projection methods coupled to level set interface techniques in this work we consider hydrodynamic problems with cold flame propagation by merging a second order projection method for viscous navier stokes equations with modern techniques for computing the motion of interfaces propagating with curvature dependent speeds this is part of our efforts to try and approximate the solution of a simplified model of turbulent combustion results are given for a simple model of a flame burning in driven cavities and shear layers

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

redistancing by flow of time dependent eikonal equation construction of signed distance to a given interface is a topic of special interest to level set methods there are currently however few algorithms that can efficiently produce highly accurate solutions we introduce an algorithm for constructing an approximate signed distance function through manipulation of values calculated from flow of time dependent eikonal equations we provide operation counts and experimental results to show that this algorithm can efficiently generate solutions with a high order of accuracy comparison with the standard level set reinitialization algorithm shows ours is superior in terms of predictability and local construction which for example are critical in local level set methods we further apply the same ideas to extension of values off interfaces together our proposed approaches can be used to advance the level set method for fast and accurate computations of the latest scientific problems

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

the credible regions on the approximate stability boundaries of nonlinear dynamic systems the stability boundary of a nonlinear dynamic system may often be constructed from stable manifolds of type 1 equilibrium points these stable manifolds are typically approximated numerically with only local validity a numerical method is proposed based on 1 and 2 d characteristic invariant manifolds cims which lie on the stable manifold and can be computed with greater accuracy than approximations to the entire manifold because of their lower dimension to estimate credible regions in which the mismatch between the approximate and actual boundaries is below a threshold

a viscosity solutions approach to shape from shading the problem of recovering a lambertian surface from a single two dimensional image may be written as a first order nonlinear equation which presents the disadvantage of having several continuous and even smooth solutions a new approach based on hamilton jacobi bellman equations and viscosity solutions theories enables one to study non uniqueness phenomenon and thus to characterize the surface among the various solutions a consistent and monotone scheme approximating the surface is constructed thanks to the dynamic programming principle and numerical results are presented

a method for convex curve approximation abstract in this paper a new sandwich method is introduced to approximate a convex curve in r2 this method requires only function evaluation and the solution of a number of scalar optimization problems a quadratic convergence property of the method is established that is the total number of optimization problems solved is bounded by a constant multiple of the square root of the inverse of the given error an application to approximation of the efficient frontier of a bi criteria convex quadratic network program is given

approximation methods for non convex curves based on the triangle sandwich and block sandwich methods two approximation methods for non convex curves are considered in this paper numerical algorithms corresponding to these methods are developed and quadratic convergence properties of two methods in the original space are established application is given to the approximation of an efficient frontier of a bi criteria optimization problem

crystal growth and dendritic solidification we present a numerical method which computes the motion of complex solid liquid boundaries in crystal growth the model we solve includes physical effects such as crystalline anisotropy surface tension molecular kinetics and undercooling the method is based on two ideas first the equations of motion are recast as a single history dependent boundary integral equation on the solid liquid boundary a fast algorithm is used to solve the integral equation efficiently second the boundary is moved by solving a hamiltonjacobi type equation on a fixed domain formulated by osher and sethian for a function in which the boundary is a particular level set this equation is solved by finite difference schemes borrowed from the technology of hyperbolic conservation laws the two ideas are combined by constructing a smooth extension of the normal velocity off the moving boundary in a way suggested by the physics of the problem our numerical experiments show the evolution of complex crystalline shapes development of large spikes and corners dendrite formation and side branching and pieces of solid merging and breaking off freely

analyzing and synthesizing images by evolving curves with theosher sethian method numerical analysis of conservation laws plays an important role in the implementation of curve evolution equations this paper reviews the relevant concepts in numerical analysis and the relation between curve evolution hamilton jacobi partial differential equations and differential conservation laws this close relation enables us to introduce finite difference approximations based on the theory of conservation laws into curve evolution it is shown how curve evolution serves as a powerful tool for image analysis and how these mathematical relations enable us to construct efficient and accurate numerical schemes some examples demonstrate the importance of the cfl condition as a necessary condition for the stability of the numerical schemes

