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 fast marching algorithm for the factored eikonal equation the eikonal equation is instrumental in many applications in several fields ranging from computer vision to geoscience this equation can be efficiently solved using the iterative fast sweeping fs methods and the direct fast marching fm methods however when used for a point source the original eikonal equation is known to yield inaccurate numerical solutions because of a singularity at the source in this case the factored eikonal equation is often preferred and is known to yield a more accurate numerical solution one application that requires the solution of the eikonal equation for point sources is travel time tomography this inverse problem may be formulated using the eikonal equation as a forward problem while this problem has been solved using fs in the past the more recent choice for applying it involves fm methods because of the efficiency in which sensitivities can be obtained using them however while several fs methods are available for solving the factored equation the fm method is available only for the original eikonal equation in this paper we develop a fast marching algorithm for the factored eikonal equation using both first and second order finite difference schemes our algorithm follows the same lines as the original fm algorithm and requires the same computational effort in addition we show how to obtain sensitivities using this fm method and apply travel time tomography formulated as an inverse factored eikonal equation numerical results in two and three dimensions show that our algorithm solves the factored eikonal equation efficiently and demonstrate the achieved accuracy for computing the travel time we also demonstrate a recovery of a 2d and 3d heterogeneous medium by travel time tomography using the eikonal equation for forward modeling and inversion by gauss newton

upwind finite difference calculation of traveltimes seismic traveltimes can be computed efficiently on a regular grid by an upwind finite difference method the method solves a conservation law that describes changes in the gradient components of the traveltime field the traveltime field itself is easily obtained from the solution of the conservation law by numerical integration the conservation law derives from the eikonal equation and its solution depicts the first arrival time field the upwind finite difference scheme can be implemented in fully vectorized form in contrast to a similar scheme proposed recently by vidale the resulting traveltime field is useful both in kirchhoff migration and modeling and in seismic tomography many reliable methods exist for the numerical solution of conservation laws which appear in fluid mechanics as statements of the conservation of mass momentum etc a first order upwind finite difference scheme proves accurate enough for seismic applications upwind schemes are stable because they mimic the behavior of flu

ccs with priority choice abstract this paper investigates an extension of milner s ccs with a priority choice operator called prisum this operator is very similar to the prialt construct of occam the new binary prisum operator only allows execution of its second component in the case where the environment is not ready to allow the first component to proceed this dependency on the set of actions the environment is ready to perform goes beyond that encountered in traditional ccs its expression leads to a novel operational semantics in which transitions carry ready sets of the environment as well as the normal action symbols from ccs a notion of strong bisimulation is defined on agents with priority via this semantics it is a congruence and satisfies new equational laws including a new expansion law which are shown to be complete for finite agents with prisum the laws are conservative over agents of traditional ccs

fast sweeping methods for static hamilton jacobi equations we propose a new sweeping algorithm which discretizes the legendre transform of the numerical hamiltonian using an explicit formula this formula yields the numerical solution at a grid point using only its immediate neighboring grid values and is easy to implement numerically the minimization that is related to the legendre transform in our sweeping scheme can either be solved analytically or numerically we illustrate the efficiency and accuracy approach with several numerical examples in two and three dimensions

biharmonic hypersurfaces of the 4 dimensional semi euclidean space es4 abstract a submanifold m r n of a semi euclidean space e s m is said to have harmonic mean curvature vector field if h 0 where h denotes the mean curvature vector submanifolds with harmonic mean curvature vector are also known as biharmonic submanifolds in this paper we prove that every nondegenerate hypersurface of e s 4 the shape operator of which is diagonalizable with harmonic mean curvature vector field is minimal

