discretization on non orthogonal quadrilateral grids for inhomogeneous anisotropic media two classes of discretization methods are proposed for control volume formulations on quadrilateral grids in two space dimensions curvilinear grids are considered as a special case the methods are applicable for any system of conservation laws where the flux is defined by a gradient law like darcy s law for porous media flow a strong feature of the methods is the ability to handle media inhomogeneities in combination with full tensor anisotropy and or non orthogonality of the grid further properties of the methods will be discussed and examples of their use will be presented

isoperimetry and integrability of the sum of independent banach space valued random variables we develop a new method to study the tails of a sum of independent mean zero banach space valued random variables x i i leq n it relies on a new isoperimetric inequality for subsets of a product of probability spaces in particular we prove that for p geq 1 bigg sum i leq n x i bigg p leq frac kp 1 log p bigg bigg sum i leq n x i bigg 1 max i leq n x i p bigg where k is a universal constant other optimal inequalities for exponential moments are obtained

fully implicit mixed hybrid finite element discretization for general purpose subsurface reservoir simulation abstract we present a new fully implicit mixed hybrid finite element mhfe discretization scheme for general purpose compositional reservoir simulation the locally conservative scheme solves the coupled momentum and mass balance equations simultaneously and the fluid system is modeled using a cubic equation of state we introduce a new conservative flux approach for the mass balance equations for this fully implicit approach we discuss the nonlinear solution procedure for the proposed approach and we present extensive numerical tests to demonstrate the convergence and accuracy of the mhfe method using tetrahedral elements we also compare the method to other advanced discretization schemes for unstructured meshes and tensor permeability finally we illustrate the applicability and robustness of the method for highly heterogeneous reservoirs with unstructured grids

best constants in martingale version of rosenthal s inequality the following generalization of rosenthal s inequality was proved by burkholder a 1 p s f p d ast p leq f ast p leq b p s f p d ast p for all martingales f n it is known that a p grows like sqrt p as p rightarrow infty in this paper we prove that the growth rate of b p as p rightarrow infty is p ln p

numerical simulations for radiation hydrodynamics a finite difference scheme is proposed for two dimensional radiation hydrodynamical equations in the transport limit the scheme is of godunov type in which the set of time averaged flux needed in the scheme is calculated through riemann problems solved in the scheme flow signals are explicitly treated while radiation signals are implicitly treated flow fields and radiation fields are updated simultaneously an iterative approach is proposed to solve the set of nonlinear algebraic equations arising from the implicitness of the scheme the sweeping method used in the scheme significantly reduces the number of iterations or computer cpu time needed a new approach to further accelerate the convergence is proposed which further reduces the number of iterations needed by more than one order no matter how many cells radiation signals propagate in one time step only an extremely small number of iterations are needed in the scheme and each iteration costs only about 0 8 percent of computer cpu time which is needed for one time step of a second order accurate and fully explicit scheme two dimensional problems are treated through a dimensionally split technique therefore iterations for solving the set of algebraic equations are carried out only in each one dimensional sweep through numerical examples it is shown that the scheme keeps the principle advantages of godunov schemes for flow motion in the time scale of flow motion numerical results are the same as those obtained from a second order accurate and fully explicit scheme the acceleration of the convergence proposed in this paper may be directly applied to other hyperbolic systems

piecewise linear discretization of symbolic implicit monte carlo radiation transport in the difference formulation we describe a monte carlo solution for time dependent photon transport in the difference formulation with the material in local thermodynamic equilibrium that is piecewise linear in its treatment of the material state variable our method employs a galerkin solution for the material energy equation while using symbolic implicit monte carlo to solve the transport equation in constructing the scheme one has the freedom to choose between expanding the material temperature or the equivalent black body radiation energy density at the material temperature in terms of finite element basis functions the former provides a linear treatment of the material energy while the latter provides a linear treatment of the radiative coupling between zones subject to the conditional use of a lumped material energy in the vicinity of strong gradients possible with a linear treatment of the material energy our approach provides a robust solution for time dependent transport of thermally emitted radiation that can address a wide range of problems it produces accurate results in thick media

validation of the shnc time dependent transport code based on the spherical harmonics method for complex nuclear fuel assemblies abstract the diffusion approximation to the time dependent boltzmann transport equation gives accurate results for traditional nuclear reactor designs but new reactor designs and new fuel elements require neutron transport methods we develop a numerical approximation to the time dependent transport equation coupled to delayed neutron precursors based on the spherical harmonics p l equations for odd l and on the backward euler finite difference discretization of time the resulting scheme can be written as a stationary form of diffusive second order p l equations this allows a reduction by half to the number of unknowns and also to apply a nodal collocation method to the spatial discretization of the problem using coarse spatial grids to further reduce memory requirements this scheme is validated with several transient benchmarks where the convergence properties are established and compared with the simplified p l approximation a more realistic transient benchmark based on the two group c5 mox problem is finally introduced showing the need of high order p l approximation for complex fuel geometries

a comparison of implicit time integration methods for nonlinear relaxation and diffusion several time integration methods for nonlinear systems are compared all of the time discretizations are based on the method but differ in their treatment of the implicit nonlinear terms one method converges the implicit nonlinear terms to a small tolerance and is often referred to as nonlinearly consistent nc newton s method or its approximation newton krylov is used to converge the nonlinearities the other methods considered are linearized and comparisons are made for a relaxation problem and a radiation diffusion problem the linearized one step method that uses the full jacobian is shown to have similar accuracy as nc methods the lagged linearization method and an extension that is second order accurate are also studied a trunction error analysis complements the numerical results for the relaxation problem it is shown that each of the second order accurate linearized methods may be more accurate than an nc method depending on the degree of nonlinearity in the problem for the radiation diffusion problem in general the nc method is most accurate and allows a larger time step however the linearized methods perfom surprisingly well

enriched multi point flux approximation for general grids it is well known that the two point flux approximation a numerical scheme used in most commercial reservoir simulators has o 1 error when grids are not k orthogonal in the last decade the multi point flux approximations have been developed as a remedy however non physical oscillations can appear when the anisotropy is really strong we found out the oscillations are closely related to the poor approximation of pressure gradient in the flux computation in this paper we propose the control volume enriched multi point flux approximation empfa for general diffusion problems on polygonal and polyhedral meshes non physical oscillations are not observed for realistic and strongly anisotropic heterogeneous material properties described by a full tensor exact linear solutions are recovered for grids with non planar interfaces and a first and second order convergence are achieved for the flux and scalar unknowns respectively