approximation of convex curves with application to the bicriterial minimum cost flow problem abstract an approximation of an explicity or implicity given convex curve in the plane is given by two piecewise linear outer and inner curves to compute these three rules for choosing the supporting points are proposed and it is shown for two of them that the projective distance between inner and outer approximation decreases quadratically with the number of supporting points this method is applied to approximate the efficient point curve of the bicriterial minimum cost flow problem which is a piecewise linear convex curve that may have an exponential number of breakpoints in the worst case

a multigrid method for shape from shading the shape from shading problem has received much attention in the computer vision literature in recent years the basic problem is to recover the em shape z x y of a surface from a given map of its em shading i e its variation of brightness over a given domain mathematically one has to solve approximately the em image irradiance equation begin center e x y end center relating a given image irradiance em e x y to the radiance of the surface at each point em x y with em r p q a given em reflectance z x and q z y na possible presence of noise and lack of adequate boundary conditions adds to the difficulty of this problem a number of different approaches towards its solution have been proposed in the vision literature including various regularization models however a reliable efficient solution method for practical instances has remained elusive so far nin this paper we analyze the various solution models proposed with the aim of applying an efficient multigrid solver a combination of an fmg continuation technique with an appropriate discretization of one such solution model proposed by b horn yields an efficient solver our results are demonstrated by examples

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

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

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

uniqueness in shape from shading for general images of smooth objects wholly contained in the field of view and for illumination symmetric around the viewing direction it is proven that shape is uniquely determined by shading thus shape from shading is a well posed problem under these illumination conditions and regularization is unnecessary for surface reconstruction and should be avoided generic properties of surfaces and images are established questions of existence are also discussed under the conditions above it is argued that most images are effectively impossible with no corresponding physically reasonable surface and that any image can be rendered effectively impossible by a small perturbation of its intensities this is explicitly illustrated for a synthetic image the proofs are based on ideas of dynamical systems theory and global analysis

convergence analysis for a class of high order semi lagrangian advection schemes the convergence properties of a class of high order semi lagrangian schemes for pure advection equations are studied here in the framework of the theory of viscosity solutions we review the general convergence results for discrete time approximation schemes belonging to that class and we prove some a priori estimates in l infty and l2 for the rate of convergence of fully discrete schemes we prove then that a careful coupling of time and space discretizations can allow large time steps in the numerical integration still preserving the accuracy of the solutions several examples of schemes and numerical tests are presented

equivalence of three quadratic approximation methods of stability boundaries of nonlinear dynamic systems approximation of stable manifolds of unstable equilibrium points is a key problem in stability region estimation of nonlinear dynamic systems different quadratic approximation methods have been proposed in past literature then what is their relationship and which has better performance are of concern the mathematical equivalence of three quadratic approximation methods is proved in this note their computation burdens of numerical implementation are compared and the one with the least computation time is recommended for application

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

object recognition using shape from shading investigates whether surface topography information extracted from intensity images using a shape from shading sfs algorithm can be used for the purposes of 3d object recognition we consider how curvature and shape index information delivered by this algorithm can be used to recognize objects based on their surface topography we explore two contrasting object recognition strategies the first of these is based on a low level attribute summary and uses histograms of curvature and orientation measurements the second approach is based on the structural arrangement of constant shape index maximal patches and their associated region attributes we show that region curvedness and a string ordering of the regions according to size provides recognition accuracy of about 96 percent by polling various recognition schemes including a graph matching method we show that a recognition rate of 98 99 percent is achievable

stability regions of nonlinear autonomous dynamical systems a topological and dynamical characterization of the stability boundaries for a fairly large class of nonlinear autonomous dynamic systems is presented the stability boundary of a stable equilibrium point is shown to consist of the stable manifolds of all the equilibrium points and or closed orbits on the stability boundary several necessary and sufficient conditions are derived to determine whether a given equilibrium point or closed orbit is on the stability boundary a method for finding the stability region on the basis of these results is proposed the method when feasible will find the exact stability region rather than a subset of it as in the lyapunov theory approach several examples are given to illustrate the theoretical prediction