imaging complex structures with semirecursive kirchhoff migration i present a semirecursive kirchhoff migration algorithm that is capable of obtaining accurate images of complex structures by combining wave equation datuming and kirchhoff migration the method is successful because breaking up the complex velocity structure into small depth regions allows traveltimes to be calculated in regions where the computation is well behaved and where the computation corresponds to energetic arrivals the traveltimes computed in such a region are used first for imaging and second for downward continuation of the entire survey shots and receivers to the boundary of the next region this process results in images comparable to those obtained by shot profile migration but at reduced computational cost because traveltimes are computed for small depth domains the adverse effects of caustics headwaves and multiple arrivals do not develop in principle this method requires only the same number of traveltime calculations as a standard migration tests on the marmousi data set produce excellent results

quasi minimal surfaces of pseudo riemannian space forms with positive relative nullity abstract in this paper we consider surfaces in 4 dimensional pseudo riemannian space forms with index 2 first we obtain some of geometrical properties of such surfaces considering their relative null space then we get classifications of quasi minimal surfaces with positive relative nullity

a third order accurate fast marching method for the eikonal equation in two dimensions in this paper we develop a third order accurate fast marching method for the solution of the eikonal equation in two dimensions there have been two obstacles to extending the fast marching method to higher orders of accuracy the first obstacle is that using one sided difference schemes is unstable for orders of accuracy higher than two the second obstacle is that the points in the difference stencil are not available when the gradient is closely aligned with the grid we overcome these obstacles by using a two dimensional 2d finite difference approximation to improve stability and by locally rotating the grid 45 degrees i e using derivatives along the diagonals to ensure all the points needed in the difference stencil are available we show that in smooth regions the full difference stencil is used for a suitably small enough grid size and that the difference scheme satisfies the von neumann stability condition for the linearized eikonal equation our method reverts to first order accuracy near caustics without developing oscillations by using a simple switching scheme the efficiency and high order of the method are demonstrated on a number of 2d test problems

lax friedrichs sweeping scheme for static hamilton jacobi equations we propose a simple fast sweeping method based on the lax friedrichs monotone numerical hamiltonian to approximate viscosity solutions of arbitrary static hamilton jacobi equations in any number of spatial dimensions by using the lax friedrichs numerical hamiltonian we can easily obtain the solution at a specific grid point in terms of its neighbors so that a gauss seidel type nonlinear iterative method can be utilized furthermore by incorporating a group wise causality principle into the gauss seidel iteration by following a finite group of characteristics we have an easy to implement sweeping type and fast convergent numerical method however unlike other methods based on the godunov numerical hamiltonian some computational boundary conditions are needed in the implementation we give a simple recipe which enforces a version of discrete min max principle some convergence analysis is done for the one dimensional eikonal equation extensive 2 d and 3 d numerical examples illustrate the efficiency and accuracy of the new approach to our knowledge this is the first fast numerical method based on discretizing the hamilton jacobi equation directly without assuming convexity and or homogeneity of the hamiltonian

a tutorial on empa a theory of concurrent processes with nondeterminism priorities probabilities and time we give an overview of empa a process algebra for modeling and analyzing concurrent systems where nondeterminism priorities probabilities and time are present aiming at a reasonable trade off between the expressive power of the calculus and the complexity of its underlying theory

classification of spatial surfaces with parallel mean curvature vector in pseudo euclidean spaces of arbitrary dimension spatial surfaces with parallel mean curvature vector play some important roles in general relativity theory of harmonic maps as well as in differential geometry recently spatial surfaces in four dimensional minkowski space time with parallel mean curvature vector were classified by chen and van der veken complete classification of parallel surfaces in 4 dimensional lorentzian space forms tohoku math j 61 1 2009 in this article we completely classify spatial surfaces with parallel mean curvature vector in pseudo euclidean spaces of arbitrary dimension the main result states that there exist 16 families of such surfaces as by product we achieve the complete classification of spatial surfaces with parallel mean curvature vector in minkowski space times of arbitrary dimension

priority and abstraction in process algebra more than 15 years ago cleaveland and hennessy proposed an extension of the process algebra ccs in which some actions may take priority over others the theory was equipped with a behavioral congruence based on strong bisimulation this article gives a full account of the challenges in and the solutions employed for defining a semantic theory of observation congruence for this process algebra a full abstraction result is presented whose proof relies on a novel approach based on successive approximations for identifying the largest congruence contained in an intuitive but naive equivalence prioritized observation congruence is also characterized equationally for the class of finite processes while its utility for system verification is demonstrated by an illustrative example

