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

rigorous results for the number of convex polygons on the square and honeycomb lattices the generating functions for the number of convex polygons on the square and honeycomb lattices are derived rigorously these functions were found by guttmann and enting 1988 their calculation is based on the series expansions up to the 64th order and is nonrigorous the asymptotic form of the mean squared radius of gyration of n step convex polygons on the square lattice is determined and the critical exponent v is 1

subgrid upscaling and mixed multiscale finite elements second order elliptic problems in divergence form with a highly varying leading order coefficient on the scale epsilon can be approximated on coarse meshes of spacing h gg epsilon only if one uses special techniques the mixed variational multiscale method also called subgrid upscaling can be used and this method is extended to allow oversampling of the local subgrid problems the method is shown to be equivalent to the multiscale finite element method when one uses the lowest order raviart thomas spaces and provided that there are no fine scale components in the source function f in the periodic setting a multiscale error analysis based on homogenization theory of the more general subgrid upscaling method shows that the error is o epsilon hm sqrt epsilon h where m 1 moreover m 2 if one uses the second order brezzi douglas marini or brezzi douglas duran fortin spaces and no oversampling the error bounding constant depends only on the hm 1 norm of f and so is independent of small scales when m 1 when oversampling is not used a superconvergence result for the pressure approximation is shown

a global jacobian method for mortar discretizations of nonlinear porous media flows we describe a nonoverlapping domain decomposition algorithm for nonlinear porous media flows discretized with the multiscale mortar mixed finite element method there are two main ideas 1 linearize the global system in both subdomain and interface variables simultaneously to yield a single newton iteration and 2 algebraically eliminate subdomain velocities and optionally subdomain pressures to solve linear systems for the 1st or the 2nd schur complements solving the 1st schur complement system gives the multiscale solution without the need to solve an interface iteration solving the 2nd schur complement system gives a linear interface problem for a nonlinear model the methods are less complex than a previously developed nonlinear mortar algorithm which requires two nested newton iterations and a forward difference approximation furthermore efficient linear preconditioners can be applied to speed up the iteration the methods are implemented in parallel and a numerical study is performed

tailbiting codes bounds and search results tailbiting trellis representations of linear block codes with an arbitrary sectionalization of the time axis are studied the notations of regular and irregular tailbiting codes are introduced and their maximal state complexities are lower bounded the asymptotic behavior of the derived bound is investigated furthermore for regular tailbiting codes the product state complexity is lower bounded tables of new tailbiting trellis representations of linear block codes of rates 1 2 1 3 and 1 4 are presented almost all found trellises are optimal in the sense of the new bound on the state complexity and for most codes with nonoptimal trellises there exist time varying trellises which are optimal five of our newly found tailbiting codes are better than the previously known linear codes with the same parameters four of them are also superior to any previously known nonlinear code with the same parameters also more than 40 other quasi cyclic codes have been found that improve the parameter set of previously known quasi cyclic codes

implementation of a mortar mixed finite element method using a multiscale flux basis this paper provides a new implementation of a multiscale mortar mixed finite element method for second order elliptic problems the algorithm uses non overlapping domain decomposition to reformulate a fine scale problem as a coarse scale mortar interface problem which is then solved using an iterative method the original implementation by arbogast pencheva wheeler and yotov multiscale model simul 2007 required solving one local dirichlet problem on each subdomain per interface iteration we alter this implementation by forming a multiscale flux basis this basis consists of mortar functions representing the individual flux responses for each mortar degree of freedom on each subdomain independently the computation of these basis functions requires solving a fixed number of dirichlet subdomain problems taking linear combinations of the multiscale flux basis functions replaces the need to solve any dirichlet subdomain problems during the interface iteration this new implementation yields the same solution as the original implementation and is computationally more efficient in cases where the number of interface iterations is greater than the number of mortar degrees of freedom per subdomain the gain in computational efficiency increases with the number of subdomains

polygons on the honeycomb lattice the authors enumerated the number of self avoiding polygons on the honeycomb lattice to 82 steps analysis of the resulting series gives for the connective constant mu 1 8477594 or 6 10 6 in agreement with nienhuis value 1984 of mu 2 2 12 1 84775906 the critical exponent alpha 0 4999 or 10 4 unbiased or alpha 0 49998 or 2 10 0 5 biased a non physical singularity on the negative real axis is accurately located and some curious results concerning the confluent singularity are obtained

multiscale hybrid mixed method for the stokes and brinkman equations the method abstract the multiscale hybrid mixed mhm method is extended to the stokes and brinkman equations with highly heterogeneous coefficients the approach is constructive we first propose an equivalent dual hybrid formulation of the original problem using a coarse partition of the heterogeneous domain faces may be not aligned with jumps in the data then the exact velocity and the pressure are characterized as the solution of a global face problem and the solutions of local independent stokes or brinkman problems at the continuous level owing to this decomposition the one level mhm method stems from the standard galerkin approach for the lagrange multiplier space basis functions are responsible for upscaling the unresolved scales of the medium into the global formulation they are the exact solution of the local problems with prescribed neumann boundary conditions on faces driven by the lagrange multipliers we make the mhm method effective by adopting the unusual stabilized finite element method to solve the local problems approximately as such equal order interpolation turns out to be an option for the velocity the pressure and the lagrange multipliers the numerical solutions share the important properties of the continuum such as local equilibrium with respect to external forces and local mass conservation several academic and highly heterogeneous tests infer that the method achieves super convergence for the velocity as well optimal convergence for the pressure and also for the stress tensor in their natural norms