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 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

a new incompressibility discretization for a hybrid particle mac grid representation with surface tension we take a particle based approach to incompressible free surface flow motivated by the fact that an explicit representation of the interface geometry and internal deformations gives precise feedback to an implicit solver for surface tension methods that enforce incompressibility directly on the particles are typically numerically inefficient compared to those that utilize a background grid however background grid discretizations suffer from inaccuracy near the free surface where they do not properly capture the interface geometry therefore our incompressibility discretization utilizes a particle based projection near the interface and a background mac grid based projection for efficiency in the vast interior of the liquid domain as well as a novel method for coupling these two disparate projections together we show that the overall coupled elliptic solver is second order accurate and remains second order accurate when used in conjunction with an appropriate temporal discretization for parabolic problems a similar second order accurate discretization is derived when the mac grid unknowns are located on faces as opposed to cell centers so that navier stokes viscosity can be solved for implicitly as well finally we present a fully implicit approach to surface tension that is robust enough to achieve a steady state solution in a single time step beyond stable implicit surface tension for our novel hybrid discretization we demonstrate preliminary results for both standard front tracking and the particle level set method

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

a simple explicit operator splitting method for effective hamiltonians understanding effective hamiltonians quantitatively is essential for the homogenization of hamilton jacobi equations we propose in this article a simple efficient operator splitting method for computing effective hamiltonians when the hamiltonian is either convex or nonconvex in the gradient variable to speed up our lie scheme based operator splitting method we further propose a cascadic initialization strategy so that the steady state of the underlying time dependent hamilton jacobi equation can be reached more rapidly extensive numerical examples demonstrate the efficiency and accuracy of the new algorithm

a locally extended finite element method for the simulation of multi fluid flows using the particle level set method abstract the simulation of immiscible two phase flows on eulerian meshes requires the use of special techniques to guarantee a sharp definition of the evolving fluid interface this work describes the combination of two distinct technologies with the goal of improving the accuracy of the target simulations first of all a spatial enrichment is employed to improve the approximation properties of the eulerian mesh this is done by injecting into the solution space new features to make it able to correctly resolve the solution in the vicinity of the moving interface then the lagrangian particle level set pls method is employed to keep trace of the evolving solution and to improve the mass conservation properties of the resulting method while the local enrichment can be understood in the general context of the xfem we employ an element local variant which allows preserving the matrix graph and hence highly improving the computational efficiency

shape from shading level set propagation and viscosity solutions we present a new implementation of an algorithm aimed at recovering a 3d shape from its 2d gray level picture in order to reconstruct the shape of the object an almost arbitrarily initialized 3d function is propagated on a rectangular grid so that a level set of this function tracks the height contours of the shape the method imports techniques from differential geometry fluid dynamics and numerical analysis and provides an accurate shape from shading algorithm the method solves some topological problems and gracefully handles cases of non smooth surfaces that give rise to shocks in the propagating contours real and synthetic images of 3d profiles were submitted to the algorithm and the reconstructed surfaces are presented demonstrating the effectiveness of the proposed method

