analysis of an adaptive control scheme for a partially observed controlled markov chain the authors consider an adaptive finite state controlled markov chain with partial state information motivated by a class of replacement problems they present parameter estimation techniques based on the information available after actions that reset the state to a known value are taken it is proved that the parameter estimates converge w p 1 to the true unknown parameter under the feedback structure induced by a certainty equivalent adaptive policy it is shown that the adaptive policy is self optimizing in a long run average sense for any measurable sequence of parameter estimates converging w p 1 to the true parameter

galerkin projected residual method applied to diffusion reaction problems a stabilized finite element method is presented for scalar and linear second order boundary value problems the method is obtained by adding to the galerkin formulation multiple projections of the residual of the differential equation at element level these multiple projections allow the generation of appropriate number of free stabilization parameters in the element matrix depending on the local space of approximation and on the differential operator the free parameters can be determined imposing some convergence and or stability criteria or by postulating the element matrix with the desired stability properties the element matrix of most stabilized methods such as gls and ggls methods can be obtained using this new method with appropriate choices of the stabilization parameters we applied this formulation to diffusion reaction problems optimal rates of convergency are numerically observed for regular solutions

consistent relations in the method of reducibility in the cluster variation method a redundancy is noted in the set of lagrange multipliers effective fields and effective interactions used in the expressions of the reduced density matrices given by morita 4 in some problems with less symmetry 6 the expressions of the reduced density matrices without this redundancy are given an argument showing that a certain expression of the free energy in terms of the lagrange multipliers is stationary with respect to the variations of these is presented

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

a hierarchical multiscale framework for problems with multiscale source terms this paper presents a hierarchical multiscale framework for problems that involve multiscale source terms an assumption on the additive decomposition of the source function results in consistent decoupling of the fully coupled system and constitutes the new method the structure of this decomposition is investigated and its mathematical implications are delineated this method results in variational embedding of fine scale information that is derived from the fine scale equations in the corresponding coarse scale equations it therefore provides a mathematically consistent way of bridging information between disparate spatial scales in the response function that are induced by multiscale forcing functions

numerical multiscale methods for a reaction dominated model a galerkin enriched finite element method gem is proposed for the singularly perturbed reaction diffusion equation this new method is an improvement on the petrov galerkin enriched method pgem where now the standard piecewise bi linear test space incorporates fine scales this appears as the fundamental ingredient for suppressing oscillations in the numerical solutions also new parameter free stabilized finite element methods derived from both the gem and the pgem are driven by local generalized eigenvalue problems in the process jump stabilizing terms belonging to the class of cip methods emerge as a result of the enriching procedure interestingly numerical results indicate that jump based stabilizations are unnecessary and sometimes undesirable when treating reaction dominated problems finally we establish relationships with more standard enriched and stabilized methods and show that the proposed methods outperform them numerically

upper bounds for survival probability of the contact process a precise description of the nontrivial upper invariant measure for c is still an open problem for the basic contact process which is a self dual attractive but nonreversible markov process of an interacting particle system by its self duality to identify the invariant measure is equivalent to determining the initial state dependence of the survival probability of the process a procedure to give rigorous upper bounds for the survival probability is presented based on a lemma given by harris two new bounds are given improving the simple branching process bound in the one dimensional case the present procedure can be viewed as a trial to make approximate measures by generalized markov extensions

on some variational approximations in two dimensional classical lattice systems a new derivation is presented of some variational approximations for classical lattice systems that belong to the class of cluster variation methods among them the well known bethe peierls and kramers wannier approximations the limiting behavior of a hierarchical sequence of cluster variation approximations the so calledc hierarchy is discussed it is shown that this hierarchy provides a monotonically decreasing sequence of upper boundsfn on the free energy per lattice sitef and thatfn f asn our results are based on extension theorems for states given on subsets of the lattice which might be of some independent interest and on an application of transfer matrix concepts to the variational characterization of translation invariant equilibrium states