an asymptotic preserving two dimensional staggered grid method for multiscale transport equations we propose a two dimensional asymptotic preserving scheme for linear transport equations with diffusive scalings it is an extension of the time splitting developed by jin pareschi and toscani siam j numer anal 38 2000 pp 913 936 but uses spatial discretizations on staggered grids which preserves the discrete diffusion limit with a more compact stencil the first novelty of this paper is that we propose a staggering in two dimensions that requires fewer unknowns than one could have naively expected the second contribution of this paper is that we rigorously analyze the scheme of jin pareschi and toscani we show that the scheme is ap and obtain an explicit cfl condition which couples a hyperbolic and a parabolic condition this type of condition is common for asymptotic preserving schemes and guarantees uniform stability with respect to the mean free path in addition we obtain an upper bound on the relaxation parameter which is the crucial parameter of the used time discretization se

some applications of isoperimetric methods to strong limit theorems for sums of independent random variables we develop several applications to almost sure limit theorems for sums of independent vector valued random variables of an isoperimetric inequality due to talagrand a general treatment of the classical laws of large numbers of kolmogorov and prokorov and laws of the iterated logarithm of kolmogorov and hartman and wintner is described new results as well as simpler new proofs of known ones illustrate the usefulness of isoperimetric methods in this context we show further how this approach can be used in the study of limit theorems for trimmed sums of independent and identically distributed random variables 1 introduction and theoretical statement isoperimetric methods already appeared in the study of strong limit theorems for sums of independent vector valued random variables through concentration inequalities mainly of gaussian type these inequalities are commonly used with the tool of randomization and applied conditionally in particular this was the basic idea of the main result of 24 which characterizes banach space valued random variables satisfying the classical hartman wintner strassen law of the iterated logarithm in a recent work 32 a new isoperimetric inequality for product measures was obtained and applied to the integrability of sums of independent vector valued random variables providing at the same time new perspectives in the study of strong limit theorems this inequality was indeed further applied in 25 to get extensions of kolmogorov s law of the iterated logarithm and prokorov s law of large numbers in banach space the purpose of this article is to demonstrate further the usefulness of this approach we study a general statement section 2 on sums of independent banach space valued random

m matrix flux splitting for general full tensor discretization operators on structured and unstructured grids diagonal tensor flux approximations are commonly used in fluid dynamics this approximation introduces an o 1 error in flux whenever the coordinate system is nonaligned with the principal axes of the tensor which is particularly common when employing curvilinear gridding in general a consistent full tensor flux approximation leads to a significant increase in support and consequent size of the jacobian matrix after decomposition of a general full tensor flux into a diagonal tensor flux together with cross terms time split semi implicit stable full tensor flux approximations are introduced with in a general finite volume formalism enabling the standard diagonal tensor jacobian matrix structure to be retained for single phase flow impes and standard block fully implicit formulations while ensuring spatial consistency of the discretization for both structured and unstructured grids stability of the scheme is proven for constant elliptic coefficients the results presented demonstrate the benefits of the method for multiphase flow within a fully implicit framework on structured and unstructured grids

a new optimization algorithm for the vehicle routing problem with time windows the vehicle routing problem with time windows vrptw is a generalization of the vehicle routing problem where the service of a customer can begin within the time window defined by the earliest and the latest times when the customer will permit the start of service in this paper we present the development of a new optimization algorithm for its solution the lp relaxation of the set partitioning formulation of the vrptw is solved by column generation feasible columns are added as needed by solving a shortest path problem with time windows and capacity constraints using dynamic programming the lp solution obtained generally provides an excellent lower bound that is used in a branch and bound algorithm to solve the integer set partitioning formulation our results indicate that this algorithm proved to be successful on a variety of practical sized benchmark vrptw test problems the algorithm was capable of optimally solving 100 customer problems this problem size is six times larger than any reported to date by other published research

a low rank method for two dimensional time dependent radiation transport calculations abstract the low rank approximation is a complexity reduction technique to approximate a tensor or a matrix with a reduced rank which has been applied to the simulation of high dimensional problems to reduce the memory required and computational cost in this work a dynamical low rank approximation method is developed for the time dependent radiation transport equation in 1 d and 2 d cartesian geometries using a finite volume discretization in space and a spherical harmonics basis in angle we construct a system that evolves on a low rank manifold via an operator splitting approach numerical results on five test problems demonstrate that the low rank solution requires less memory and computational time than solving the full rank equations with the same accuracy it is furthermore shown that the low rank algorithm can obtain high fidelity results by increasing the number of basis functions while keeping the rank fixed

second order discretization in space and time for radiation hydrodynamics second order accurate discretizations for radiation hydrodynamics are currently an area of interest in the high energy density laboratory physics and astrophysics communities second order methods used to solve the hydrodynamics equations and second order methods used to solve the radiation transport equation often differ fundamentally making it difficult to combine them in a second order manner here we present an implicitexplicit imex method for solving the 1d equations of radiation hydrodynamics that is second order accurate in space and time our radiation hydrodynamics model consists of the 1d euler equations coupled with a gray radiation s2 approximation our rh method combines the muscl hancock method for solving the euler equations with the tr bdf2 time integration scheme and the linear discontinuous galerkin finite element spatial discretization scheme for the s2 radiation equations the muscl hancock method is commonly used for hydrodynamic calculations and the linear discontinuous galerkin scheme is the standard for the sn equations of radiative transfer while somewhat similar these schemes vary fundamentally with respect to the treatment of spatial slopes we address the challenges inherent to coupling these different numerical methods and demonstrate how these challenges can be overcome using the method of manufactured solutions we show that the method is second order accurate in space and time for both the equilibrium diffusion and streaming limit and we show that the method is capable of computing radiative shock solutions accurately by comparing our results with semi analytic solutions