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

numerical shape from shading for discontinuous photographic images the height u x y of a continuous lambertian surface of known albedo i e grayness is related to u x y information recoverable from a black and white flash photograph of the surface by the partial differential equation sqrt u 2 x u 2 y n 0 we review the notion of a unique viscosity solution for this equation when n is continuous and a recent unique extension of the viscosity solution when n is discontinuous we prove convergence to this extension for a wide class of the numerical algorithms that converge when n is continuous after discussing the properties of the extension and the order of error in the algorithms simulating the extension we point out warning signs which when observed in the numerical solution usually indicate that the surface is not continuous or that the viscosity solution or its extension does not correspond to the actual surface finally we discuss a method that in some of these cases allows us to correct the simulation and recover the actual surface again

regular article a pde based fast local level set method we develop a fast method to localize the level set method of osher and sethian 1988 j comput phys 79 12 and address two important issues that are intrinsic to the level set method a how to extend a quantity that is given only on the interface to a neighborhood of the interface b how to reset the level set function to be a signed distance function to the interface efficiently without appreciably moving the interface this fast local level set method reduces the computational effort by one order of magnitude works in as much generality as the original one and is conceptually simple and easy to implement our approach differs from previous related works in that we extract all the information needed from the level set function or functions in multiphase flow and do not need to find explicitly the location of the interface in the space domain the complexity of our method to do tasks such as extension and distance reinitialization is o n where n is the number of points in space not o n log n as in works by sethian 1996 proc nat acad sci 93 1591 and helmsen and co workers 1996 spie microlithography ix p 253 this complexity estimation is also valid for quite general geometrically based front motion for our localized method

a method for enforcing integrability in shape from shading algorithms an approach for enforcing integrability a particular implementation of the approach an example of its application to extending an existing shape from shading algorithm and experimental results showing the improvement that results from enforcing integrability are presented a possibly nonintegrable estimate of surface slopes is represented by a finite set of basis functions and integrability is enforced by calculating the orthogonal projection onto a vector subspace spanning the set of integrable slopes the integrability projection constraint was applied to extending an iterative shape from shading algorithm of m j brooks and b k p horn 1985 experimental results show that the extended algorithm converges faster and with less error than the original version good surface reconstructions were obtained with and without known boundary conditions and for fairly complicated surfaces

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

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

shortening three dimensional curves via two dimensional flows in this paper a curve evolution approach for the computation of geodesic curves on 3d surfaces is presented the algorithm is based on deforming via the curve shortening flow an arbitrary initial curve ending at two given surface points the 3d curve shortening flow is first transformed into an equivalent 2d one this 2d flow is implemented using an efficient numerical algorithm for curve evolution with fixed end points

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

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

vials an eulerian tool based on total variation and the level set method for studying dynamical systems abstract we propose a new eulerian tool to study complicated dynamical systems based on the average growth in the surface area of a family of level surfaces represented implicitly by a level set function since this proposed quantity determines the temporal variation of the averaged surface area of all level surfaces we name the quantity the variation of the integral over area of level surfaces vials numerically all these infinitely many level surfaces are advected according to the given dynamics by solving one single linear advection equation to develop a computationally efficient approach we apply the coarea formula and rewrite the surface area integral as a simple integral relating the total variation tv of the level set function the proposed method can be easily incorporated with a recent eulerian algorithm for efficient computation of flow maps to speed up our approach we will also prove that the proposed vials is closely related to the computation of the finite time lyapunov exponent ftle in the lagrangian coherent structure lcs extraction this connects our proposed eulerian approach to widely used lagrangian techniques for understanding complicated dynamical systems

vector transport for shape from shading in this paper we describe a new shape from shading method we show how the parallel transport of surface normals can be used to impose curvature consistency and also to iteratively update surface normal directions so as to improve the brightness error we commence by showing how to make local estimates of the hessian matrix from surface normal information with the local hessian matrix to hand we develop an em like algorithm for updating the surface normal directions at each image location parallel transport is applied to the neighbouring surface normals to generate a sample of local surface orientation predictions from this sample a local weighted estimate of the image brightness is made the transported surface normal which gives the brightness prediction which is closest to this value is selected as the revised estimate of surface orientation the revised surface normals obtained in this way may in turn be used to re estimate the hessian matrix and the process iterated until stability is reached we experiment with the method on a variety of real world and synthetic data here we explore the properties of the fields of surface normals and the height data delivered by 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