a variational multiscale discontinuous galerkin formulation for both implicit and explicit dynamic modeling of interfacial fracture abstract a stabilized discontinuous galerkin method is presented for modeling interfacial debonding under dynamic conditions the method s distinctive feature is its treatment of the inelastic gap at the interface as an internal variable that is evolved through a traction separation constitutive model enabling initially perfect adhesion to be captured numerical stability is derived using variational multiscale ideas and comparisons are drawn with both the intrinsic cohesive zone method and hybrid discontinuous galerkin cohesive zone method to expose subtle differences the proposed formulation is applied to model the nonlinear response of a composite unit cell under impact loading both explicit and implicit time integration schemes provide similar accuracy and stability and the critical time step for explicit analyses is only slightly reduced below the value for the continuous galerkin mesh without an interface further studies of mesh refinement and adjustments to the initial elastic interface stiffness in the reference cohesive zone model reveal the improved accuracy of the variational multiscale discontinuous galerkin method on coarser grids

entropy and global markov properties we extend refine and give simple proofs of some recent results on the validity of global markov properties for classical spin systems one of the new results is that there is a global markov property that is satisfied by equilibrium states in general the proof of this establishes formulas for the entropy and free energy that show that these quantities are ford dimensional systems given in terms of d 1 dimensional systems furthermore we show that global markov properties imply the absence of some types of symmetry breaking

discretized maximum likelihood and almost optimal adaptive control of ergodic markov models three distinct controlled ergodic markov models are considered here the models are a discrete time controlled markov process with complete observations a controlled diffusion process with complete observations and a discrete time controlled markov process with partial observations the partial observations for the third model have the special form of complete observations in a fixed recurrent set and noisy observations in its complement for each of the models an almost self optimizing adaptive control is given these adaptive controls are constructed from a family of estimates that use a finite discretization of the parameter set and a finite family of almost optimal ergodic controls by a randomized certainty equivalence method a continuity property of the information of a model for one parameter value with respect to another is used to establish this almost optimality property

pair correlations and structure factor of the j1 j2 square lattice ising model in an external field abstract we compute the structure factor of the j 1 j 2 ising model in an external field on the square lattice within the cluster variation method we use a four point plaquette approximation which is the minimal one able to capture phases with broken orientational order in real space like the recently reported ising nematic phase in the model the analysis of different local maxima in the structure factor allows us to track the different phases and phase transitions against temperature and external field although the nematic susceptibility is not directly related to the structure factor we show that because of the close relationship between the nematic order parameter and the structure factor the latter shows unambiguous signatures of the presence of a nematic phase in agreement with results from direct minimization of a variational free energy the disorder variety of the model is identified and the possibility that the cvm four point approximation be exact on the disorder variety is discussed

variational multiscale enrichment method with mixed boundary conditions for modeling diffusion and deformation problems abstract this manuscript presents the formulation and implementation of the variational multiscale enrichment vme method using canopy shaped microscale enrichment functions obtained through the use of a new family of microscale boundary conditions the purpose of the new enrichment functions and the new boundary condition is to relax the overconstraint imposed by the homogeneous microscale boundary condition e g residual free bubbles commonly employed in the variational multiscale literature the formulation and implementation of the method are presented for diffusion and elasticity problems the performance of the proposed method is assessed by comparing with direct numerical simulations on diffusion and deformation problems a boundary parameter identification approach is proposed to obtain near optimal boundary conditions the identification approach is verified in the context of the deformation response of particle reinforced composites

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

a multiscale stabilized finite element method for the advection diffusion equation this paper presents a multiscale method that yields a stabilized finite element formulation for the advection diffusion equation the multiscale method arises from a decomposition of the scalar field into coarse resolved scale and fine unresolved scale the resulting stabilized formulation possesses superior properties like that of the supg and the gls methods a significant feature of the present method is that the definition of the stabilization term appears naturally and therefore the formulation is free of any user designed or user defined parameters another important ingredient is that since the method is residual based it satisfies consistency ab initio based on the proposed formulation a family of 2 d elements comprising 3 and 6 node triangles and 4 and 9 node quadrilaterals has been developed numerical results show the good performance of the method on uniform skewed as well as composite meshes and confirm convergence at optimal rates 2004 elsevier b v all rights reserved

variational subgrid scale formulations for the advection diffusion reaction equation abstract the exact variational multiscale vms and the subgrid scale sgs methods have been developed for the advection reaction and the advection diffusion reaction equations from the element green s function approximate intrinsic time scale parameters have been derived for these cases and are shown to be similar to other expressions obtained in the literature out of the maximum principle and convergence error analysis the methods have been compared with typical stabilized finite element methods as expected the vms is nodally exact for the one dimensional case

