employed for the solid phase while for the fluid phase the vector part of the raviart thomas nedelec mixed finite element space of order zero was used results on the uniqueness of the continuous and discrete problems as well as optimal a priori error estimates for the galerkin finite element procedure are derived numerical experiments showing the implementation of the procedure to estimate the average and variance of the fast compressional and shear phase velocities and inverse quality factors in these kind of highly heterogeneous fluid saturated porous media are presented

existence of multiple positive solutions for nonlinear m point boundary value problems abstract in this paper we afford some sufficient conditions to guarantee the existence of multiple positive solutions for the nonlinear m point boundary value problem for the one dimensional p laplacian p u a t f t u 0 t 0 1 u 0 0 u 1 i 1 m 2 a i u i

half eigenvalues and the fucik spectrum of multi point boundary value problems abstract we consider the nonlinear boundary value problem consisting of the equation 1 u f u h a e on 1 1 where h l 1 1 1 together with the multi point dirichlet type boundary conditions 2 u 1 i 1 m i u i where m 1 are integers 1 m 0 1 m 1 1 m and we suppose that i 1 m i 1 we also suppose that f r r is continuous and 0 f lim s f s s we assume that these limits exist we allow f f such a nonlinearity f is said to be jumping related to 1 is the equation 3 u a u b u on 1 1 where a b 0 and u x max u x 0 for x 1 1 the problem 2 3 is positively homogeneous and jumping regarding a b as fixed values of a b for which 2 3 has a non trivial solution u will be called half eigenvalues while the corresponding solutions u will be called half eigenfunctions we show that a sequence of half eigenvalues exists the corresponding half eigenfunctions having specified nodal properties and we obtain certain spectral and degree theoretic properties of the set of half eigenvalues these properties lead to solvability and non solvability results for the problem 1 2 the set of half eigenvalues is closely related to the fucik spectrum of the problem which we briefly describe equivalent solvability and non solvability results for 1 2 are obtained from either the half eigenvalue or the fucik spectrum approach

spectral properties of second order multi point p laplacian boundary value problems abstract we consider the multi point boundary value problem p u p u on 1 1 u 1 i 1 m i u i where p 1 p s s p 1 sgn s for s r r m 1 are integers i 1 1 1 i m and the coefficients i satisfy i 1 m i 1 a number r is said to be an eigenvalue of the above problem if there exists a non trivial solution u the spectrum is the set of eigenvalues in this paper we obtain some basic spectral and degree theoretic properties of this eigenvalue problem these results have numerous applications to more general problems as an example a rabinowitz type global bifurcation theorem is briefly described

second order three point boundary value problems with jumping non linearities abstract we consider the non linear three point boundary value problem consisting of the equation 1 u f u h a e on 0 1 where h l 1 0 1 together with the boundary conditions 2 u 0 0 u 1 u where 0 1 the function f r r is continuous and we assume that the following limits exist and are finite f lim s f s s we allow f f such a non linearity f is said to be jumping related to 1 is the equation 3 u a u b u u on 0 1 where a b r and u x max u x 0 for x 0 1 the problem 2 and 3 is positively homogeneous and jumping values of for which 2 and 3 has a non trivial solution u will be called half eigenvalues while the corresponding solutions u will be called half eigenfunctions we show that a sequence of half eigenvalues exists the corresponding half eigenfunctions having specified nodal properties and we obtain certain spectral and degree theoretic properties of the set of half eigenvalues these properties lead to solvability and non solvability results for the problem 1 and 2 the set of half eigenvalues is closely related to the fucik spectrum of the problem and equivalent solvability and non solvability results for 1 and 2 are obtained from either the half eigenvalue or the fucik spectrum approach

second order multi point problems with variable coefficients abstract in this paper we consider the eigenvalue problem consisting of the equation u r u on 1 1 where r c 1 1 1 r 0 and r together with the multi point boundary conditions u 1 i 1 m i u i where m 1 are integers and for i 1 m i r i 1 1 with i 1 i 1 we show that if the coefficients i r are sufficiently small depending on r then the spectral properties of this problem are similar to those of the usual separated problem but if the coefficients i are not sufficiently small then these standard spectral properties need not hold the spectral properties of such multi point problems have been obtained before for the constant coefficient case r 1 but the variable coefficient case has not been considered previously apart from the existence of principal eigenvalues some nonlinear multi point problems are also considered we obtain a partial rabinowitz type result on global bifurcation from the eigenvalues and various nonresonance conditions for the existence of general solutions and also of nodal solutions these results rely on the spectral properties of the linear problem

spectral properties and nodal solutions for second order m point boundary value problems abstract we consider the m point boundary value problem consisting of the equation u f u on 0 1 where f r r is c 1 with f 0 0 together with the boundary conditions u 0 0 u 1 i 1 m 2 i u i where m 3 i 0 1 and i 0 for i 1 m 2 with i 1 m 2 i 1 we first show that the spectral properties of the linearisation of this problem are similar to the well known properties of the standard sturm liouville problem with separated boundary conditions with a minor modification to deal with the multi point boundary condition these spectral properties are then used to prove a rabinowitz type global bifurcation theorem for a bifurcation problem related to the above problem finally we use the global bifurcation theorem to obtain nodal solutions that is sign changing solutions with a specified number of zeros of the above problem under various conditions on the asymptotic behaviour of f