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

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

a fast phase space method for computing creeping rays creeping rays can give an important contribution to the solution of medium to high frequency scattering problems they are generated at the shadow lines of the illuminated scatterer by grazing incident rays and propagate along geodesics on the scatterer surface continuously shedding diffracted rays in their tangential direction in this paper we show how the ray propagation problem can be formulated as a partial differential equation pde in a three dimensional phase space to solve the pde we use a fast marching method the pde solution contains information about all possible creeping rays this information includes the phase and amplitude of the field which are extracted by a fast post processing computationally the cost of solving the pde is less than tracing all rays individually by solving a system of ordinary differential equations we consider an application to mono static radar cross section problems where creeping rays from all illumination angles must be computed the numerical results of the fast phase space method and a comparison with the results of ray tracing are presented

a level set based eulerian method for paraxial multivalued traveltimes we apply the level set methodology to compute multivalued solutions of the paraxial eikonal equation in both isotropic and anisotropic metrics this paraxial equation is obtained from 2d stationary eikonal equations by using one of the spatial directions as the artificial evolution direction the advection velocity field used to move level sets is obtained by the method of characteristics therefore the motion of level sets is defined in a phase space and the zero level set yields the location of bicharacteristic strips in the reduced phase space the multivalued traveltime is obtained from solving another advection equation with a source term the complexity of the algorithm is o n3 logn in the worst case and o n3 in the average case where n is the number of the sampling points along one of the spatial directions numerical experiments including the well known marmousi synthetic model illustrate the accuracy and the efficiency of the eulerian method

a field space based level set method for computing multi valued solutions to 1d euler poisson equations we present a field space based level set method for computing multi valued solutions to one dimensional euler poisson equations the system of these equations has many applications and in particular arises in semiclassical approximations of the schrodinger poisson equation the proposed approach involves an implicit eulerian formulation in an augmented space called field space which incorporates both velocity and electric fields into the configuration both velocity and electric fields are captured through common zeros of two level set functions which are governed by a field transport equation simultaneously we obtain a weighted density f by solving again the field transport equation but with initial density as starting data the averaged density is then resolved by the integration of the obtained f against the dirac delta function of two level set functions in the field space moreover we prove that such obtained averaged density is simply a linear superposition of all multi valued densities and the averaged field quantities are weighted superposition of corresponding multi valued ones computational results are presented and compared with some exact solutions which demonstrate the effectiveness of the proposed method

a local level set method for paraxial geometrical optics we propose a local level set method for constructing the geometrical optics term in the paraxial formulation for the high frequency asymptotics of two dimensional 2 d acoustic wave equations the geometrical optics term consists of two multivalued functions a travel time function satisfying the eikonal equation locally and an amplitude function solving a transport equation locally the multivalued travel times are obtained by solving a level set equation and a travel time equation with a forcing term the multivalued amplitudes are computed by a neweulerian formula based on the gradients of travel times and takeoff angles as a byproduct the method is also able to capture the caustic locations the proposed eulerian method has complexity of o n 2 rm log n rather than o n 4 as typically seen in the lagrangian ray tracing method several examples including the well known marmousi synthetic model illustrate the accuracy and efficiency of the eulerian method

multiple reflection and transmission phases in complex layered media using a multistage fast marching method traditional grid based eikonal schemes for computing traveltimes are usually confined to obtaining first arrivals only however later arrivals can be numerous and of greater amplitude making them a potentially valuable resource for practical applications such as seismic imaging the aim of this paper is to introduce a grid based method for tracking multivalued wavefronts composed of any number of reflection and refraction branches in layered media a finite difference eikonal solver known as the fast marching method fmm is used to propagate wavefronts from one interface to the next by treating each layer that the wavefront enters as a separate computational domain one obtains a refracted branch by reinitializing fmm in the adjacent layer and a reflected branch by reinitializing fmm in the incident layer to improve accuracy a local grid refinement scheme is used in the vicinity of the source where wavefront curvature is high several examples are presented which demonstrate the viability of the new method in highly complex layered media even in the presence of velocity variations as large as 8 1 and interfaces of high curvature wavefronts composed of many reflection and transmission events are tracked rapidly and accurately this is because the scheme retains the two desirable properties of a single stage fmm computational speed and stability local grid refinement about the source also can increase accuracy by an order of magnitude with little increase in computational cost

information geometry on hierarchy of probability distributions an exponential family or mixture family of probability distributions has a natural hierarchical structure this paper gives an orthogonal decomposition of such a system based on information geometry a typical example is the decomposition of stochastic dependency among a number of random variables in general they have a complex structure of dependencies pairwise dependency is easily represented by correlation but it is more difficult to measure effects of pure triplewise or higher order interactions dependencies among these variables stochastic dependency is decomposed quantitatively into an orthogonal sum of pairwise triplewise and further higher order dependencies this gives a new invariant decomposition of joint entropy this problem is important for extracting intrinsic interactions in firing patterns of an ensemble of neurons and for estimating its functional connections the orthogonal decomposition is given in a wide class of hierarchical structures including both exponential and mixture families as an example we decompose the dependency in a higher order markov chain into a sum of those in various lower order markov chains

efficient level set methods for constructing wavefronts in three spatial dimensions wavefront construction in geometrical optics has long faced the twin difficulties of dealing with multi valued forms and resolution of wavefront surfaces a recent change in viewpoint however has demonstrated that working in phase space on bicharacteristic strips using eulerian methods can bypass both difficulties the level set method for interface dynamics makes a suitable choice for the eulerian method unfortunately in three dimensional space the setting of interest for most practical applications the advantages of this method are largely offset by a new problem the high dimension of phase space in this work we present new types of level set algorithms that remove this obstacle and demonstrate their abilities to accurately construct wavefronts under high resolution these results propel the level set method forward significantly as a competitive approach in geometrical optics under realistic conditions