lennard jones triangular lattice gas in the kikuchi approximation abstract the triangular lattice gas problem for a model with infinitely repulsive nearest neighbor and attractive second j 2 and third neighbor j 3 interactions is studied in the kikuchi approximation with a triangle of nearest neighbor sites as the basic cluster values of the ratio j 3 j 2 correspond to a lennard jones interaction in a model for commensurate phases of submonolayers of inert gases adsorbed on graphite a method of calculation is developed which allows a detailed study of the evolution of the phase diagrams in particular the location of the liquid gas critical end point and of the solid liquid gas triple point distinct liquid and gas phases are found to be stable for 0 9481

a note on the cluster variation method kikuchi s cluster variation method cvm is reformulated as the truncation of a mobius inversion an attempt is made to explicate and simplify the various approaches to the cvm this formulation makes apparent the connection of the method with other types of cluster approximation an illustration of the procedure is provided

application of group theory to the calculation of the configurational entropy in the cluster variation method abstract the counting and classification of clusters required for the entropy calculation in the cluster variation method is systematically carried out for general crystal structures the approach explicitly uses the symmetry properties of both the set of clusters and the crystal structure in conjunction with simple group theoretical considerations as an illustration of the method several new configurational entropy expressions are calculated for hcp diamond cubic and spinel structures for each approximation the critical temperature of the corresponding ising ferromagnet has been calculated

discontinuous subgrid formulations for transport problems in this paper we develop two discontinuous galerkin formulations within the framework of the two scale subgrid method for solving advection diffusion reaction equations we reformulate using broken spaces the nonlinear subgrid scale nsgs finite element model in which a nonlinear eddy viscosity term is introduced only to the subgrid scales of a finite element mesh here two new subgrid formulations are built by introducing subgrid stabilized terms either at the element level or on the edges by means of the residual of the approximated resolved scale solution inside each element and the jump of the subgrid solution across interelement edges the amount of subgrid viscosity is scaled by the resolved scale solution at the element level yielding a self adaptive method so that no additional stabilization parameter is required numerical experiments are conducted in order to demonstrate the behavior of the proposed methodology in comparison with some discontinuous galerkin methods

stabilized finite element methods for steady advection diffusion with production abstract finite element methods for solving equations of steady advection diffusion with production are constructed employing a linear source to model production the galerkin method requires relatively fine meshes to retain an acceptable degree of accuracy for wide ranges of values of the physical coefficients numerous criteria for improved accuracy via galerkin least squares are proposed and examined in the entire parameter space of these several provide the lowest error in nodal amplification factors each in a certain range of ratios of physical coefficients guidelines for required mesh resolutions are presented for galerkin and galerkin least squares showing that in large parts of the parameter space substantial savings may be obtained by employing galerkin least squares galerkin least squares gradient least squares a new variation of the galerkin method is designed to provide exact nodal amplification factors offering accurate solutions at any resolution numerical tests validate these conclusions

on the stabilization parameter in the subgrid scale approximation of scalar convection diffusion reaction equations on distorted meshes abstract in this paper we revisit the definition of the stabilization parameter in the finite element approximation of the convection diffusion reaction equation the starting point is the decomposition of the unknown into its finite element component and a subgrid scale that needs to be approximated in order to incorporate the distortion of the mesh into this approximation we transform the equation for the subgrid scale within each element to the shape regular reference domain the expression for the subgrid scale arises from an approximate fourier analysis and the identification of the wave number direction where instabilities are most likely to occur the final outcome is an expression for the stabilization parameter that accounts for anisotropy and the dominance of either convection or reaction terms in the equation

stability of semidiscrete formulations for parabolic problems at small time steps solutions of direct time integration schemes that converge in time to conventional semidiscrete formulations may be polluted at small time steps by spurious spatial oscillations along with attendant overshoot in time this degradation is not an artifact of the time marching scheme but rather a property of the solution of the semidiscrete formulation itself a critical time step for the onset of spatial oscillations is derived by analogy to singularly perturbed elliptic problems we then propose a simple procedure of spatial stabilization to remove this pathology from implicit time integration schemes without affecting unconditional temporal stability spatially stabilized implicit time integration is free of spurious spatial oscillations at small time steps and numerical experience points to improved temporal accuracy as well