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

homothetic and conformal symmetries of solutions to einstein s equations we present several results about the nonexistence of solutions of einstein s equations with homothetic or conformal symmetry we show that the only spatially compact globally hyperbolic spacetimes admitting a hypersurface of constant mean extrinsic curvature and also admitting an infinitesimal proper homothetic symmetry are everywhere locally flat this assumes that the matter fields either obey certain energy conditions or are the yang mills or massless klein gordon fields we find that the only vacuum solutions admitting an infinitesimal proper conformal symmetry are everywhere locally flat spacetimes and certain plane wave solutions we show that if the dominant energy condition is assumed then minkowski spacetime is the only asymptotically flat solution which has an infinitesimal conformal symmetry that is asymptotic to a dilation in other words with the exceptions cited homothetic or conformal killing fields are in fact killing in spatially compact or asymptotically flat spactimes in the conformal procedure for solving the initial value problem we show that data with infinitesimal conformal symmetry evolves to a spacetime with full isometry

screw invariant marginally trapped surfaces in minkowski 4 space a spacelike surface in a lorentzian manifold whose mean curvature vector is lightlike everywhere is called marginally trapped the classification of marginally trapped surfaces in minkowski 4 space which are invariant under a subgroup of the lorentz group that leaves invariant a lightlike direction i e the so called screw invariant surfaces is obtained as corollaries the screw invariant marginally trapped surfaces with harmonic mean curvature vector and with prescribed gaussian curvature are explicitly described

fast sweeping methods for eikonal equations on triangular meshes the original fast sweeping method which is an efficient iterative method for stationary hamilton jacobi equations relies on natural ordering provided by a rectangular mesh we propose novel ordering strategies so that the fast sweeping method can be extended efficiently and easily to any unstructured mesh to that end we introduce multiple reference points and order all the nodes according to their l p metrics to those reference points we show that these orderings satisfy the two most important properties underlying the fast sweeping method 1 these orderings can cover all directions of information propagating efficiently 2 any characteristic can be decomposed into a finite number of pieces and each piece can be covered by one of the orderings we prove the convergence of the new algorithm the computational complexity of the algorithm is nearly optimal in the sense that the total computational cost consists of o m flops for iteration steps and o m rm log m flops for sorting at the predetermined initialization step which can be efficiently optimized by adopting a linear time sorting method where m is the total number of mesh points extensive numerical examples demonstrate that the new algorithm converges in a finite number of iterations independent of mesh size

addendum to classification of marginally trapped lorentzian flat surfaces in e24 and its application to biharmonic surfaces abstract due to omission an additional family shall be added to the list of marginally trapped lorentzian flat surfaces in e 2 4 given in theorem 4 1 of b y chen classification of marginally trapped lorentzian flat surfaces in e 2 4 and its application to biharmonic surfaces j math anal appl 340 2008 861 875

efficient algorithms for globally optimal trajectories we present serial and parallel algorithms for solving a system of equations that arises from the discretization of the hamilton jacobi equation associated to a trajectory optimization problem of the following type a vehicle starts at a prespecified point x sub o and follows a unit speed trajectory x t inside a region in spl rscr sup m until an unspecified time t that the region is exited a trajectory minimizing a cost function of the form spl int sub 0 sup t r x t dt q x t is sought the discretized hamilton jacobi equation corresponding to this problem is usually solved using iterative methods nevertheless assuming that the function r is positive we are able to exploit the problem structure and develop one pass algorithms for the discretized problem the first algorithm resembles dijkstra s shortest path algorithm and runs in time o n log n where n is the number of grid points the second algorithm uses a somewhat different discretization and borrows some ideas from a variation of dial s shortest path algorithm 1969 that we develop here it runs in time o n which is the best possible under some fairly mild assumptions finally we show that the latter algorithm can be efficiently parallelized for two dimensional problems and with p processors its running time becomes o n p provided that p o spl radic n log n