height and gradient from shading the method described here for recovering the shape of a surface from a shaded image can deal with complex wrinkled surfaces integrability can be enforced easily because both surface height and gradient are represented the robustness of the method stems in part from linearization of the reflectance map about the current estimate of the surface orientation at each picture cell the new scheme can find an exact solution of a given shape from shading problem even though a regularizing term is included this is a reflection of the fact that shape from shading problems are it not ill posed when boundary conditions are available or when the image contains singular points

a graph spectral approach to shape from shading in this paper we explore how graph spectral methods can be used to develop a new shape from shading algorithm we characterize the field of surface normals using a weight matrix whose elements are computed from the sectional curvature between different image locations and penalize large changes in surface normal direction modeling the blocks of the weight matrix as distinct surface patches we use a graph seriation method to find a surface integration path that maximizes the sum of curvature dependent weights and that can be used for the purposes of height reconstruction to smooth the reconstructed surface we fit quadrics to the height data for each patch the smoothed surface normal directions are updated ensuring compliance with lambert s law the processes of height recovery and surface normal adjustment are interleaved and iterated until a stable surface is obtained we provide results on synthetic and real world imagery

space time discontinuous galerkin finite element method for two fluid flows a novel numerical method for two fluid flow computations is presented which combines the space time discontinuous galerkin finite element discretization with the level set method and cut cell based interface tracking the space time discontinuous galerkin stdg finite element method offers high accuracy an inherent ability to handle discontinuities and a very local stencil making it relatively easy to combine with local hp refinement the front tracking is incorporated via cut cell mesh refinement to ensure a sharp interface between the fluids to compute the interface dynamics the level set method lsm is used because of its ability to deal with merging and breakup also the lsm is easy to extend to higher dimensions small cells arising from the cut cell refinement are merged to improve the stability and performance the interface conditions are incorporated in the numerical flux at the interface and the stdg discretization ensures that the scheme is conservative as long as the numerical fluxes are conservative the numerical method is applied to one and two dimensional two fluid test problems using the euler equations

finite element approximations of general fully nonlinear second order elliptic partial differential equations based on the vanishing moment method the vanishing moment method was introduced by the authors in feng and neilan 2009 as a reliable methodology for computing viscosity solutions of fully nonlinear second order partial differential equations pdes it is based on the simple idea of approximating a hard to handle fully nonlinear second order pde by a family parameterized by a small parameter e of easy to handle quasilinear fourth order pdes the primary objective of this article is to present a comprehensive finite element analysis for the vanishing moment approximation of general fully nonlinear second order elliptic pdes which fulfill some structure conditions abstract methodological and convergence analysis frameworks of conforming finite element methods are first developed for fully nonlinear second order pdes in a general setting the abstract framework is then applied to three prototypical nonlinear equations namely the monge ampere equation the equation of prescribed gauss curvature and the infinity laplacian equation numerical experiments are presented for each problem to validate the theoretical error estimate results and to gauge the efficiency of the proposed numerical methods and the vanishing moment methodology

numerical shape from shading and occluding boundaries an iterative method for computing shape from shading using occluding boundary information is proposed some applications of this method are shown we employ the stereographic plane to express the orientations of surface patches rather than the more commonly used gradient space use of the stereographic plane makes it possible to incorporate occluding boundary information but forces us to employ a smoothness constraint different from the one previously proposed the new constraint follows directly from a particular definition of surface smoothness we solve the set of equations arising from the smoothness constraints and the image irradiance equation iteratively using occluding boundary information to supply boundary conditions good initial values are found at certain points to help reduce the number of iterations required to reach a reasonable solution numerical experiments show that the method is effective and robust finally we analyze scanning electron microscope sem pictures using this method other applications are also proposed