the incorporation of the semi implicit linear equations into newton s method to solve radiation transfer equations abstract for large time steps the nonlinear equations of radiation transfer may not be solved adequately by the semi implicit linear approximation to yield physical solutions this deficiency is rectified in three steps the equations of the semi implicit linear method are modified the modified equations are incorporated into newton s method to solve nonlinear equations and the transfer equations are solved by the resulting method the new method also uses the photon free method to search for the solution in a lower dimensional space than the space of the underlying transfer equations two algorithms are developed from the new method they solve the modified semi implicit linear equations by different approaches the first is a physics approach it solves the linear equations approximately by the grey transport approximation the second is a mathematical approach it solves the linear equations exactly by the sherman morrison woodbury formula of linear algebra however both algorithms yield the solution to the nonlinear system derived by the photon free method therefore their solutions are equal to within the specified tolerance of the nonlinear solver moreover both methods can take advantage of the unconditional stability which comes with the implicit differencing of the time derivative the time step which both methods can take is much larger than the time step in which time discretization error is discernible we shall relate the mathematical approach to the photon free method numerical results for three test problems are presented

spatial discretizations for self adjoint forms of the radiative transfer equations there are three commonly recognized second order self adjoint forms of the neutron transport equation the even parity equations the odd parity equations and the self adjoint angular flux equations because all of these equations contain second order spatial derivatives and are self adjoint for the mono energetic case standard continuous finite element discretization techniques have proved quite effective when applied to the spatial variables we first derive analogs of these equations for the case of time dependent radiative transfer the primary unknowns for these equations are functions of the angular intensity rather than the angular flux hence the analog of the self adjoint angular flux equation is referred to as the self adjoint angular intensity equation then we describe a general arbitrary order continuous spatial finite element approach that is applied to each of the three equations in conjunction with backward euler differencing in time we refer to it as the standard technique we also introduce an alternative spatial discretization scheme for the self adjoint angular intensity equation that requires far fewer unknowns than the standard method but appears to give comparable accuracy computational results are given that demonstrate the validity of both of these discretization schemes

laws of large numbers and tail inequalities for random tries and patricia trees we consider random tries and random patricia trees constructed from n independent strings of symbols drawn from any distribution on any discrete space if hn is the height of this tree we show that hn e hn tends to one in probability additional tail inequalities are given for the height depth size internal path length and profile of these trees and ordinary tries that apply without any conditions on the string distributions they need not even be identically distributed

efficient asymptotic preserving ap schemes for some multiscale kinetic equations many kinetic models of the boltzmann equation have a diffusive scaling that leads to the navier stokes type parabolic equations as the small scaling parameter approaches zero in practical applications it is desirable to develop a class of numerical schemes that can work uniformly with respect to this relaxation parameter from the rarefied kinetic regimes to the hydrodynamic diffusive regimes an earlier approach in s jin l pareschi and g toscani siam j numer anal 35 1998 pp 2405 2439 reformulates such systems into the common hyperbolic relaxation system by jin and xin for hyperbolic conservation laws used to construct the relaxation schemes and then uses a multistep time splitting method to solve the relaxation system here we observe that the combination of the two time split steps may yield hyperbolic parabolic systems that are more advantageous in both stability and efficiency for numerical computations we show that such an approach yields a class of asymptotic preserving ap schemes which are suitable for the computation of multiscale kinetic problems we use the goldstein taylor and carleman models to illustrate this approach

adaptive moving grid methods for two phase flow in porous media a b s t r a c t in this paper we present an application of the moving mesh method for approximating numerical solutions of the two phase flow model in porous media the numerical schemes combine a mixed finite element method and a finite volume method which can handle the nonlinearities of the governing equations in an efficient way the adaptive moving grid method is then used to distribute more grid points near the sharp interfaces which enables us to obtain accurate numerical solutions with fewer computational resources the numerical experiments indicate that the proposed moving mesh strategy could be an effective way to approximate two phase flows in porous media

on extremal distributions and sharp l sub p bounds for sums of multilinear forms in this paper we present a study of the problem of approximating the expectations of functions of statistics in independent and dependent random variables in terms of the expectations of functions of the component random variables we present results providing sharp analogues of the burkholder rosenthal inequalities and related estimates for the expectations of functions of sums of dependent nonnegative r v s and conditionally symmetric martingale differences with bounded conditional moments as well as for sums of multilinear forms among others we obtain the following sharp inequalities e nk 1xk t 2max nk 1extk nk 1ak t for all nonnegative r v s x1 xn with e xk x1 xk 1 ak extk k 1 n 1 x003c t x003c 2undefined e nk 1xk t e t 1 max nk 1bk nk 1ask t s for all nonnegative r v s x1 xn with e xsk x1 xk 1 ask e xtk x1 xk 1 bk k 1 n 1 x003c t x003c 2undefined 0 x003c s le t 1undefined or t 2 0 x003c s le 1undefined where 1 is a poisson random variable with parameter 1 as applications new decoupling inequalities for sums of multilinear forms are presented and sharp khintchine marcinkiewicz zygmund inequalities for generalized moving averages are obtained the results can also be used in the study of a wide class of nonlinear statistics connected to problems of long range dependence and in an econometric setup in particular in stabilization policy problems and in the study of properties of moving average and autocorrelation processes the results are based on the iteration of a series of key lemmas that capture the essential extremal properties of the moments of the statistics involved

numerical approximation of the euler maxwell model in the quasineutral limit we derive and analyze an asymptotic preserving scheme for the euler maxwell system in the quasi neutral limit we prove that the linear stability condition on the time step is independent of the scaled debye length l when l 0 numerical validation performed on riemann initial data and for a model plasma opening switch device show that the ap scheme is convergent to the euler maxwell solution when dx l 0 where dx is the spatial discretization but when l dx 0 the ap scheme is consistent with the quasi neutral euler maxwell system the scheme is also perfectly consistent with the gauss equation the possibility of using large time and space steps leads to several orders of magnitude reductions in computer time and storage

specular reflection treatment for the 3d radiative transfer equation solved with the discrete ordinates method the contribution of this paper relies in the development of numerical algorithms for the mathematical treatment of specular reflection on borders when dealing with the numerical solution of radiative transfer problems the radiative transfer equation being integro differential the discrete ordinates method allows to write down a set of semi discrete equations in which weights are to be calculated the calculation of these weights is well known to be based on either a quadrature or on angular discretization making the use of such method straightforward for the state equation also the diffuse contribution of reflection on borders is usually well taken into account however the calculation of accurate partition ratio coefficients is much more tricky for the specular condition applied on arbitrary geometrical borders this paper presents algorithms that calculate analytically partition ratio coefficients needed in numerical treatments the developed algorithms combined with a decentered finite element scheme are validated with the help of comparisons with analytical solutions before being applied on complex geometries