parallel surfaces in lorentzian three manifolds admitting a parallel null vector field we classify parallel surfaces in lorentzian three manifolds admitting a parallel null vector field some families of semi parallel surfaces are also described

legendre transform based fast sweeping methods for static hamilton jacobi equations on triangulated meshes we propose a new sweeping algorithm which utilizes the legendre transform of the hamiltonian on triangulated meshes the algorithm is a general extension of the previous proposed algorithm by kao et al c y kao s j osher y h tsai fast sweeping method for static hamilton jacobi equations siam j numer anal 42 2005 2612 2632 the algorithm yields the numerical solution at a grid point using only its one ring neighboring grid values and is easy to implement numerically the minimization that is related to the legendre transform in the sweeping algorithm can either be solved analytically or numerically the scheme is shown to be monotone and consistent we illustrate the efficiency and accuracy of the new method with several numerical examples in two and three dimensions

a fast sweeping algorithm for accurate solution of the tilted transversely isotropic eikonal equation using factorization abstracttraveltime computation is essential for many seismic data processing applications and velocity analysis tools high resolution seismic imaging requires eikonal solvers to account for anisotropy whenever it significantly affects the seismic wave kinematics moreover computation of auxiliary quantities such as amplitude and take off angle relies on highly accurate traveltime solutions however the finite difference based eikonal solution for a point source initial condition has upwind source singularity at the source position because the wavefront curvature is large near the source point therefore all finite difference solvers even the high order ones show inaccuracies because the errors due to source singularity spread from the source point to the whole computational domain we address the source singularity problem for tilted transversely isotropic tti eikonal solvers using factorization we solve a sequence of factored tilted elliptically anisotropic tea eikonal equations iteratively

fast sweeping method for the factored eikonal equation we develop a fast sweeping method for the factored eikonal equation by decomposing the solution of a general eikonal equation as the product of two factors the first factor is the solution to a simple eikonal equation such as distance or a previously computed solution to an approximate eikonal equation the second factor is a necessary modification correction appropriate discretization and a fast sweeping strategy are designed for the equation of the correction part the key idea is to enforce the causality of the original eikonal equation during the gauss seidel iterations using extensive numerical examples we demonstrate that 1 the convergence behavior of the fast sweeping method for the factored eikonal equation is the same as for the original eikonal equation i e the number of iterations for the gauss seidel iterations is independent of the mesh size 2 the numerical solution from the factored eikonal equation is more accurate than the numerical solution directly computed from the original eikonal equation especially for point sources

3 d traveltime computation using the fast marching method we present a fast algorithm for solving the eikonal equation in three dimensions based on the fast marching method the algorithm is of the order o n log n where n is the total number of grid points in the computational domain the algorithm can be used in any orthogonal coordinate system and globally constructs the solution to the eikonal equation for each point in the coordinate domain the method is unconditionally stable and constructs solutions consistent with the exact solution for arbitrarily large gradient jumps in velocity in addition the method resolves any overturning propagation wavefronts we begin with the mathematical foundation for solving the eikonal equation using the fast marching method and follow with the numerical details we then show examples of traveltime propagation through the seg eage salt model using point source and planewave initial conditions and analyze the error in constant velocity media the algorithm allows for any shape of the initial wavefront while a point source is the most commonly used initial condition initial plane waves can be used for controlled illumination or for downward continuation of the traveltime field from one depth to another or from a topographic depth surface to another the algorithm presented here is designed for computing first arrival traveltimes nonetheless since it exploits the fast marching method for solving the eikonal equation we believe it is the fastest of all possible consistent schemes to compute first arrivals

high order factorization based high order hybrid fast sweeping methods for point source eikonal equations the solution for the eikonal equation with a point source condition has an upwind singularity at the source point as the eikonal solution behaves like a distance function at and near the source as