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 graph spectral method for surface height recovery this paper describes a graph spectral method for 3d surface integration the algorithm takes as its input a 2d field of surface normal estimates delivered for instance by a shape from shading or shape from texture procedure we commence by using the surface normals to obtain an affinity weight matrix whose elements are related to the surface curvature the weight matrix is used to compute a row normalized transition probability matrix and we pose the recovery of the integration path as that of finding the steady state random walk for the markov chain defined by this matrix the steady state random walk is given by the leading eigenvector of the original affinity weight matrix by threading the surface normals together along the path specified by the magnitude order of the components of the leading eigenvector we perform surface integration the height increments along the path are simply related to the traversed path length and the slope of the local tangent plane the method is evaluated on needle maps delivered by a shape from shading algorithm applied to real world data and also on synthetic data the method is compared with the local geometric height reconstruction method of bors hancock and wilson and the global methods of horn and brooks and frankot and chellappa

shape from shading using the heat equation this paper offers two new directions to shape from shading namely the use of the heat equation to smooth the field of surface normals and the recovery of surface height using a low dimensional embedding turning our attention to the first of these contributions we pose the problem of surface normal recovery as that of solving the steady state heat equation subject to the hard constraint that lambert s law is satisfied we perform our analysis on a plane perpendicular to the light source direction where the z component of the surface normal is equal to the normalized image brightness the x y or azimuthal component of the surface normal is found by computing the gradient of a scalar field that evolves with time subject to the heat equation we solve the heat equation for the scalar potential and hence recover the azimuthal component of the surface normal from the average image brightness making use of a simple finite difference method the second contribution is to pose the problem of recovering the surface height function as that of embedding the field of surface normals on a manifold so as to preserve the pattern of surface height differences and the lattice footprint of the surface normals we experiment with the resulting method on a variety of real world image data where it produces qualitatively good reconstructed surfaces

curvature and the evolution of fronts the evolution of a front propagating along its normal vector field with speedf dependent on curvaturek is considered the change in total variation of the propagating front is shown to depend only ondf dk only wherek changes sign analysis of the casef k 1 ek where e is a constant shows that curvature plays a role similar to that of viscosity in burgers equation for e 0 and non convex initial data the curvature blows up corners develop and an entropy condition can be formulated to provide an explicit construction for a weak solution beyond the singularity we then numerically show that the solution as e goes to zero converges to the constructed weak solution numerical methods based on finite difference schemes for marker particles along the front are shown to be unstable in regions where the curvature builds as a remedy we show that front tracking based on volume of fluid techniques can be used together with the entropy condition to provide transition from the classical to weak solution

an optimal adaptive algorithm for the approximation of concave functions motivated by the study of parametric convex programs we consider approximation of concave functions by piecewise affine functions using dynamic programming we derive a procedure for selecting the knots at which an oracle provides the function value and one supergradient the procedure is adaptive in that the choice of a knot is dependent on the choice of the previous knots it is also optimal in that the approximation error in the integral sense is minimized in the worst case

factored singularities and high order lax friedrichs sweeping schemes for point source traveltimes and amplitudes in the high frequency regime the geometrical optics approximation for the helmholtz equation with a point source results in an eikonal equation for traveltime and a transport equation for amplitude because the point source traveltime field has an upwind singularity at the source point all formally high order finite difference eikonal solvers exhibit first order convergence and relatively large errors in this paper we propose to first factor out the singularities of traveltimes takeoff angles and amplitudes and then we design high order lax friedrichs sweeping schemes for point source traveltimes takeoff angles and amplitudes numerical examples are presented to demonstrate the performance of our new method