linear multifrequency grey acceleration recast for preconditioned krylov iterations the linear multifrequency grey acceleration lmfga technique is used to accelerate the iterative convergence of multigroup thermal radiation diffusion calculations in high energy density simulations although it is effective and efficient in one dimensional calculations the lmfga method has recently been observed to significantly degrade under certain conditions in multidimensional calculations with large discontinuities in material properties to address this deficiency we recast the lmfga method in terms of a preconditioned system that is solved with a krylov method lmfgk results are presented demonstrating that the new lmfgk method always requires fewer iterations than the original lmfga method the reduction in iteration count increases with both the size of the time step and the inhomogeneity of the problem however for reasons later explained the lmfgk method can cost more per iteration than the lmfga method resulting in lower but comparable efficiency in problems with small time steps and weak inhomogeneities in problems with large time steps and strong inhomogeneities the lmfgk method is significantly more efficient than the lmfga method

asymmetry matters dynamic half way points in bidirectional labeling for solving shortest path problems with resource constraints faster abstract with their paper symmetry helps bounded bi directional dynamic programming for the elementary shortest path problem with resource constraints discrete optimization 3 2006 pp 255 273 righini and salani introduced bounded bidirectional dynamic programming dp as an acceleration technique for solving variants of the shortest path problem with resource constraints spprc spprcs must be solved iteratively when vehicle routing and scheduling problems are tackled via lagrangian relaxation or column generation techniques righini and salani and several subsequent works have shown that bounded bidirectional dp algorithms are often superior to their monodirectional counterparts since the former can mitigate the fact that the number of labels increases strongly with the path length bidirectional dp has become a quasi standard for solving spprcs with general resource extension functions in computational experiments however one can still observe that the number of forward and backward label extensions is very unbalanced despite a symmetric bounding of a critical resource in the middle of its feasible domain we exploit this asymmetry in forward and backward label extensions to reduce the overall workload by introducing a so called dynamic half way point which is a dynamic bounding criterion based on the current state of the simultaneously solved forward and backward dps experiments with the standard and the electric vehicle routing problem with time windows as well as the vehicle routing and truck driver scheduling problem confirm that dynamic half way points better balance forward and backward labeling and reduce the overall runtime

symbolic implicit monte carlo radiation transport in the difference formulation a piecewise constant discretization the equations of radiation transport for thermal photons are notoriously difficult to solve in thick media without resorting to asymptotic approximations such as the diffusion limit one source of this difficulty is that in thick absorbing media thermal emission and absorption are almost completely balanced a new formulation for thermal radiation transport called the difference formulation was recently introduced in order to remove the stiff balance between emission and absorption in the new formulation thermal emission is replaced by derivative terms that become small in thick media it was proposed that the difficulties of solving the transport equation in thick media would be ameliorated by the difference formulation while preserving full rigor and accuracy of the transport solution in the streaming limit in this paper the transport equation is solved by the symbolic implicit monte carlo method and comparisons are made between the standard formulation and the difference formulation the method is easily adapted to the derivative source terms of the difference formulation and a remarkable reduction in noise is obtained when the difference formulation is applied to problems involving thick media

nonoscillatory high order accurate self similar maximum principle satisfying shock capturing schemes i this is the first paper in a series in which a class of nonoscillatory high order accurate self similar local maximum principle satisfying in scalar conservation law shock capturing schemes for solving multidimensional systems of conservation laws are constructed and analyzed in this paper a scheme which is of third order of accuracy in the sense of flux approximation is presented using scalar one dimensional initial value problems as a model for this model the schemes are made to satisfy a local maximum principle and a nonoscillatory property the method uses a simple centered stencil with quadratic reconstruction followed by two modifications imposed as needed the first enforces a local maximum principle the second guarantees that no new extrema develop the schemes are self similar in the sense that the numerical flux does not depend explicitly on the grid size i e there are no grid size dependent limits involving free parameters as in e g math comp 49 1987 pp 105 121 math comp

a second order radiative transfer equation and its solution by meshless method with application to strongly inhomogeneous media a new second order form of radiative transfer equation named msorte is proposed which overcomes the singularity problem of a previously proposed second order radiative transfer equation j e morel b t adams t noh j m mcghee t m evans t j urbatsch spatial discretizations for self adjoint forms of the radiative transfer equations j comput phys 214 1 2006 12 40 where it was termed saai j m zhao l h liu second order radiative transfer equation and its properties of numerical solution using finite element method numer heat transfer b 51 2007 391 409 in dealing with inhomogeneous media where some locations have very small zero extinction coefficient the msorte contains a naturally introduced diffusion or second order term which provides better numerical property than the classic first order radiative transfer equation rte the stability and convergence characteristics of the msorte discretized by central difference scheme is analyzed theoretically and the better numerical stability of the second order form radiative transfer equations than the rte when discretized by the central difference type method is proved a collocation meshless method is developed based on the msorte to solve radiative transfer in inhomogeneous media several critical test cases are taken to verify the performance of the presented method the collocation meshless method based on the msorte is demonstrated to be capable of stably and accurately solve radiative transfer in strongly inhomogeneous media media with void region and even with discontinuous extinction coefficient

a quasi positive family of continuous darcy flux finite volume schemes with full pressure support a new family of flux continuous locally conservative finite volume schemes is presented for solving the general tensor pressure equation of subsurface flow in porous media the new schemes have full pressure continuity imposed across control volume faces previous families of flux continuous schemes are point wise continuous in pressure and flux when applying the earlier point wise flux continuous schemes to strongly anisotropic full tensor fields their failure to satisfy a maximum principle as with other fem and finite volume methods can result in loss of local stability for high anisotropy ratios which can cause strong spurious oscillations in the numerical pressure solution an m matrix analysis reveals the upper limits for guaranteeing a maximum principle for general 9 point schemes and aids in the design of schemes that minimize the occurrence of spurious oscillations in the discrete pressure field the full pressure continuity schemes are shown to possess a larger range of flux continuous schemes than the previous point wise counter parts for strongly anisotropic full tensor cases it is shown that the full quadrature range possessed by the new schemes permits these schemes to exploit quadrature points previously out of range that are shown to minimize spurious oscillations in discrete pressure solutions the new formulation leads to a more robust quasi positive family of flux continuous schemes applicable to general discontinuous full tensor fields