a bloch band based level set method for computing the semiclassical limit of schrodinger equations a novel bloch band based level set method is proposed for computing the semiclassical limit of schrodinger equations in periodic media for the underlying equation subject to a highly oscillatory initial data a hybrid of the wkb approximation and homogenization leads to the bloch eigenvalue problem and an associated hamilton jacobi system for the phase in each bloch band with the bloch eigenvalue be part of the hamiltonian we formulate a level set description to capture multi valued solutions to the band wkb system and then evaluate total homogenized density over a sample set of bands a superposition of band densities is established over all bands and solution branches when away from caustic points the numerical approach splits the solution process into several parts i initialize the level set function from the band decomposition of the initial data ii solve the bloch eigenvalue problem to compute bloch waves iii evolve the band level set equation to compute multi valued velocity and density on each bloch band iv evaluate the total position density over a sample set of bands using bloch waves and band densities obtained in steps ii and iii respectively numerical examples with different number of bands are shown to demonstrate the capacity of the method

level set approach to mean curvature flow in arbitrary codimension we develop a level set theory for the mean curvature evolution of surfaces with arbitrary co dimension thus generalizing the previous work 8 15 on hypersurfaces the main idea is to surround the evolving surface of co dimension k in r by a family of hypersurfaces the level sets of a function evolving with normal velocity equal to the sum of the d k smallest principal curvatures the existence and the uniqueness of a weak level set solution is easily established using mainly the results of 8 and the theory of viscosity solutions for second order nonlinear parabolic equations the level set solutions coincide with the classical solutions whenever the latter exist the proof of this connection uses a careful analysis of the squared distance from the surfaces it is also shown that varifold solutions constructed by brakke 7 are included in the level set solutions the idea of surrounding the evolving surface by a family of hypersurfaces with a certain property is related to the barriers of de giorgi an introduction to the theory of barriers and his connection to the level set solutions is also provided

vector level sets for description of propagating cracks in finite elements a new level set method is developed for describing surfaces that are frozen behind a moving front such as cracks in this formulation the level set is described in two dimensions by a three tuple the sign of the level set function and the components of the closest point projection to the surface the update of the level set is constructed by geometric formulas which are easily implemented results are given for growth of lines in two dimensions that show the method is very accurate the method combines very naturally with the extended finite element method xfem where the discontinuous enrichment for cracks is best described in terms of level set functions examples of crack growth simulations obtained by combining this level set method with the extended finite element method are given copyright 2003 john wiley sons ltd

information geometric approach to inferring causal directions while conventional approaches to causal inference are mainly based on conditional in dependences recent methods also account for the shape of conditional distributions the idea is that the causal hypothesis x causes y imposes that the marginal distribution p x and the conditional distribution p y x represent independent mechanisms of nature recently it has been postulated that the shortest description of the joint distribution p x y should therefore be given by separate descriptions of p x and p y x since description length in the sense of kolmogorov complexity is uncomputable practical implementations rely on other notions of independence here we define independence via orthogonality in information space this way we can explicitly describe the kind of dependence that occurs between p y and p x y making the causal hypothesis y causes x implausible remarkably this asymmetry between cause and effect becomes particularly simple if x and y are deterministically related we present an inference method that works in this case we also discuss some theoretical results for the non deterministic case although it is not clear how to employ them for a more general inference method

the use of a legendre pseudospectral viscosity technique to solve a class of nonlinear dynamic hamilton jacobi equations in this research we study the problem of finding the approximate solution of a class of hamilton jacobi equations namely the eikonal equation we employ the legendre pseudospectral viscosity method to solve this problem this method basically consists of adding a spectral viscosity to the equation this spectral viscosity which is sufficiently small to retain the formal spectral accuracy is large enough to stabilize the numerical scheme several test problems are considered and the numerical results are given to show the efficiency of the proposed method

numerical solution of the high frequency asymptotic expansion for the scalar wave equation new numerical methods are derived for calculation of high frequency asymptotic expansion of the scalar wave equation the nonlinear partial differential equations defining the terms in the expansio

multi phase computations of the semiclassical limit of the schrodinger equation and related problems whitham vs wigner abstract we present and compare two different techniques to obtain the multi phase solutions for the schrodinger equation in the semiclassical limit the first is whitham s averaging method which gives the modulation equations governing the evolution of multi phase solutions the second is the wigner transform a convenient tool to derive the semiclassical limit equation in the phase space the vlasov equation for the linear schrodinger equation motivated by the linear superposition principle we derive and prove the multi phase ansatz for the wigner function by the stationary phase method and then use the ansatz to close the moment equations of the vlasov equation and obtain the multi phase equations in the physical space we show that the multi phase equations so derived agree with those derived by whitham s averaging method which can be proved using different arguments generic way of obtaining and computing the multi phase equations by the wigner function is given and kinetic schemes are introduced to solve the multi phase equations the numerical schemes based on multiphase closure of the liouville equation are purely eulerian and only operate in the physical space several numerical examples are given to explore the validity of this approach similar studies are conducted for the linearized korteweg de vries equation and the linear wave equation

high order numerical methods to a type of delta function integrals we study second to fourth order numerical methods to a type of delta function integrals in one to three dimensions these delta function integrals arise from recent efficient level set methods for computing the multivalued solutions of nonlinear pdes we show that the natural quadrature approach with usual discrete delta functions and support size formulas to the two dimensional delta function integrals suffer from nonconvergence we then design high order numerical methods to this type of delta function integrals based on interpolation approach numerical examples are presented to verify the efficiency and accuracy of our methods