global optimization of mixed integer nonlinear programs a theoretical and computational study this work addresses the development of an efficient solution strategy for obtaining global optima of continuous integer and mixed integer nonlinear programs towards this end we develop novel relaxation schemes range reduction tests and branching strategies which we incorporate into the prototypical branch and bound algorithm in the theoretical algorithmic part of the paper we begin by developing novel strategies for constructing linear relaxations of mixed integer nonlinear programs and prove that these relaxations enjoy quadratic convergence properties we then use lagrangian linear programming duality to develop a unifying theory of domain reduction strategies as a consequence of which we derive many range reduction strategies currently used in nonlinear programming and integer linear programming this theory leads to new range reduction schemes including a learning heuristic that improves initial branching decisions by relaying data across siblings in a branch and bound tree finally we incorporate these relaxation and reduction strategies in a branch and bound algorithm that incorporates branching strategies that guarantee finiteness for certain classes of continuous global optimization problems in the computational part of the paper we describe our implementation discussing wherever appropriate the use of suitable data structures and associated algorithms we present computational experience with benchmark separable concave quadratic programs fractional 0 1 programs and mixed integer nonlinear programs from applications in synthesis of chemical processes engineering design just in time manufacturing and molecular design

deformable pedal curves and surfaces hybrid geometric active models for shape recovery in this paper we propose significant extensions to the snake pedal model a powerful geometric shape modeling scheme introduced in vemuri and guo 1998 the extension allows the model to automatically cope with topological changes and for the first time introduces the concept of a compact global shape into geometric active models the ability to characterize global shape of an object using very few parameters facilitates shape learning and recognition in this new modeling scheme object shapes are represented using a parameterized function called the generator which accounts for the global shape of an object and the pedal curve surface of this global shape with respect to a geometric snake to represent any local detail traditionally pedal curves surfaces are defined as the loci of the feet of perpendiculars to the tangents of the generator from a fixed point called the pedal point local shape control is achieved by introducing a set of pedal points lying on a snake for each point on the generator the model dubbed as a snake pedal allows for interactive manipulation via forces applied to the snake in this work we replace the snake by a geometric snake and derive all the necessary mathematics for evolving the geometric snake when the snake pedal is assumed to evolve as a function of its curvature automatic topological changes of the model may be achieved by implementing the geometric snake in a level set framework we demonstrate the applicability of this modeling scheme via examples of shape recovery from a variety of 2d and 3d image data

local shading analysis local analysis of image shading in the absence of prior knowledge about the viewed scene may be used to provide information about the scene the following has been proved every image point has the same image intensity and first and second derivatives as the image of some point on a lambertian surface with principal curvatures of equal magnitude further if the principal curvatures are assumed to be equal there is a unique combination of image formation parameters up to a mirror reversal that will produce a particular set of image intensity and first and second derivatives a solution for the unique combination of surface orientation etc is presented this solution has been extended to natural imagery by using general position and regional constraints to obtain estimates of the following surface orientation at each image point the qualitative type of the surface i e whether the surface is planar cylindrical convex concave or saddle the illuminant direction within a region algorithms to recover illuminant direction and estimate surface orientation have been evaluated on both natural and synthesized images and have been found to produce useful information about the scene

linear shape from shading in many situations the reflectance function of a surface is approximately linear and there is an effielent closed form solution to the shape from shading problem when boundary conditions e g edges singular points are not available good estimates of shape may still be extracted by using the assumption of general viewing position an improved method for estimating the illuminant direction is also presented

shapes shocks and deformations i the components of two dimensional shape and the reaction diffusion space we undertake to develop a general theory of two dimensional shape by elucidating several principles which any such theory should meet the principles are organized around two basic intuitions first if a boundary were changed only slightly then in general its shape would change only slightly this leads us to propose an operational theory of shape based on incremental contour deformations the second intuition is that not all contours are shapes but rather only those that can enclose physical material a theory of contour deformation is derived from these principles based on abstract conservation principles and hamilton jacobi theory these principles are based on the work of sethian 1985a c the osher sethian 1988 level set formulation the classical shock theory of lax 1971 1973 as well as curve evolution theory for a curve evolving as a function of the curvature and the relation to geometric smoothing of gage hamilton grayson 1986 1989 the result is a characterization of the computational elements of shape deformations parts bends and seeds which show where to place the components of a shape the theory unifies many of the diverse aspects of shapes and leads to a space of shapes the reaction diffusion space which places shapes within a neighborhood of similar ones such similarity relationships underlie descriptions suitable for recognition