asymptotic diffusion limit of cell temperature discretisation schemes for thermal radiation transport abstract this paper attempts to unify the asymptotic diffusion limit analysis of thermal radiation transport schemes for a linear discontinuous representation of the material temperature reconstructed from cell centred temperature unknowns in a process known as source tilting the asymptotic limits of both monte carlo continuous in space and deterministic approaches based on linear discontinuous finite elements for solving the transport equation are investigated in slab geometry the resulting discrete diffusion equations are found to have nonphysical terms that are proportional to any cell edge discontinuity in the temperature representation based on this analysis it is possible to design accurate schemes for representing the material temperature for coupling thermal radiation transport codes to a cell centred representation of internal energy favoured by ale arbitrary lagrange eulerian hydrodynamics schemes

asymptotic analysis of a computational method for time and frequency dependent radiative transfer we consider a time dependent energy dependent nonlinear radiative transfer problem in which opacities are large o 1 and interior sources are small o an asymptotic analysis of this problem as 0 leads to the equilibrium diffusion equation in the interior of the system along with boundary conditions and initial conditions for this equation we apply the same asymptotic analysis to a discrete version of the problem in which the frequency variable is discretized by the multigroup method the direction variable by the discrete ordinates method the time variable by the fully implicit method and the spatial variable by a subcell balance method we find that as 0 the discrete solution satisfies a robust discretized version of the correct equilibrium diffusion equation with boundary conditions and initial conditions that are remarkably accurate the analysis thus predicts that if a spatial grid is chosen that resolves interior temperature gradients then the numerical method obtains an accurate solution in the interior of the system even though the optical thickness of the spatial cells tends to and boundary layers in the transport solution are not resolved we go a step further to analyze problems that are optically thin at some photon frequencies but thick at others and show that once again the discrete solution is remarkably accurate we present numerical results that verify these and other predictions of the analyses

semi implicit time integration for pn thermal radiative transfer implicit time integration involving the solution of large systems of equations is the current paradigm for time dependent radiative transfer in this paper we present a semi implicit linear discontinuous galerkin method for the spherical harmonics p n equations for thermal radiative transfer in planar geometry our method is novel in that the material coupling terms are treated implicitly via linearizing the emission source and the streaming operator is treated explicitly using a second order accurate runge kutta method the benefit of this approach is that each time step only involves the solution of equations that are local to each cell this benefit comes at the cost of having the time step limited by a cfl condition based on the speed of light to guarantee positivity and avoid artificial oscillations we use a slope limiting technique we present analysis and numerical results that show the method is robust in the diffusion limit when the photon mean free path is not resolved by the spatial mesh also in the diffusion limit the time step restriction relaxes to a less restrictive explicit diffusion cfl condition we demonstrate with numerical results that away from the diffusion limit our method demonstrates second order error convergence as the spatial mesh is refined with a fixed cfl number

asymptotic analysis of the spatial discretization of radiation absorption and re emission in implicit monte carlo we perform an asymptotic analysis of the spatial discretization of radiation absorption and re emission in implicit monte carlo imc a monte carlo technique for simulating nonlinear radiative transfer specifically we examine the approximation of absorption and re emission by a spatially continuous artificial scattering process and either a piecewise constant or piecewise linear emission source within each spatial cell we consider three asymptotic scalings representing i a time step that resolves the mean free time ii a courant limit on the time step size and iii a fixed time step that does not depend on any asymptotic scaling for the piecewise constant approximation we show that only the third scaling results in a valid discretization of the proper diffusion equation which implies that imc may generate inaccurate solutions with optically large spatial cells if time steps are refined however we also demonstrate that for a certain class of problems the piecewise linear approximation yields an appropriate discretized diffusion equation under all three scalings we therefore expect imc to produce accurate solutions for a wider range of time step sizes when the piecewise linear instead of piecewise constant discretization is employed we demonstrate the validity of our analysis with a set of numerical examples

new route relaxation and pricing strategies for the vehicle routing problem in this paper we describe an effective exact method for solving both the capacitated vehicle routing problem cvrp and the vehicle routing problem with time windows vrptw that improves the method proposed by baldacci et al baldacci r n christofides a mingozzi 2008 an exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts math programming115 2 351 385 for the cvrp the proposed algorithm is based on the set partitioning sp formulation of the problem we introduce a new route relaxation called ng route used by different dual ascent heuristics to find near optimal dual solutions of the lp relaxation of the sp model we describe a column and cut generation algorithm strengthened by valid inequalities that uses a new strategy for solving the pricing problem the new ng route relaxation and the different dual solutions achieved allow us to generate a reduced sp problem containing all routes of any optimal solution that is finally solved by an integer programming solver the proposed method solves four of the five open solomon s vrptw instances and significantly improves the running times of state of the art algorithms for both vrptw and cvrp

discretization on unstructured grids for inhomogeneous anisotropic media part i derivation of the methods discretization methods are proposed for control volume formulations on polygonal and triangular grid cells in two space dimensions the methods are applicable for any system of conservation laws where the flow density is defined by a gradient law like darcy s law for porous media flow a strong feature of the methods is the ability to handle media inhomogeneities in combination with full tensor anisotropy this paper gives a derivation of the methods and the relation to previously published methods is also discussed a further discussion of the methods including numerical examples is given in the companion paper part ii siam j sci comput pp 1717 1736

on stable parametric finite element methods for the stefan problem and the mullins sekerka problem with applications to dendritic growth we introduce a parametric finite element approximation for the stefan problem with the gibbs thomson law and kinetic undercooling which mimics the underlying energy structure of the problem the proposed method is also applicable to certain quasi stationary variants such as the mullins sekerka problem in addition fully anisotropic energies are easily handled the approximation has good mesh properties leading to a well conditioned discretization even in three space dimensions several numerical computations including for dendritic growth and for snow crystal growth are presented

a homotopy perturbation method for solving a neutron transport equation the purpose of this paper consists in the finding of the solution for a stationary transport equation using the techniques of homotopy perturbation method hpm the results of a numerical example illustrate the accuracy and computational efficiency of the new proposed method

a high order asymptotic preserving scheme for kinetic equations using projective integration we investigate a high order fully explicit asymptotic preserving scheme for a kinetic equation with linear relaxation both in the hydrodynamic and diffusive scalings in which a hyperbolic resp parabolic limiting equation exists the scheme first takes a few small inner steps with a simple explicit method such as direct forward euler to damp out the stiff components of the solution and estimate the time derivative of the slow components these estimated time derivatives are then used in an outer runge kutta method of arbitrary order we show that with an appropriate choice of inner step size the time step restriction on the outer time step is similar to the stability condition for the limiting macroscopic equation moreover the number of inner time steps is also independent of the scaling parameter we analyze stability and consistency and illustrate with numerical results