on the evolution of vector distance functions of closed curves inspired by the work by gomes et al we describe and analyze a vector distance function approach for the implicit evolution of closed curves of codimension larger than one the approach is set up in complete generality and then applied to the evolution of dynamic geometric active contours in mathbb r 4 codimension three case in order to carry this out one needs an explicit expression for the zero level set for which we propose a discrete connectivity method this leads us to make connections with the new theory of cubical homology we provide some explicit simulation results in order to illustrate the methodology

a new eulerian method for the computation of propagating short acoustic and electromagnetic pulses a new method is described to compute short acoustic or electromagnetic pulses that propagate according to geometrical optics the pulses are treated as zero thickness sheets that can propagate over long distances through inhomogeneous media with multiple reflections the method has many of the advantages of lagrangian ray tracing but is completely eulerian typically using a uniform cartesian grid accordingly it can treat arbitrary configurations of pulses that can reflect from surfaces and pass through each other without requiring special computational marker arrays for each pulse also information describing the pulses which are treated as continuous surfaces can be available throughout the computational grid rather than only at isolated individual markers the method uses a new type of representation which we call dynamic surface extension the basic idea is to propagate or broadcast defining fields from each pulse surface through a computational grid these fields carry information about a nearby pulse surface that is used at each node to compute the location of the pulse surfaces and other attributes such as amplitude thus the emphasis is on the dynamics of these propagating defining fields which obey only local eulerian equations at each node the dynamic surface extension representation can be thought of as dual to level set representation the defining fields involve single valued variables which are constant at each time along lines that are normal to the evolving surface whereas level set techniques involve a function which has constant values on the evolving surface and neighboring surfaces in this way the new method overcomes the inability of level set or eikonal methods to treat intersecting pulses that obey a wave equation and can pass through each other while still using only single valued variables propagating thin pulse surfaces in 1 d 2 d and 3 d that can reflect from boundaries and pass through each other are computed using the new method the method was first presented as a new general representation of surfaces filaments and particles by j steinhoff and m fan 1998 eulerian computation of evolving surfaces curves and discontinuous fields utsi preprint

geometric optics in a phase space based level set and eulerian framework we introduce a level set approach for ray tracing and the construction of wavefronts in geometric optics this is important in a wide variety of applications in wave propagation our approach automatically handles the multivalued solutions that appear and automatically resolves the wavefronts this is achieved through solving for the bicharacteristic strips whose projection to spatial space gives the wavefronts in a reduced phase space under a eulerian and partial differential equation based framework the bicharacteristic strips are represented using a level set approach for handling higher codimensional objects and the partial differential equations responsible for the evolution are reduced forms of the liouville equations results for the two dimensional case for constant and variable indices of refraction are shown and compared to those of other current methods in the field results are also introduced to show the ability to handle reflection and to extend the method to the three dimensional case

a simple level set method for solving stefan problems a simple level set method for solving stefan problems is presented this method can be applied to problems involving dendritic solidification our method consists of an implicit finite difference scheme for solving the heat equation and a level set approach for capturing the front between solid and liquid phases of a pure substance our method is accurate with respect to some exact solutions of the stefan problem results indicate that this method can handle topology changes and complicated interfacial shapes and that it can numerically simulate many of the physical features of dendritic solidification

computing multi valued physical observables for the high frequency limit of symmetric hyperbolic systems we develop a level set method for the computation of multi valued physical observables density velocity energy etc for the high frequency limit of symmetric hyperbolic systems in any number of space dimensions we take two approaches to derive the method the first one starts with a weakly coupled system of an eikonal equation for phase s and a transport equation for density r ts h x s 0 t x r xr n email protected email protected x email protected ph x xs 0 the main idea is to evolve the density near the n dimensional bi characteristic manifold of the eikonal hamiltonian jacobi equation which is identified as the common zeros of n level set functions in phase space x k r 2 n these level set functions are generated from solving the liouville equation with initial data chosen to embed the phase gradient simultaneously we track a new quantity email protected t x k det email protected by solving again the liouville equation near the obtained zero level set f 0 but with initial density as initial data the multi valued density and higher moments are thus resolved by integrating f along the bi characteristic manifold in the phase directions the second one uses the high frequency limit of symmetric hyperbolic systems derived by the wigner transform this gives rise to liouville equations in the phase space with measure valued solution in its initial data due to the linearity of the liouville equation we can decompose the density distribution into products of function each of which solves the liouville equation with l initial data on any bounded domain it yields higher order moments such as energy and energy flux the main advantages of these new approaches in contrast to the standard kinetic equation approach using the liouville equation with a dirac measure initial data include 1 the liouville equations are solved with l initial data and a singular integral involving the email protected function is evaluated only in the post processing step thus avoiding oscillations and excessive numerical smearing 2 a local level set method can be utilized to significantly reduce the computation in the phase space these methods can be used to compute all physical observables for multi dimensional problems our method applies to the wave fields corresponding to simple eigenvalues of the dispersion matrix one such example is the wave equation which will be studied numerically in this paper

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

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

bloch decomposition based gaussian beam method for the schrodinger equation with periodic potentials the linear schrodinger equation with periodic potentials is an important model in solid state physics the most efficient direct simulation using a bloch decomposition based time splitting spectral method 18 requires the mesh size to be o e where e is the scaled semiclassical parameter in this paper we generalize the gaussian beam method introduced in jin et al 23 to solve this problem asymptotically we combine the technique of bloch decomposition and the eulerian gaussian beam method to arrive at an eulerian computational method that requires mesh size of o e the accuracy of this method is demonstrated via several numerical examples