certainty equivalence control with forcing revisited abstract certainty equivalence control with forcing has been shown to be optimal for several stochastic adaptive control problems with the average cost per unit time criterion recently researchers have started looking at stochastic adaptive control problems with a view to minimizing the rate of increase of the learning loss this criterion is stronger than the average cost per unit time criterion certainty equivalence control with forcing does not usually suffice for the learning loss criterion and one has to develop fairly complicated schemes in order to achieve optimality the objective of this paper is to see how well one might be able to do with a certainty equivalence control with forcing type of scheme in particular we construct a class of such schemes whose learning loss is o log n 1 for 0 whereas optimal schemes typically have a o log n learning loss

a hierarchical classification of cellular automata abstract a central issue in the theory of cellular automata is classification a classification is a structure imposed on the space of automata which groups together cellular automata with related properties in this paper a classification based on the action of cellular automata on n step markov measures is presented this classification is hierarchical and parametric an algorithm is described which efficiently constructs all rules in a given class at a given level of the hierarchy the utility of the classification is explored in the study of the invariant statistical properties of next nearest neighbor cellular automata on two states per cell

applications of the pseudo residual free bubbles to the stabilization of convection diffusion problems abstract residual free bubbles have been recently introduced in order to compute optimal values for the stabilization methods a la hughes franca however unless in very special situations one dimensional problems limit cases etc they require the actual solution of pde problems the bubble problems in each element thus they are very difficult to be used in practice in this paper we present for the special case of convection dominated elliptic problems a cheap way to compute approximately the solution of the bubble problem in each element this provides as a consequence a cheap way to compute good approximations for the optimal values of the stabilization parameters

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100

numerical limit of the spinodal point abstract the relation is studied between the phase boundary c pb and the spinodal point c s of the ferromagnetic ising model in the two dimensional triangular lattice using a series of approximations in the cvm cluster variation method hierarchy we keep the magnetization m fixed and extrapolate the temperature difference t pb m n t s m n for n 1 3 5 7 and 9 triangle approximations of the cvm it is found that the limiting value of the difference is practically zero our analysis indicates i the spinodal decomposition is to be treated as a system changing from the quenched initial state and the free energy curve used in the spinodal analysis is not a fixed curve but is changing in time and ii cases can occur in which a quenched system is initially decomposing in the nucleation mode but switches to the spinodal decomposition mode in time

a new finite element formulation for computational fluid dynamics viii the galerkin least squares method for advective diffusive equations abstract galerkin least squares finite element methods are presented for advective diffusive equations galerkin least squares represents a conceptual simplification of supg and is in fact applicable to a wide variety of other problem types a convergence analysis and error estimates are presented

variational multiscale analysis the fine scale green s function projection optimization localization and stabilized methods we derive an explicit formula for the fine scale green s function arising in variational multiscale analysis the formula is expressed in terms of the classical green s function and a projector which defines the decomposition of the solution into coarse and fine scales the theory is presented in an abstract operator format and subsequently specialized for the advection diffusion equation it is shown that different projectors lead to fine scale green s functions with very different properties for example in the advection dominated case the projector induced by the h 1 0 seminorm produces a fine scale green s function which is highly attenuated and localized these are very desirable properties in a multiscale method and ones that are not shared by the l 2 projector by design the coarse scale solution attains optimality in the norm associated with the projector this property combined with a localized fine scale green s function indicates the possibility of effective methods with local character for dominantly hyperbolic problems the constructs lead to a new class of stabilized methods and the relationship between h 1 0 optimality and the streamline upwind petrov galerkin supg method is described

mean field renormalization group for the boundary magnetization of strip clusters we analyse in some detail a recently proposed transfer matrix mean field approximation which yields the exact critical point for several two dimensional nearest neighbour ising models for the square lattice model we show explicitly that this approximation yields not only the exact critical point but also the exact boundary magnetization of a semi infinite ising model independent of the size of the strips used then we develop a new mean field renormalization group strategy based on this approximation and make connections with finite size scaling applying our strategy to the quadratic ising and three state potts models we obtain results for the critical exponents which are in very good agreement with the exact ones in this way we also clarify some advantages and limitations of the mean field renormalization group approach