numerical schemes for hyperbolic conservation laws with stiff relaxation terms hyperbolic systems often have relaxation terms that give them a partially conservative form and that lead to a long time behavior governed by reduced systems that are parabolic in nature in this article it is shown by asymptotic analysis and numerical examples that semidiscrete high resolution methods for hyperbolic conservation laws fail to capture this asymptotic behavior unless the small relaxation rate is resolved by a fine spatial grid we introduce a modification of higher order godunov methods that possesses the correct asymptotic behavior allowing the use of coarse grids large cell peclet numbers the idea is to build into the numerical scheme the asymptotic balances that lead to this behavior numerical experiments on 2 2 systems verify our analysis

an optimal bound on the tail distribution of the number of recurrences of an event in product spaces let x 1 x 2 be independent random variables and a a positive real number for the sake of illustration suppose a is the event that xi 1 x j greater than or equal to a for some integers 0 less than or equal to i 0 without further assumptions we prove that if 0 p a 1 then p l a greater than or equal to k p n ln p ac greater than or equal to k for all integers k greater than or equal to 2 an application to sums of independent banach space random elements in l infinity is given showing how to extend our theorem to situations having dependent components less

a monte carlo synthetic acceleration method for solving the thermal radiation diffusion equation we present a novel synthetic acceleration based monte carlo method for solving the equilibrium thermal radiation diffusion equation in three spatial dimensions the algorithm performance is compared against traditional solution techniques using a marshak benchmark problem and a more complex multiple material problem our results show that our monte carlo method is an effective solver for sparse matrix systems for solutions converged to the same tolerance it performs competitively with deterministic methods including preconditioned conjugate gradient and gmres we also discuss various aspects of preconditioning the method and its general applicability to broader classes of problems

discretization on unstructured grids for inhomogeneous anisotropic media part ii discussion and numerical results a companion paper part i siam j sci comput 19 1998 pp 1700 1716 presents discretization methods for control volume formulations in two space dimensions properties of the developed methods are further discussed including rotational invariance and stability under the influence of anisotropy strong anisotropy may have to be handled by use of stretched grid cells numerical examples that demonstrate strengths and weaknesses of the methods are presented the examples focus on inhomogeneity anisotropy and rotational invariance

central schemes for balance laws of relaxation type several models in mathematical physics are described by quasi linear hyperbolic systems with source term and in several cases the production term can become stiff here suitable central numerical schemes for such problems are developed and applications to the broadwell model and extended thermodynamics are presented the numerical methods are a generalization of the nessyahu tadmor scheme to the nonhomogeneous case by including the cell averages of the production terms in the discrete balance equations a second order scheme uniformly accurate in the relaxation parameter is derived and its properties analyzed numerical tests confirm the accuracy and robustness of the scheme

bidirectional labeling in column generation algorithms for pickup and delivery problems abstract for the exact solution of many types of vehicle routing problems column generation based algorithms have become predominant the column generation subproblems are then variants of the shortest path problem with resource constraints which can be solved well with dynamic programming labeling algorithms for vehicle routing problems with a pickup and delivery structure the strongest known dominance between two labels requires the delivery triangle inequality dti for reduced costs to hold when the direction of labeling is altered from forward labeling to backward labeling the dti requirement becomes the pickup triangle inequality pti dti and pti cannot be guaranteed at the same time the consequence seemed to be that bidirectional labeling one of the most successful acceleration techniques developed over the last years is not applicable with the strong dominance on both directions for problems with a pickup and delivery structure in this paper we show that bidirectional labeling with the strong dominance rules in forward as well as backward direction is possible by using different cost matrices in the two directions we adopt the both sided strong bidirectional labeling approach to integrate the standard robust and non robust cuts moreover a recent acceleration technique that dynamically adjusts the bidirectional half way point is implemented full fledged branch cut and price algorithms are tested on the pickup and delivery problem with time windows pdptw in particular an in depth analysis of different mono and bidirectional labeling algorithms is presented overall compared to the standard pure forward labeling based algorithm we obtain average reductions in computation time of more than 40 percent when solving pdptw instances exactly

iterative methods for neutron transport eigenvalue problems we discuss iterative methods for computing criticality in nuclear reactors in general this requires the solution of a generalized eigenvalue problem for an unsymmetric integro differential operator in six independent variables modeling transport scattering and fission where the dependent variable is the neutron angular flux in engineering practice this problem is often solved iteratively using some variant of the inverse power method because of the high dimension matrix representations for the operators are often not available and the inner solves needed for the eigenvalue iteration are implemented by matrix free inner iterations this leads to technically complicated inexact iterative methods for which there appears to be no published rigorous convergence theory for the monoenergetic homogeneous model case with isotropic scattering and vacuum boundary conditions we show that before discretization the general nonsymmetric eigenproblem for the angular flux is equivalent to a certain related eigenproblem for the scalar flux involving a symmetric positive definite weakly singular integral operator in space only this correspondence to a symmetric problem in a space of reduced dimension permits us to give a convergence theory for inexact inverse iteration and related methods in particular this theory provides rather precise criteria on how accurate the inner solves need to be in order for the whole iterative method to converge we also give examples of discretizations which have a corresponding symmetric finite dimensional reduced form the theory is illustrated with numerical examples for several test problems of physical relevance using gmres as the inner solver

a deterministic photon free method to solve radiation transfer equations a new method to solve radiation transfer equations is presented in the absence of scattering material motion and heat conduction the photon variables can be eliminated from the fully implicit multi group discrete ordinate finite difference finite element equations of continuum radiation transfer to yield a smaller set of equations which depends only on temperature the solution to this smaller set of equations is used to generate the solution to the original set of equations from which the reduced set is derived the reduced system simplifies to a nonlinear heat equation in the regime of strong absorption and strong emission we solve the reduced set of equations by the newton gmres method in which the jacobian update is preconditioned by a linearization of this nonlinear heat equation the performances of this new method and of the semi implicit linear method which is preconditioned by grey transport acceleration combined with diffusion synthetic acceleration are compared on two test problems the test results indicate that the new method can take larger time steps requires less memory is more accurate and is competitive in speed with the semi implicit linear method