computing multivalued physical observables for the semiclassical limit of the schrodinger equation we develop a level set method for the computation of multivalued physical observables density velocity etc for the semiclassical limit of the schrodinger equation this method uses an eulerian formulation and applies directly to arbitrary number of space dimensions the main idea is to evolve the density near an n dimensional manifold that is identified as the common zeros of n level set functions in phase space these level set functions are generated from solving the liouville equation with initial data chosen to embed the phase gradient simultaneously we track a new quantity f by solving again the liouville equation near the obtained zero level set but with initial density as initial data the multivalued density and higher moments are thus resolved by integrating f along the n dimensional manifold in the phase directions we show that this is equivalent to using the wigner approach but decomposing the velocity from the density each of which evolves by the same liouville equation the main advantages of this approach in contrast to the standard kinetic equation approach using the liouville equation with a dirac measure initial data include 1 the liouville equations are solved with l initial data and a singular integral involving the email protected function is evaluated only in the post processing step thus avoiding oscillations and excessive numerical smearing 2 a local level set method can be utilized to significantly reduce the computation in the phase space these advantages allow us to compute for the first time all physical observables for multidimensional problems in an eulerian framework

simplicial isosurfacing in arbitrary dimension and codimension the level set method has been successfully used for moving interface problems the final step of the method is to construct and visualize the isosurface of a discrete function 0 n n rm there have existed many practical isosurfacing algorithms when n 3 m 1 or n 2 m 1 recently we have begun to see the development of isosurfacing algorithms for higher dimensions and codimensions this paper introduces a unified theory and an efficient isosurfacing algorithm that works in arbitrary number of dimensions and codimensions the isosurface of a discrete function is defined as the isosurface of its simplicial interpolant 0 n n rm with this simplicial definition is geometrically a piecewise intersection of a simplex and m hyperplanes is constructed as the union of simplices the construction costs o nn with a uniform grid and o nn m log n with a dyadic grid in numerical space and time when n m 1 or m 2 is projected down into r3 and can be visualized for surface visualizations a simple formula is presented calculating the normal vector field of the projection of into r3 which gives light shadings

universal optimality of balanced uniform crossover designs kunert ann statist 12 1984 1006 1017 proved that in the class of repeated measurement designs based on t treatments p t periods and n t experimental units a balanced uniform design is universally optimal for direct treatment effects if t 3 and 1 or if t 6 and 2 this result is generalized to t 3 as long as t 1 2

on the entropy production of time series with unidirectional linearity there are non gaussian time series that admit a causal linear autoregressive moving average arma model when regressing the future on the past but not when regressing the past on the future the reason is that in the latter case the regression residuals are not statistically independent of the regressor in previous work we have experimentally verified that many empirical time series indeed show such a time inversion asymmetry

a phase based hybridizable discontinuous galerkin method for the numerical solution of the helmholtz equation we introduce a new hybridizable discontinuous galerkin hdg method for the numerical solution of the helmholtz equation over a wide range of wave frequencies our approach combines the hdg methodology with geometrical optics in a fashion that allows us to take advantage of the strengths of these two methodologies the phase based hdg method is devised as follows first we enrich the local approximation spaces with precomputed phases which are solutions of the eikonal equation in geometrical optics second we propose a novel scheme that combines the hdg method with ray tracing to compute multivalued solution of the eikonal equation third we utilize the proper orthogonal decomposition to remove redundant modes and obtain locally orthogonal basis functions which are then used to construct the global approximation spaces of the phase based hdg method and fourth we propose an appropriate choice of the stabilization parameter to guarantee stability and accuracy for the proposed method numerical experiments presented show that optimal orders of convergence are achieved that the number of degrees of freedom to achieve a given accuracy is independent of the wave number and that the number of unknowns required to achieve a given accuracy with the proposed method is orders of magnitude smaller than that with the standard finite element method

a grid based particle method for moving interface problems we propose a novel algorithm for modeling interface motions the interface is represented and is tracked using quasi uniform meshless particles these particles are sampled according to an underlying grid such that each particle is associated to a grid point which is in the neighborhood of the interface the underlying grid provides an eulerian reference and local sampling rate for particles on the interface it also renders neighborhood information among the meshless particles for local reconstruction of the interface the resulting algorithm which is based on lagrangian tracking using meshless particles with eulerian reference grid can naturally handle control topological changes moreover adaptive sampling of the interface can be achieved easily through local grid refinement with simple quad oct tree data structure extensive numerical examples are presented to demonstrate the capability of our new algorithm

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

universal optimality of patterson s crossover designs we show that the balanced crossover designs given by patterson biometrika 39 1952 32 48 are a universally optimal uo for the joint estimation of direct and residual effects when the competing class is the class of connected binary designs and b uo for the estimation of direct residual effects when the competing class of designs is the class of connected designs which includes the connected binary designs in which no treatment is given to the same subject in consecutive periods in both results the formulation of uo is as given by shah and sinha unpublished manuscript 2002 further we introduce a functional of practical interest involving both direct and residual effects and establish c optimality of patterson s designs with respect to this functional when the class of competing designs is as in b above

big ray tracing different traveltimes are identified as viscosity solutions of the eikonal equation on locally selected domains called big rays big rays are constructed by an ad hoc algorithm based on a coarse ray tracing viscosity solutions in the big rays are computed using an upwind finite difference scheme this hybrid method allows an easy and fast computation of multivalued traveltime fields

traveltime and amplitude estimation using wavefront construction we have developed and implemented a new method for estimating traveltimes and amplitudes in a general smooth two dimensional 2 d model the basic idea of this wavefront wf construction approach is to use ray tracing to estimate a new wf from the old one the wf is defined as a curve in 2 d of constant traveltime from the source the ray direction and amplitude will then be a function of s the distance along the front to maintain a sufficiently small sampling distance along the wf it is scanned at every time step and new rays are interpolated whenever the distance between two rays becomes larger than a predefined limit as the wavefronts are constructed the data i e traveltimes amplitude coefficients etc are transferred to the receivers by interpolation within the ray cells advantages of the wf construction method are its flexibility robustness and accuracy first second and later arrivals may be found at any point in the model any shape of the initial wavefront is possible the drawb