minimal surfaces a geometric three dimensional segmentation approach a novel geometric approach for three dimensional object segmentation is presented the scheme is based on geometric deformable surfaces moving towards the objects to be detected we show that this model is related to the computation of surfaces of minimal area local minimal surfaces the space where these surfaces are computed is induced from the three dimensional image in which the objects are to be detected the general approach also shows the relation between classical deformable surfaces obtained via energy minimization and geometric ones derived from curvature flows in the surface evolution framework the scheme is stable robust and automatically handles changes in the surface topology during the deformation results related to existence uniqueness stability and correctness of the solution to this geometric deformable model are presented as well based on an efficient numerical algorithm for surface evolution we present a number of examples of object detection in real and synthetic images

the effect of transformations on the approximation of univariate convex functions with applications to pareto curves in the literature methods for the construction of piecewise linear upper and lower bounds for the approximation of univariate convex functions have been proposed we study the effect of the use of increasing convex or increasing concave transformations on the approximation of univariate convex functions in this paper we show that these transformations can be used to construct upper and lower bounds for nonconvex functions moreover we show that by using such transformations of the input variable or the output variable we obtain tighter upper and lower bounds for the approximation of convex functions than without these approximations we show that these transformations can be applied to the approximation of a convex pareto curve that is associated with a convex bi objective optimization problem

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

computing minimal surfaces via level set curvature flow given a fixed closed curve or curves in r sp3 there is an associated problem of finding a surface with mean curvature zero which has that curve or curves as its boundary in this paper a new approach to numerically solving the minimal surface problem is introduced the surface is represented as a level set of a global function phi r sp3 to r the family of surfaces evolve in time according to mean curvature flow a solution is found when a steady state condition is achieved a new system of interpolatory boundary conditions are used to maintain the connection between the surface and the fixed boundary contour several computer surfaces are shown including examples of topological changes

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

new high resolution semi discrete central schemes for hamilton jacobi equations we introduce a new high resolution central scheme for multidimensional hamilton jacobi equations the scheme retains the simplicity of the non oscillatory central schemes developed by c t lin and e tadmor in press siam j sci comput yet it enjoys a smaller amount of numerical viscosity independent of 1 t by letting t 0 we obtain a new second order central scheme in the particularly simple semi discrete form along the lines of the new semi discrete central schemes recently introduced by the authors in the context of hyperbolic conservation laws fully discrete versions are obtained with appropriate runge kutta solvers the smaller amount of dissipation enables efficient integration of convection diffusion equations where the accumulated error is independent of a small time step dictated by the cfl limitation the scheme is non oscillatory thanks to the use of nonlinear limiters here we advocate the use of such limiters on second discrete derivatives which is shown to yield an improved high resolution when compared to the usual limitation of first derivatives numerical experiments demonstrate the remarkable resolution obtained by the proposed new central scheme

on the evolution of curves via a function of curvature i the classical case abstract the problem of curve evolution as a function of its local geometry arises naturally in many physical applications a special case of this problem is the curve shortening problem which has been extensively studied here we consider the general problem and prove an existence theorem for the classical solution the main theorem rests on lemmas that bound the evolution of length curvature and how far the curve can travel

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

multiple objective minimum cost flow problems a review in this paper theory and algorithms for solving the multiple objective minimum cost flow problem are reviewed for both the continuous and integer case exact and approximation algorithms are presented in addition a section on compromise solutions summarizes corresponding results the reference list consists of all papers known to the authors which deal with the multiple objective minimum cost flow problem