generalized cluster description of multicomponent systems a general formalism for the description of configurational cluster functions in multicomponent systems is developed the approach is based on the description of configurational cluster functions in terms of an orthogonal basis in the multidimensional space of discrete spin variables the formalism is used to characterize the reduced density matrices or cluster probability densities and the free energy functional obtained in the cluster variation method approximation for the particular representation chosen the expectation values of the base functions are the commonly used multisite correlation functions the latter form an independent set of variational parameters for the free energy which in general facilitates the minimization procedure a new interpretation of the cluster variation method as a self consistency relation on the renormalized cluster energies is also presented

pair correlations in the cluster variation approximation abstract the cluster variation method is used to obtain approximate free energy functions for a binary system in an inhomogeneous field in order to calculate the fourier transform of the pair correlation functions in the two dimensional square lattice using the nearest neighbors square as the basic cluster the correlation functions are correctly given up to the order 7 in the inverse temperature the effect of second neighbor pairs and many body interactions is investigated

convergence of the cluster variation method for a system on a triangular lattice abstract the paper studies the convergence of the cluster variation method to the rigorous result as the cluster size increases the calculation is done on the phase boundary at t 0 between the a 2 b type ordered phase and the disordered phase on a two dimensional triangular lattice with nearest neighbor interaction it is shown that the phase boundary at t 0 is obtained by maximizing the entropy under the constraint that only a limited number of atomic configurations are allowed formulations are developed for clusters of n 3 5 7 9 11 and 13 points when thermodynamic quantities which are calculated using these clusters are plotted against 1 n they approach the known rigorous n results more or less linearly but with a pseudo period of n 6 an exception is the square of the long range order which bends down as 1 n tends to zero

exact variational methods and cluster variation approximations it is shown that for classical d dimensional lattice models with finite range interactions the translation invariant equilibrium states have the property that their mean entropy is completely determined by their restriction to a subset of the lattice that is infinite in d 1 dimensions and has a width equal to the range of the interaction in the dth dimension this property is used to show proper convergence toward the exact result for a hierarchy of approximations of the cluster variation method that uses one dimensionally increasing basis clusters in a two dimensional lattice

bubble functions prompt unusual stabilized finite element methods a second order linear scalar differential equation including a zero th order term is approximated using first the standard galerkin method enriched with bubble functions static condensation of the bubbles suggest an unusual stabilized finite element method for which we establish a convergence study and obtain successful numerical simulations the method is generalized to allow for a convection operator in the equation this work may be employed as a starting point for simulation of nonlinear equations governing turbulence phenomena flows with chemical reactions and other important problems

the two dimensional annni model in the cvm approximation the phase diagram of the two dimensional annni model is investigated by the cluster variation method cvm we confirm the stability of the disordered phase down tot 0 and the absence of a lifshitz point at finite temperature fork 1 2 wherek the ratio of the second to the first neighbor pair interactions two different modulation regimes for the correlation functions of the disordered phase are separated by a disorder line along which theq vector of the susceptibility maximum undergoes a lock in transition the study in reciprocal space of the stability of the disordered phase allows us to define a critical line in the phase diagram along which theq vector characterizing the instability is incommensurate finally we show the existence of another lifshitz point fork tending to infinity

accurate fic fem formulation for the multidimensional steady state advection diffusion absorption equation abstract in this paper we present an accurate stabilized fic fem formulation for the multidimensional steady state advection diffusion absorption equation the stabilized formulation is based on the galerkin fem solution of the governing differential equations derived via the finite increment calculus fic method using two stabilization parameters the value of the two stabilization parameters ensuring an accurate nodal fem solution using uniform meshes of linear elements is obtained from the optimal values for the 1d problem the accuracy of the new fic fem formulation is demonstrated in the solution of 2d steady state advection diffusion absorption problems for a range of physical parameters and boundary conditions

modeling of steep layers in singularly perturbed diffusion reaction equation via flexible fine scale basis abstract we present a new stabilized method for the diffusion reaction equation which develops sharp boundary and or internal layers for the reaction dominated case i e singularly perturbed case the method relies on an improved expression for the stabilization parameter that is derived via the fine scale variational formulation facilitated by the variational multiscale vms framework the proposed fine scale basis consists of enrichment functions which may be nonzero at element edges the derived stabilization parameter enjoys spatial variation over element interiors and along inter element boundaries that helps model the rapid variation of the residual of the euler lagrange equations over the domain this feature also facilitates consistent stabilization across boundary and internal layers as well as capturing anisotropic refinement effects the method is able to better satisfy the maximum principle as compared to other existing methods we present a priori mathematical analysis of the stability and convergence of the method for the diffusion reaction equation optimal convergence on meshes comprised of linear triangles and bilinear quadrilateral elements are presented for smooth problems as well as for problems with steep boundary layers stability and accuracy features of the method for problems with discontinuous forcing function internal layers and boundary layers are shown and its performance on unstructured and distorted meshes comprised of quadrilateral and triangular elements is highlighted