optimal crossover designs for the comparison of two treatments in the presence of carryover effects and autocorrelated errors summary a model for responses in a crossover experiment that includes direct and first order carryover treatment effects together with an autocorrelated error term is considered when there are two treatments a method for generating designs that minimize the variance of the estimated treatment effects is proposed and explicit results are given for the three and four period cases efficiencies of some practical designs are calculated

universal optimality for selected crossover designs hedayat and yang earlier proved that balanced uniform designs in the entire class of crossover designs based on t treatments n subjects and p t periods are universally optimal when n t t 1 2 surprisingly in the class of crossover designs with t treatments and p t periods a balanced uniform design may not be universally optimal if the number of subjects exceeds t t 1 2 this article among other results shows that a a balanced uniform design is universally optimal in the entire class of crossover designs withp t as long as n is not greater than t t 2 2 and 3 t 12 b a balanced uniform design with n 2t t 3 and p t is universally optimal in the entire class of crossover designs with n 2t and p t and c for the case where p t the design suggested by stufken is universally optimal thus completing kushner s result that a stufken design is universally optimal if n is divisible by t p 1

computational high frequency waves through curved interfaces via the liouville equation and geometric theory of diffraction we construct a class of numerical schemes for the liouville equation of geometric optics coupled with the geometric theory of diffractions to simulate the high frequency linear waves with a discontinuous index of refraction in this work s jin x wen a hamiltonian preserving scheme for the liouville equation of geometric optics with partial transmissions and reflections siam j numer anal 44 2006 1801 1828 a hamiltonian preserving scheme for the liouville equation was constructed to capture partial transmissions and reflections at the interfaces this scheme is extended by incorporating diffraction terms derived from geometric theory of diffraction into the numerical flux in order to capture diffraction at the interface we give such a scheme for curved interfaces this scheme is proved to be positive under a suitable time step constraint numerical experiments show that it can capture diffraction phenomena without fully resolving the wave length of the original wave equation

causal inference using the algorithmic markov condition inferring the causal structure that links n observables is usually based upon detecting statistical dependences and choosing simple graphs that make the joint measure markovian here we argue why causal inference is also possible when the sample size is one we develop a theory how to generate causal graphs explaining similarities between single objects to this end we replace the notion of conditional stochastic independence in the causal markov condition with the vanishing of conditional algorithmic mutual information and describe the corresponding causal inference rules we explain why a consistent reformulation of causal inference in terms of algorithmic complexity implies a new inference principle that takes into account also the complexity of conditional probability densities making it possible to select among markov equivalent causal graphs this insight provides a theoretical foundation of a heuristic principle proposed in earlier work we also sketch some ideas on how to replace kolmogorov complexity with decidable complexity criteria this can be seen as an algorithmic analog of replacing the empirically undecidable question of statistical independence with practical independence tests that are based on implicit or explicit assumptions on the underlying distribution

high order numerical methods to two dimensional delta function integrals in level set methods in this paper we design and analyze a class of high order numerical methods to delta function integrals appearing in level set methods in two dimensional case the methods comprise approximating the mesh cell restrictions of the delta function integral in each mesh cell the two dimensional delta function integral can be rewritten as a one dimensional ordinary integral with the smooth integrand being a one dimensional delta function integral and thus is approximated by applying standard one dimensional high order numerical quadratures and high order numerical methods to one dimensional delta function integrals proposed in x wen high order numerical methods to a type of delta function integrals j comput phys 226 2007 1952 1967 we establish error estimates for the method which show that the method can achieve any desired accuracy by assigning the corresponding accuracy to the sub algorithms and has better accuracy under an assumption on the zero level set of the level set function which holds generally numerical examples are presented showing that the second to fourth order methods implemented in this paper achieve or exceed the expected accuracy and demonstrating the advantage of using our high order numerical methods

quantum thermodynamics with missing reference frames decompositions of free energy into non increasing components if an absolute reference frame with respect to time position or orientation is missing one can only implement quantum operations which are covariant with respect to the corresponding unitary symmetry group g extending observations of vaccaro et al i argue that the free energy of a quantum system with g invariant hamiltonian then splits up into the holevo information of the orbit of the state under the action of g and the free energy of its orbit average these two kinds of free energy cannot be converted into each other the first component is subadditive and the second superadditive in the limit of infinitely many copies only the usual free energy matters

big ray tracing and eikonal solver on unstructured grids application to the computation of a multivalued traveltime field in the marmousi model this paper presents a numerical computation of the multivalued traveltime field generated by a point source experiment in the marmousi model two methods are combined to achieve this goal a method called big ray tracing used to compute multivalued traveltime fields and an eikonal solver designed to work on unstructured meshes big ray tracing is based on a combination of ray tracing and local solutions of the eikonal equation classical ray tracing first discretizes the phase space and defines local zones that possibly overlap where the traveltime field is multivalued then an eikonal solver computes traveltimes in these zones called big rays it acts as an exact interpolation process between rays associated with different branches of the traveltime field the geometry of big rays may be complicated and is better discretized using unstructured meshes an eikonal solver designed to work on unstructured meshes is used

a variational approach to path planning in three dimensions using level set methods in this paper we extend the two dimensional methods set forth in t cecil d marthaler a variational approach to search and path planning using level set methods ucla cam report 04 61 2004 proposing a variational approach to a path planning problem in three dimensions using a level set framework after defining an energy integral over the path we use gradient flow on the defined energy and evolve the entire path until a locally optimal steady state is reached we follow the framework for motion of curves in three dimensions set forth in p burchard l t cheng b merriman s osher motion of curves in three spatial dimensions using a level set approach j comput phys 170 2 2001 720 741 modified appropriately to take into account that we allow for paths with positive varying widths applications of this method extend to robotic motion and visibility problems for example numerical methods and algorithms are given and examples are presented