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

operator splitting based fast sweeping methods for isotropic wave propagation in a moving fluid wave propagation in an isotropic acoustic medium occupied by a moving fluid is governed by an anisotropic eikonal equation since this anisotropic eikonal equation is associated with an inhomogeneous hamiltonian most of existing anisotropic eikonal solvers are either inapplicable or of unpredictable behavior in convergence realizing that this anisotropic eikonal equation is defined by a sum of two well understood first order differential operators we propose novel operator splitting based fast sweeping methods to solve this generalized eikonal equation we develop various operator splitting methods relying on the peaceman rachford scheme the douglas rachford scheme the theta scheme and the regularized theta scheme after applying the operator splitting strategy each splitting step corresponds to a much simpler hamilton jacobi equation so that we can apply the lax friedrichs sweeping method to solve these splitted equations efficiently and easily two and three dimensional examples demons

weakly trapped submanifolds immersed in generalized robertson walker spacetimes abstract we deal with codimension two weakly trapped submanifolds that is the mean curvature vector field is causal immersed in a generalized robertson walker grw spacetime under suitable hypothesis we are able to prove that such a spacelike submanifold is immersed into a slice of the ambient space for this we use three main core concepts the well known generalized maximum principle of omori and yau stochastic completeness and another appropriate maximum principle at infinity due to yau we also construct a nontrivial example of weakly trapped submanifold in the static grw spacetime r h 2 r illustrating the importance of our results

fast huygens sweeping methods for multiarrival green s functions of helmholtz equations in the high frequency regime abstractmultiarrival green s functions are essential in seismic modeling migration and inversion huygens kirchhoff hk integrals provide a bridge to integrate locally valid first arrival green s functions into a globally valid multiarrival green s function we have designed robust and accurate finite difference methods to compute first arrival traveltimes and amplitudes so that first arrival green s functions can be constructed rapidly we adapted a fast butterfly algorithm to evaluate discretized hk integrals the resulting fast huygens sweeping method has the following unique features 1 it precomputes a set of local traveltime and amplitude tables 2 it automatically takes care of caustics 3 it constructs green s functions of the helmholtz equation for arbitrary frequencies and for many point sources and 4 for a fixed number of points per wavelength it constructs each green s function in nearly optimal complexity o n log n in terms of the total number of mesh points n where the prefa

classification of lorentzian surfaces with parallel mean curvature vector in pseudo euclidean spaces spatial surfaces with parallel mean curvature vector in pseudo euclidean spaces of arbitrary dimension and index were classified by b y chen in this work we give a complete classification of lorentzian surfaces with parallel mean curvature vector in pseudo euclidean spaces of arbitrary dimension and index consequently the problem to classify all the surfaces with parallel mean curvature vector in pseudo euclidean spaces has been solved

spatial and lorentzian surfaces in robertson walker space times let l14 f c i fs gfc be a robertson walker space time which does not contain any open subset of constant curvature in this paper we provide a general study of nondegenerate surfaces in l14 f c first we prove the nonexistence of marginally trapped surfaces with positive relative nullity then we classify totally geodesic submanifolds finally we classify the family of surfaces with parallel second fundamental form and the family of totally umbilical surfaces with parallel mean curvature vector

3 d eikonal solvers first arrival traveltimes the article is concerned with the development and comparison of three different algorithms for the computation of first arrival traveltimes the fast marching method fmm the group marching method gmm and a second order finite difference eikonal solver gmm is introduced as a variant of fmm it proceeds the solution by advancing a selected group of grid points at a time rather than sorting the solution in the narrow band to march forward a single grid point the second order eikonal solver studied in the article is an expanding box essentially nonoscillatory scheme for which the stability is enforced by the introduction of a down n out marching and a post sweeping iteration techniques such as the maximum angle condition the average normal velocity and cache based implementation are introduced for the algorithms to improve the numerical accuracy and efficiency the algorithms are implemented for solving the eikonal equation in 3 d isotropic media and their performances are compared gmm is nume