comparison of some finite element methods for solving the diffusion convection reaction equation abstract in this paper we describe several finite element methods for solving the diffusion convection reaction equation none of them is new although the presentation is non standard in an effort to emphasize the similarities and differences between them in particular it is shown that the classical supg method is very similar to an explicit version of the characteristic galerkin method whereas the taylor galerkin method has a stabilization effect similar to a sub grid scale model which is in turn related to the introduction of bubble functions

primal interface formulation for coupling multiple pdes a consistent derivation via the variational multiscale method abstract this paper presents a primal interface formulation that is derived in a systematic manner from a lagrange multiplier method to provide a consistent framework to couple different partial differential equations pde as well as to tie together nonconforming meshes the derivation relies crucially on concepts from the variational multiscale vms approach wherein an additive multiscale decomposition is applied to the primary solution field modeling the fine scales locally at the interface using bubble functions consistent residual based terms on the boundary are obtained that are subsequently embedded into the coarse scale problem the resulting stabilized lagrange multiplier formulation is converted into a robust discontinuous galerkin dg method by employing a discontinuous interpolation of the multipliers along the segments of the interface as a byproduct analytical expressions are derived for the stabilizing terms and weighted numerical flux that reflect the jump in material properties governing equation or element geometry across the interface also a procedure is proposed for automatically generating the fine scale bubble functions that is inspired by a performance study of residual free bubbles for the interface problem a series of numerical tests confirms the robustness of the method for solving interface problems with heterogeneous elements materials and or governing equations and also highlights the benefit and importance of deriving the flux and stabilization terms

a weak galerkin finite element method for second order elliptic problems this paper introduces a finite element method by using a weakly defined gradient operator over generalized functions the use of weak gradients and their approximations results in a new concept called discrete weak gradients which is expected to play an important role in numerical methods for partial differential equations this article intends to provide a general framework for managing differential operators on generalized functions as a demonstrative example the discrete weak gradient operator is employed as a building block in the design of numerical schemes for a second order elliptic problem in which the classical gradient operator is replaced by the discrete weak gradient the resulting numerical scheme is called a weak galerkin wg finite element method it can be seen that the weak galerkin method allows the use of totally discontinuous functions in the finite element procedure for the second order elliptic problem an optimal order error estimate in both a discrete h 1 and l 2 norms are established for the corresponding weak galerkin finite element solutions a superconvergence is also observed for the weak galerkin approximation

a variational multiscale method for inelasticity application to superelasticity in shape memory alloys this paper presents a variational multiscale method for developing stabilized finite element formulations for small strain inelasticity the multiscale method arises from a decomposition of the displacement field into coarse resolved and fine unresolved scales the resulting finite element formulation allows arbitrary combinations of interpolation functions for the displacement and the pressure fields and thus yields a family of stable and convergent elements specifically equal order interpolations that are easy to implement but violate the celebrated babuska brezzi condition become stable and convergent an important feature of the present method is that it does not lock in the incompressible limit a nonlinear constitutive model for the superelastic behavior of shape memory alloys is integrated in the multiscale formulation numerical tests of the performance of the elements are presented and representative simulations of the superelastic behavior of shape memory alloys are shown 2005 elsevier b v all rights reserved

tiling problems and undecidability in the cluster variation method in cluster approximations for lattice systems the thermodynamic behavior of the infinite system is inferred from that of a relatively small finite subsystem cluster approximations being made for the influence of the surrounding system in this context we study for translation invariant classical lattice systems the conditions under which a state for a cluster admits an extension to a global translation invariant state this extension problem is related to undecidable tiling problems the implication is that restrictions of global translation invariant states cannot be characterized purely locally in general this means that there is an unavoidable element of uncertainty in the application of a cluster approximation