simplex free adaptive tree fast sweeping and evolution methods for solving level set equations in arbitrary dimension we introduce simplex free adaptive tree numerical methods for solving static and time dependent hamilton jacobi equations arising in level set problems in arbitrary dimension the data structure upon which our method is built in a generalized n dimensional binary tree but it does not require the complicated splitting of cubes into simplices aka generalized n dimensional triangles or hypertetrahedrons that current tree based methods require it has enough simplicity that minor variants of standard numerical hamiltonians developed for uniform grids can be applied yielding consistent monotone convergent schemes combined with the fast sweeping strategy the resulting tree based methods are highly efficient and accurate thus without changing more than a few lines of code when changing dimension we have obtained results for calculations in up to n 7 dimensions

jacobi s last geometric statement extends to a wider class of liouville surfaces numerical evidence is presented which strongly suggests that jacobi s last geometric statement that the conjugate locus from a point has exactly four cusps and the corresponding cut locus consists of only one topological segment holds for compact real analytic liouville surfaces diffeomorphic to s 2 if the gaussian curvature is everywhere positive and has exactly six critical points these being two saddles two global minima and two global maxima as is the case for an ellipsoid our experiments suggest that this is a sufficient rather than a necessary condition furthermore for compact real analytic liouville surfaces diffeomorphic to s 2 upon which the gaussian curvature can be negative but has exactly six critical points these being two saddles two global minima and two global maxima it appears that the cut locus is always a subarc of a line given by x 1 const or x 2 const where x 1 x 2 are canonical coordinates with respect to which the metric has the form f 1 x 1 f 2 x 2 dx 2 1 dx 2 2 in the case of an ellipsoid these curves are lines of curvature

optimal crossover designs for two treatments in the presence of mixed and self carryover effects the main purpose of this article is to identify optimal crossover designs for two treatments under a model that includes mixed and self carryover effects in addition results are reported for optimal two treatment crossover designs under several closely related models and the performance of various designs for three and four periods is studied under the different models

direct computation of multivalued phase space solutions for hamilton jacobi equations a method is presented for the exact and complete resolution of hamiltonian systems only using the corresponding hamilton jacobi equation we show that it possible to get the ode solution from the corresponding pde

a variational approach to optimal two treatment crossover designs application to carryover effect models abstract a variational method is used to obtain sufficient conditions for an optimal two treatment p period design given a repeated measures model with arbitrary within subject covariance the design minimizes the variance of the estimator of treatment differences over all possible allocations of n subjects to the 2 p possible treatment sequences the method is applied to a model that includes carryover effects when subject effects are random with equicorrelated observations designs are optimal if they are a strongly balanced b uniform on the rows and c either uniform on the subjects p is even and greater than 2 or uniform on the first p 1 periods if p is odd for two periods the optimal design is ab ba aa bb the same designs are optimal when baseline measurements are obtained in each period baseline observations improve efficiency considerably for p 2 only slightly for p odd and are of no help for p even and greater than 2 a second model of errors in which a subject s response ha

optimal and efficient repeated measurements designs for uncorrelated observations abstract in repeated measurements crossover or changeover designs experimental subjects units are repeatedly exposed to treatments effects of treatments from one period that extend into the next are called carryover or residual effects in exact design theory designs that are universally optimal in the presence of carryover effects have been found when a subject s responses are uncorrelated however the results are valid only for restricted values of t the number of treatments p the number of periods and n the number of subjects in other optimality results the competing class of designs is a subset of the full class of repeated measurements designs these restrictions hinder applicability in the setting of approximate design theory kushner 1997b developed a new approach valid even for correlated responses here the results are specialized to the case of uncorrelated responses linear equations are given that determine all universally or a and d optimal repeated measurements desig

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

optimal repeated measurements designs the linear optimality equations in approximate design theory necessary and sufficient conditions that a repeated measurements design be universally optimal are given as linear equations whose unknowns are the proportions of subjects on the treatment sequences both the number of periods and the number of treatments in the designs are arbitrary as is the covariance matrix of the normal response model the existence of universally optimal symmetric designs is proved the single linear equation which the proportions satisfy is given a formula for the information matrix of a universally optimal design is derived

a level set method for the semiclassical limit of the schrodinger equation with discontinuous potentials we propose a level set method for the semiclassical limit of the schrodinger equation with discontinuous potentials the discontinuities in the potential corresponds to potential barriers at which incoming waves can be partially transmitted and reflected previously such a problem was handled by jin and wen using the liouville equation which arises as the semiclassical limit of the schrodinger equation with an interface condition to account for partial transmissions and reflections s jin x wen siam j num anal 44 2006 1801 1828 however the initial data are dirac delta functions which are difficult to approximate numerically with a high accuracy in this paper we extend the level set method introduced in s jin h liu s osher r tsai j comp phys 210 2005 497 518 for this problem instead of directly discretizing the delta functions our proposed method decomposes the initial data into finite sums of smooth functions that remain smooth in finite time along the phase flow and hence can be solved much more easily using conventional high order discretization schemes two ideas are introduced here 1 the solutions of the problems involving partial transmissions and partial reflections are decomposed into a finite sum of solutions solving problems involving only complete transmissions and those involving only complete reflections for problems involving only complete transmission or complete reflection the method of jlot applies and is used in our simulations 2 a reinitialization technique is introduced so that waves coming from multiple transmissions and reflections can be combined seamlessly as new initial value problems this is implemented by rewriting the sum of several delta functions as one delta function with a suitable weight which can be easily implemented numerically we carry out numerical experiments in both one and two space dimensions to verify this new algorithm