subset row inequalities applied to the vehicle routing problem with time windows this paper presents a branch and cut and price algorithm for the vehicle routing problem with time windows the standard dantzig wolfe decomposition of the arc flow formulation leads to a set partitioning problem as the master problem and an elementary shortest path problem with resource constraints as the pricing problem we introduce the subset row inequalities which are chvatal gomory rank 1 cuts based on a subset of the constraints in the master problem applying a subset row inequality in the master problem increases the complexity of the label setting algorithm used to solve the pricing problem because an additional resource is added for each inequality we propose a modified dominance criterion that makes it possible to dominate more labels by exploiting the step like structure of the objective function of the pricing problem computational experiments have been performed on the solomon benchmarks where we were able to close several instances the results show that applying subset row inequalities in the master problem significantly improves the lower bound and in many cases makes it possible to prove optimality in the root node

a linear discontinuous spatial differencing scheme forsnradiative transfer calculations various types of linear discontinuous spatial differencing schemes have been developed for thesn discrete ordinates equations approximating the linear boltzmann transport equation it has been shown through an asymptotic analysis that the 1d slab geometry lumped linear discontinuous scheme not only goes over to a convergent and robust differencing of the diffusion equation in the monoenergetic thick diffusion limit but it also yields the correct interior solution even when boundary layers are left unresolved by the spatial mesh in the present work we generalize this scheme to obtain a 1d slab geometry lumped linear discontinuous scheme for the nonlinear radiative transfer equation and the associated material temperature equation we present a full nonlinear energy dependent asymptotic analysis of the behavior of this scheme in the thick equilibrium diffusion limit we find that this scheme goes over to a convergent and robust differencing of the equilibrium diffusion equation on the interior of the mesh but it does not yield the exact interior solution when boundary layers are left unresolved by the spatial mesh nevertheless the interior solution obtained with spatially unresolved boundary layers is always well behaved and fairly accurate computational results are presented which test the predictions of our asymptotic analysis and demonstrate the efficiency of our solution technique

remarks on strong exponential integrability of vector valued random series and triangular arrays some optimal refinements of the results on strong exponential integrability of sums of independent zero mean uniformly bounded random vectors are given

asymptotic diffusion limit of the symbolic monte carlo method for the transport equation we use asymptotic analysis to study the diffusion limit of the symbolic implicit monte carlo simc method for the transport equation for standard simc with piecewise constant basis functions we demonstrate mathematically that the solution converges to the solution of a wrong diffusion equation nevertheless a simple extension to piecewise linear basis functions enables to obtain the correct solution we present numerical examples which illustrate the analysis

unstructured grid optimization for improved monotonicity of discrete solutions of elliptic equations with highly anisotropic coefficients multipoint flux approximation mpfa techniques are commonly applied for discretizing the porous media flow equations within the context of finite volume numerical procedures although these methods can be applied to heterogeneous anisotropic systems on generally unstructured grids the inverse of the resulting linear operator can suffer from a loss of monotonicity at high permeability anisotropy ratios resulting in spurious oscillations of the pressure solution the purpose of this paper is to develop a method for optimizing unstructured grids in two and three dimensions such that the monotonicity behavior of the mpfa technique is significantly improved the method employs anisotropic triangulation and can be readily combined with permeability upscaling procedures results are presented for a variety of examples and the technique is shown to perform well on problems involving complex grid point distributions and heterogeneous permeability fields with strong anisotropy ratios of o 100 oscillation free pressure solutions are achieved in all cases considered even for examples in which the original grid shows large oscillations

the shortest path problem with resource constraints with k 2 loop elimination and its application to the capacitated arc routing problem abstract in many branch and price algorithms the column generation subproblem consists of computing feasible constrained paths in the capacitated arc routing problem carp elementarity constraints concerning the edges to be serviced and additional constraints resulting from the branch and bound process together impose two types of loop elimination constraints to fulfill the former constraints it is common practice to rely on a relaxation where loops are allowed in a k loop elimination approach all loops of length k and smaller are forbidden following bode and irnich 2012 for solving the carp branching on followers and non followers is the only known approach to guarantee integer solutions within branch and price however it comes at the cost of additional task 2 loop elimination constraints in this paper we show that a combined k 2 loop elimination in the shortest path subproblem can be accomplished in a computationally efficient way overall the improved branch and price often allows the computation of tighter lower bounds and integer optimal solutions for several instances from standard benchmark sets

algorithms for the vehicle routing and scheduling problems with time window constraints this paper considers the design and analysis of algorithms for vehicle routing and scheduling problems with time window constraints given the intrinsic difficulty of this problem class approximation methods seem to offer the most promise for practical size problems after describing a variety of heuristics we conduct an extensive computational study of their performance the problem set includes routing and scheduling environments that differ in terms of the type of data used to generate the problems the percentage of customers with time windows their tightness and positioning and the scheduling horizon we found that several heuristics performed well in different problem environments in particular an insertion type heuristic consistently gave very good results

asymptotic equilibrium diffusion analysis of time dependent monte carlo methods for grey radiative transfer the equations of nonlinear time dependent radiative transfer are known to yield the equilibrium diffusion equation as the leading order solution of an asymptotic analysis when the mean free path and mean free time of a photon become small we apply this same analysis to the fleck cummings carter forest and n kaoua monte carlo approximations for grey frequency independent radiative transfer although monte carlo simulation usually does not require the discretizations found in deterministic transport techniques monte carlo methods for radiative transfer require a time discretization due to the nonlinearities of the problem if an asymptotic analysis of the equations used by a particular monte carlo method yields an accurate time discretized version of the equilibrium diffusion equation the method should generate accurate solutions if a time discretization is chosen that resolves temperature changes even if the time steps are much larger than the mean free time of a photon this analysis is of interest because in many radiative transfer problems it is a practical necessity to use time steps that are large compared to a mean free time our asymptotic analysis shows that i the n kaoua method has the equilibrium diffusion limit ii the carter forest method has the equilibrium diffusion limit if the material temperature change during a time step is small and iii the fleck cummings method does not have the equilibrium diffusion limit we include numerical results that verify our theoretical predictions