cluster variation method in statistical physics and probabilistic graphical models the cluster variation method cvm is a hierarchy of approximate variational techniques for discrete ising like models in equilibrium statistical mechanics improving on the mean field approximation and the bethe peierls approximation which can be regarded as the lowest level of the cvm in recent years it has been applied both in statistical physics and to inference and optimization problems formulated in terms of probabilistic graphical models the foundations of the cvm are briefly reviewed and the relations with similar techniques are discussed the main properties of the method are considered with emphasis on its exactness for particular models and on its asymptotic properties the problem of the minimization of the variational free energy which arises in the cvm is also addressed and recent results about both provably convergent and message passing algorithms are discussed

the discontinuous enrichment method we propose a finite element based discretization method in which the standard polynomial field is enriched within each element by a nonconforming field that is added to it the enrichment contains free space solutions of the homogeneous differential equation that are not represented by the underlying polynomial field continuity of the enrichment across element interfaces is enforced weakly by lagrange multipliers in this manner we expect to attain high coarse mesh accuracy without significant degradation of conditioning assuring good performance of the computation at any mesh resolution examples of application to acoustics and advection diffusion are presented

a multiscale finite element method for the incompressible navier stokes equations this paper presents a new multiscale finite element method for the incompressible navier stokes equations the proposed method arises from a decomposition of the velocity field into coarse resolved scales and fine unresolved scales modeling of the unresolved scales corrects the lack of stability of the standard galerkin formulation and yields a method that possesses superior properties like that of the streamline upwind petrov galerkin supg method and the galerkin least squares gls method the multiscale method allows arbitrary combinations of interpolation functions for the velocity and the pressure fields specifically the equal order interpolations that are easy to implement but violate the celebrated babuska brezzi condition a significant feature of the present method is that the structure of the stabilization tensor s appears naturally via the solution of the fine scale problem a family of 2 d elements comprising 3 and 6 node triangles and 4 and 9 node quadrilaterals has been developed convergence studies for the method on uniform skewed as well as composite meshes are presented numerical simulations of the nonlinear steady and transient flow problems are shown that exhibit the good stability and accuracy properties of the method 2005 elsevier b v all rights reserved

on an improved unusual stabilized finite element method for theadvective reactive diffusive equation an improved unusual finite element method is studied herein for a second order linear scalar differential equation including a zero order term the method consists in subtracting from the standard galerkin method a mesh dependent term suggested by static condensation of the bubbles based on this idea a new stabilized parameter is constructed which improves accuracy of the solution several numerical tests attest the efficiency of the present method and an error analysis is perfomed

two dimensional translation invariant probability distributions approximations characterizations and no go theorems we study the properties of the set of marginal distributions of infinite translation invariant systems in the two dimensional square lattice in cases where the local variables can only take a smal

the global markov property for equilibrium states which are determined by correlations in a strip consider an interaction round face potential on the lattice d which may include nearest neighbor and next nearest neighbor pair interactions as well as more general plaquette terms assuming some periodicity of the potential it is shown that under the condition that equilibrium states can be distinguished by expectation values of sufficiently local observables equilibrium states possess a global markov property the condition under which this markov property is shown to hold is met in particular in case the equilibrium state is unique or determined by the magnetization the proof is based on an application of the variational principle to states which are constructed like markov chains

a variational multiscale method for incompressible turbulent flows bubble functions and fine scale fields this paper presents a residual based turbulence model for the incompressible navier stokes equations the method is derived employing the variational multiscale vms framework a multiscale decomposition of the continuous solution and a priori unique decomposition of the admissible spaces of functions lead to two coupled nonlinear problems termed as the coarse scale and the fine scale sub problems the fine scale velocity field is assumed to be nonlinear and time dependent and is modeled via the bubble functions approach applied directly to the fine scale sub problem a significant contribution in this paper is a systematic and consistent derivation of the fine scale variational operator commonly termed as the stabilization tensor that possesses the right order in the advective and diffusive limits and variationally projects the fine scale solution onto the coarse scale space a direct treatment of the fine scale problem via bubble functions offers several fine scale approximation options with varying degrees of mathematical sophistication that are investigated via benchmark problems numerical accuracy of the proposed method is shown on a forced isotropic turbulence problem statistically stationary turbulent channel flow problems at ret 395 and 590 and non equilibrium turbulent flow around a cylinder at re 3 900