hamiltonian preserving schemes for the liouville equation of geometrical optics with discontinuous local wave speeds in this paper we construct two classes of hamiltonian preserving numerical schemes for a liouville equation with discontinuous local wave speed this equation arises in the phase space description of geometrical optics and has been the foundation of the recently developed level set methods for multivalued solution in geometrical optics we extend our previous work in s jin x wen hamiltonian preserving schemes for the liouville equation with discontinuous potentials commun math sci 3 2005 285 315 for the semiclassical limit of the schrodinger equation into this system the designing principle of the hamiltonian preservation by building in the particle behavior at the interface into the numerical flux is used here and as a consequence we obtain two classes of schemes that allow a hyperbolic stability condition when a plane wave hits a flat interface the hamiltonian preservation is shown to be equivalent to snell s law of refraction in the case when the ratio of wave length over the width of the interface goes to zero when both length scales go to zero positivity and stabilities in both l 1 and l norms are established for both schemes the approach also provides a selection criterion for a unique solution of the underlying linear hyperbolic equation with singular discontinuous and measure valued coefficients benchmark numerical examples are given with analytic solution constructed to study the numerical accuracy of these schemes

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

optimal crossover designs in the presence of carryover effects under either the random patient effect model with sequence effects or the fixed patient effect model the usual two period two treatment crossover design ab ba cannot be used to estimate the contrast between direct treatment effects when unequal carryover effects are present if baseline observations are available the design ab ba can validly be used to estimate a treatment contrast however the design ab ba aa bb with baseline observations is more efficient in fact we show that this design is optimal whether or not baseline observations are available for experiments with more than two periods universally optimal designs are found for both models with and without carryover effects it is shown that uncertainty about the presence of carryover effects is of little or no consequence and the addition of baseline observations is of little or no added value for designs with three or more periods however if the experiment is limited to only two periods the investigator pays a heavy penalty

the theory of legendrian unfoldings and first order differential equations we consider some properties of completely integrable first order differential equations for real valued functions in order to study this subject we introduce the theory of legendrian unfoldings we give a characterisation of equations with classical complete solutions in terms of legendrian unfoldings and also assert that the set of equations with singular solutions is an open set in the space of completely integrable equations even though such a set is thin in the space of all equations

optimal change over designs when carry over effects are proportional to direct effects of treatments summary we investigate optimal change over designs for a model in which treatment carry over effects are proportional to their direct effects and the proportionality parameter i is unknown a combination of analytical results and computer search is used to identify optimal designs for the estimation of direct effects neighbour balanced designs are shown to have good optimality properties designs with self adjacencies perform better for positive i while those without self adjacencies are better when i is negative the application of the designs to sensory studies is discussed

visibility and its dynamics in a pde based implicit framework we investigate the problem of determining visible regions given a set of moving obstacles and a moving vantage point our approach to this problem is through an implicit framework where the obstacles are represented by a level set function the visibility problem is formally formulated as a boundary value problem bvp of a first order partial differential equation it is based on the continuation of values along the given ray field we propose a one pass multi level algorithm for the construction of the solution on a grid furthermore we study the dynamics of shadow boundaries on the surfaces of the obstacles when the vantage point moves along a given trajectory in all of these situations topological changes such as merging and breaking occur in the regions of interest these are automatically handled by the level set framework proposed here finally we obtain additional useful information through simple operations in the level set framework

optimal and efficient crossover designs when subject effects are random most studies on optimal crossover designs are based on models that assume subject effects to be fixed effects in this article we identify and study optimal and efficient designs for a model with random subject effects with the number of periods not exceeding the number of treatments we find that totally balanced designs are universally optimal for treatment effects in a large subclass of competing designs however in the entire class of designs totally balanced designs are in general not optimal and their efficiency depends on the ratio of the subject effects variance and the error variance we develop tools to study the efficiency of totally balanced designs and to identify designs with higher efficiency

high frequency wave propagation by the segment projection method geometrical optics is a standard technique used for the approximation of high frequency wave propagation computational methods based on partial differential equations instead of the traditional ray tracing have recently been applied to geometrical optics these new methods have a number of advantages but typically exhibit difficulties with linear superposition of waves in this paper we introduce a new partial differential technique based on the segment projection method in phase space the superposition problem is perfectly resolved and so is the problem of computing amplitudes in the neighborhood of caustics the computational complexity is of the same order as that of ray tracing the new algorithm is described and a number of computational examples are given including a simulation of waveguides

eulerian gaussian beams for schrodinger equations in the semi classical regime we propose gaussian beam based eulerian methods to compute semi classical solutions of the schrodinger equation traditional gaussian beam type methods for the schrodinger equation are based on the lagrangian ray tracing based on the first eulerian gaussian beam framework proposed in leung et al s leung j qian r burridge eulerian gaussian beams for high frequency wave propagation geophysics 72 2007 sm61 sm76 we develop a new eulerian gaussian beam method which uses global cartesian coordinates level set based implicit representation and liouville equations the resulting method gives uniformly distributed phases and amplitudes in phase space simultaneously to obtain semi classical solutions to the schrodinger equation with different initial wave functions we only need to slightly modify the summation formula this yields a very efficient method for computing semi classical solutions to the schrodinger equation for instance in the one dimensional case the proposed algorithm requires only o snm2 operations to compute s different solutions with s different initial wave functions under the influence of the same potential where n o 1 is the planck constant and m n is the number of computed beams which depends on weakly numerical experiments indicate that this eulerian gaussian beam approach yields accurate semi classical solutions even at caustics

markov chain approximations for deterministic control problems with affine dynamics and quadratic cost in the control we consider the construction of markov chain approximations for an important class of deterministic control problems the emphasis is on the construction of schemes that can be easily implemented and which possess a number of highly desirable qualitative properties the class of problems covered is that for which the control is affine in the dynamics and with quadratic running cost this class covers a number of interesting areas of application including problems that arise in large deviations risk sensitive and robust control robust filtering and certain problems in computer vision examples are given as well as a proof of convergence