on solutions to the pn equations for thermal radiative transfer we present results for the spherical harmonics p n method for solving problems of time dependent thermal radiative transport we prove a theorem that demonstrates that in the streaming limit the spatially and temporally continuous p n equations will allow negative energy densities for any finite order of n we also develop an implicit numerical method for solving the p n equations to explore the impact of the theorem the numerical method uses a high resolution riemann solver to produce an upwinded discretization we employ a quasi linear approach to integrate the nonlinearites added to make the scheme non oscillatory we use the backward euler method for time integration and treat the material interaction terms fully nonlinearly reflecting boundary conditions for the p n equations are presented and we show how to implement this boundary condition using ghost cells the implicit method was able to produce robust results to thermal transport problems in one and two dimensions the numerical method is used to analyze the accuracy of various p n expansion orders on several problems in two dimensional problems the numerical p n solutions contained negative radiation energy densities as predicted by our theorem the numerical results showed that the material temperature also became negative a result outside the scope of the theorem our numerical method can handle these negative values but they would cause problems in a radiation hydrodynamics calculation

optimum bounds for the distributions of martingales in banach spaces a general device is proposed which provides for extension of exponential inequalities for sums of independent real valued random variables to those for martingales in the 2 smooth banach spaces this is used to obtain optimum bounds of the rosenthal burkholder and chung types on moments of the martingales in 2 smooth banach spaces in turn it leads to best order bounds on moments of sums of independent random vectors in any separable banach spaces although the emphasis is put on infinite dimensional martingales most of the results seem to be new even for one dimensional martingales moreover the bounds on moments of the rosenthal burkholder type seem to be to a certain extent new even for sums of independent real valued random variables analogous inequalities for one dimensional supermartingales are given

robust and accurate filtered spherical harmonics expansions for radiative transfer we present a novel application of filters to the spherical harmonics p n expansion for radiative transfer problems in the high energy density regime the filter we use is based on non oscillatory spherical splines and a filter strength chosen to i preserve the equilibrium diffusion limit and ii vanish as the expansion order tends to infinity our implementation is based on modified equations that are derived by applying the filter after every time step in a simple first order time integration scheme the method is readily applied to existing codes that solve the p n equations numerical results demonstrate that the solution to the filtered p n equations are i more robust and less oscillatory than standard p n solutions and ii more accurate than discrete ordinates solutions of comparable order in particular the filtered p 7 solution demonstrates comparable accuracy to an implicit monte carlo solution for a benchmark hohlraum problem in 2d cartesian geometry

a residual monte carlo method for discrete thermal radiative diffusion residual monte carlo methods reduce statistical error at a rate of exp bn where b is a positive constant and n is the number of particle histories contrast this convergence rate with 1 n which is the rate of statistical error reduction for conventional monte carlo methods thus residual monte carlo methods hold great promise for increased efficiency relative to conventional monte carlo methods previous research has shown that the application of residual monte carlo methods to the solution of continuum equations such as the radiation transport equation is problematic for all but the simplest of cases however the residual method readily applies to discrete systems as long as those systems are monotone i e they produce positive solutions given positive sources we develop a residual monte carlo method for solving a discrete id non linear thermal radiative equilibrium diffusion equation and we compare its performance with that of the discrete conventional monte carlo method upon which it is based we find that the residual method provides efficiency gains of many orders of magnitude part of the residual gain is due to the fact that we begin each timestep with an initial guess equal to the solution from the previous timestep moreover fully consistent non linear solutions can be obtained in a reasonable amount of time because of the effective lack of statistical noise we conclude that the residual approach has great potential and that further research into such methods should be pursued for more general discrete and continuum systems

discretization on quadrilateral grids with improved monotonicity properties multi point flux approximation mpfa discretization methods have been applied in the oil industry since the mid 1990s the discretizations are based on a control volume formulation and the finite difference structure makes general skew grids and unstructured grids feasible in a fully implicit formulation mpfa methods are therefore suitable for flow problems in realistic reservoirs monotonicity issues are known to arise for high aspect ratios combined with skewness of computational grids for mpfa methods in this paper we improve the mpfa discretization techniques for general quadrilateral grids such that the above difficulties are handled to give a more robust discretization of the governing equations for fluid flow in porous media comparisons to the mpfa o method are made and the suggested discretization is shown to be an improvement in regards to monotonicity for smooth solutions the method performs equally well as the o method when the convergence is examined

the effects of slope limiting on asymptotic preserving numerical methods for hyperbolic conservation laws many hyperbolic systems of equations with stiff relaxation terms reduce to a parabolic description when relaxation dominates an asymptotic preserving numerical method is a discretization of the hyperbolic system that becomes a valid discretization of the parabolic system in the asymptotic limit we explore the consequences of applying a slope limiter to the discontinuous galerkin dg method with linear elements for hyperbolic systems with stiff relaxation terms without a limiter the dg method gives an accurate discretization of the chapman enskog approximation of the system when the relaxation length scale is not resolved it is well known that the first order upwind or step method fails to obtain the proper asymptotic limit we show that using the minmod slope limiter also fails but that using double minmod gives the proper asymptotic limit despite its effectiveness in the asymptotic limit the double minmod limiter allows artificial extrema at cell interfaces referred to as sawteeth we present a limiter that eliminates the sawteeth but maintains the proper asymptotic limit the systems that we analyze are the hyperbolic heat equation and the p n thermal radiation equations numerical examples are used to verify our analysis

an extension of implicit monte carlo diffusion multigroup and the difference formulation implicit monte carlo imc and implicit monte carlo diffusion imd are approaches to the numerical solution of the equations of radiative transfer imd was previously derived and numerically tested on grey or frequency integrated problems 1 in this research we extend implicit monte carlo diffusion imd to account for frequency dependence and we implement the difference formulation 2 as a source manipulation variance reduction technique we derive the relevant probability distributions and present the frequency dependent imd algorithm with and without the difference formulation the imd code with and without the difference formulation was tested using both grey and frequency dependent benchmark problems the su and olson semi analytic marshak wave benchmark was used to demonstrate the validity of the code for grey problems 3 the su and olson semi analytic picket fence benchmark was used for the frequency dependent problems 4 the frequency dependent imd algorithm reproduces the results of both su and olson benchmark problems frequency group refinement studies indicate that the computational cost of refining the group structure is likely less than that of group refinement in deterministic solutions of the radiation diffusion methods our results show that applying the difference formulation to the imd algorithm can result in an overall increase in the figure of merit for frequency dependent problems however the creation of negatively weighted particles from the difference formulation can cause significant numerical instabilities in regions of the problem with sharp spatial gradients in the solution an adaptive implementation of the difference formulation may be necessary to focus its use in regions that are at or near thermal equilibrium