optimality and efficiency of two treatment repeated measurements designs for an arbitrary covariance matrix and any number of treatments and periods necessary and sufficient conditions for the optimality of a repeated measurements i e crossover or changeover design were given in the as yet unpublished report by h b kushner optimal repeated measurements designs the linear optimality equations kushner report here we specialise the results to the two treatment case we give linear equations whose solutions are the proportions of subjects allocated to each treatment sequence in the most general optimal design the treatment effects information matrix of an optimal design is also presented the results are applied to several situations for arbitrary covariance matrices we show that designs consisting of at most two treatment sequences and their duals must necessarily be dual balanced to be optimal for autoregressive covariance matrices we confirm that designs of matthews 1987 for three and four periods are optimal and we derive new optimal designs for five and six periods some optimality results of laska meisner 1985 are extended a simple formula for the efficiency of a dual balanced nonoptimal design is provided and used to compute the exact efficiencies of some of kunert s 1991 designs

the phase flow method this paper introduces the phase flow method a novel accurate and fast approach for constructing phase maps for nonlinear autonomous ordinary differential equations the method operates by initially constructing the phase map for small times using a standard ode integration rule and builds up the phase map for larger times with the help of a local interpolation scheme together with the group property of the phase flow the computational complexity of building up the complete phase map is usually that of tracing a few rays in addition the phase flow method is provably and empirically very accurate once the phase map is available integrating the ode for initial conditions on the invariant manifold only makes use of local interpolation thus having constant complexity the paper develops applications in the field of high frequency wave propagation and shows how to use the phase flow method to 1 rapidly propagate wave fronts 2 rapidly calculate wave amplitudes along these wave fronts and 3 rapidly evaluate multiple wave arrival times at arbitrary locations

semi supervised interpolation in an anticausal learning scenario according to a recently stated independence postulate the distribution pcause contains no information about the conditional peffect cause while peffect may contain information about pcause effect since semi supervised learning ssl attempts to exploit information from px to assist in predicting y from x it should only work in anticausal direction i e when y is the cause and x is the effect in causal direction when x is the cause and y the effect unlabelled x values should be useless to shed light on this asymmetry we study a deterministic causal relation y f x as recently assayed in information geometric causal inference igci within this model we discuss two options to formalize the independence of px and f as an orthogonality of vectors in appropriate inner product spaces we prove that unlabelled data help for the problem of interpolating a monotonically increasing function if and only if the orthogonality conditions are violated which we only expect for the anticausal direction here performance of ssl and its supervised baseline analogue is measured in terms of two different loss functions first the mean squared error and second the surprise in a bayesian prediction scenario

an accurate spectral discontinuous finite element formulation of a phase space based level set approach to geometrical optics in this paper we introduce a new numerical procedure for simulations in geometrical optics that based on the recent development of eulerian phase space formulations of the model can deliver very accurate uniformly resolved solutions which can be made to converge with arbitrarily high orders in general geometrical configurations following previous treatments the scheme is based on the evolution of a wavefront in phase space defined as the intersection of level sets satisfying the relevant liouville equation in contrast with previous work however our numerical approximation is specifically designed i to take full advantage of the smoothness of solutions ii to facilitate the treatment of scattering obstacles all while retaining high order convergence characteristics indeed to incorporate the full regularity of solutions that results from the unfolding of singularities our method is based on their spectral representation to enable a simple high order treatment of scattering boundaries on the other hand we resort to a discontinuous galerkin finite element method for the solution of the resulting system of equations the procedure is complemented with the use of a recently derived strong stability preserving runge kutta ssp rk scheme for the time integration that as we demonstrate allows for overall approximations that are rapidly convergent

motion of curves in three spatial dimensions using a level set approach the level set method was originally designed for problems dealing with codimension one objects where it has been extremely succesful especially when topological changes in the interface i e merging and breaking occur attempts have been made to modify it to handle objects of higher codimension such as vortex filaments while preserving the merging and breaking property we present numerical simulations of a level set based method for moving curves in r3 the model problem for higher codimension that allows for topological changes a vector valued level set function is used with the zero level set representing the curve our results show that this method can handle many types of curves moving under all types of geometrically based flows while automatically enforcing merging and breaking

optimal design and refinement of the linera model with applications to repeated measurements designs the information matrices of one design in a finer and a simpler linear model are compared to each other the orthogonality condition ensuring that both matrices are equal is examined in the model for repeated measurements designs which was considered e g by cheng and wu 1980 examples of unbalanced designs fulfilling the orthogonality condition are shown to be optimum moreover nearly strongly balanced generalized latin squares are introduced and their universal optimality is proved if the numbers of units and periods are sufficiently large

a hamiltonian preserving scheme for the liouville equation of geometrical optics with partial transmissions and reflections we construct a class of hamiltonian preserving numerical schemes for the liouville equation of geometrical optics with partial transmissions and reflections this equation arises in the high frequency limit of the linear wave equation with a discontinuous index of refraction in our previous work hamiltonian preserving schemes for the liouville equation of geometrical optics with discontinuous local wave speeds j comput phys 214 2006 pp 672 697 we introduced the hamiltonian preserving schemes for the same equation when only complete transmissions or reflections occur at the interfaces these schemes are extended in this paper to the general case of partial transmissions and reflections the key idea is to build into the numerical flux the behavior of waves at the interface namely partial transmissions and reflections that satisfy snell s law of refraction with the correct transmission and reflection coefficients this scheme allows a hyperbolic stability condition under which positivity and stabilities in both l 1 and l infty norms are established numerical experiments are carried out to study the numerical accuracy