a new two variable generating function for convex self avoiding polygons the width of a self avoiding polygon on the square lattice is defined as the minimal horizontal or vertical distance between two of its parallel edges if the polygons are convex this distance is internal the perimeter generating functions for such convex polygons whose widths exceed a threshold can be given explicitly from these expressions a two variable width and perimeter generating function can be constructed the corresponding phase diagram shows two types of critical behaviour which meet at a tricritical point

mixed finite elements for second order elliptic problems in three variables two families of mixed finite elements one based on simplices and the other on cubes are introduced as alternatives to the usual raviart thomas nedelec spaces these spaces are analogues of those introduced by brezzi douglas and marini in two space variables error estimates inl2 andh s are derived

a mixed multiscale finite element method for elliptic problems with oscillating coefficients the recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the large scale structure of the solutions without resolving all the fine scale structures motivated by the numerical simulation of flow transport in highly heterogeneous porous media we propose a mixed multiscale finite element method with an over sampling technique for solving second order elliptic equations with rapidly oscillating coefficients the multiscale finite element bases are constructed by locally solving neumann boundary value problems we provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain the asymptotic structure of the solutions numerical experiments are carried out for flow transport in a porous medium with a random log normal relative permeability to demonstrate the efficiency and accuracy of the proposed method

high temperature expansions for classical spin models iii series and phase diagrams for the square lattice abstract the concepts of duality and broken symmetry are discussed briefly it is shown how generalized colouring polynomials can be used to calculate the terms in a high temperature expansion effectively these results and those from two previous papers are used to obtain high temperature series for a great variety of models these are given explicitly for all models with two boltzmann factors and for the ashkin teller model many phase diagrams obtained from the series by pade analysis are presented and their salient features discussed a principle suggested by these phase diagrams is applied to the dilute potts model

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

number of almost convex polygons on the square lattice the generating function for the number nc n of almost convex polygons on the square lattice with concavity index c 1 and perimeter n is derived rigorously the asymptotic behaviour of nc n for large n is determined and this result confirms a conjecture by enting et al 1992

a spectral multiscale hybridizable discontinuous galerkin method for second order elliptic problems abstract we design a multiscale model reduction framework within the hybridizable discontinuous galerkin finite element method our approach uses local snapshot spaces and local spectral decomposition following the concept of generalized multiscale finite element methods we propose several multiscale finite element spaces on the coarse edges that provide a reduced dimensional approximation for numerical traces within the hdg framework we provide a general framework for systematic construction of multiscale trace spaces using local snapshots we avoid high dimensional representation of trace spaces and use some local features of the solution space in constructing a low dimensional trace space we investigate the solvability and numerically study the performance of the proposed method on a representative number of numerical examples

an updated table of minimum distance bounds for binary linear codes in 1973 helgert and sfinaff published a table of upper and lower bounds on the maximum minimum distance for binary linear error correcting codes up to length 127 this article presents an updated table incorporating numerous improvements that have appeared since then to simplify the updating task the author has developed a computer program that systematically investigates the consequences of each improvement by applying several well known general code construction techniques this program also made it possible to check the original table furthermore it offers a quick and reliable update service for future improvements

convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients we propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients the main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution this is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions as a consequence the base functions are adapted to the local properties of the differential operator in this paper we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain a useful asymptotic solution structure the issue of boundary conditions for the base functions will be discussed our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution independently of the small scale in the homogenization limit application of our method to problems with continuous scales is also considered

high temperature expansions for classical spin models ii the square and hypercubic lattices and the potts model abstract all graphs and their contributions needed for a high temperature expansion of the partition function of a completely permissible spin model up to order 16 for the square lattice and up to order 11 for the hypercubic lattice are listed for the potts model the corresponding series are derived and analyzed by means of pade approximants in the square lattice case the phase transition of the m state potts model is reproduced to within 1 for m 2

on the complexity of minimum distance decoding of long linear codes we suggest a decoding algorithm of q ary linear codes which we call supercode decoding it ensures the error probability that approaches the error probability of minimum distance decoding as the length of the code grows for n spl rarr spl infin the algorithm has the maximum likelihood performance the asymptotic complexity of supercode decoding is exponentially smaller than the complexity of all other methods known the algorithm develops the ideas of covering set decoding and split syndrome decoding

the number of convex polygons on the square and honeycomb lattices a subset of the set of self avoiding polygons sap embeddable on the square lattice which display the property of convexity is defined an algorithm for their enumeration is developed and from the available series coefficients the exact generating function is found the singularity structure appears similar to that of the unsolved sap problem but with different critical exponents and critical points the enumeration of convex polygons allows the extension of the existing series for the square lattice sap by one term for the honeycomb lattice similar results have been obtained despite a less natural definition of convexity

mixed multiscale finite element methods for stochastic porous media flows in this paper we propose a stochastic mixed multiscale finite element method the proposed method solves the stochastic porous media flow equation on the coarse grid using a set of precomputed basis functions the precomputed basis functions are constructed based on selected realizations of the stochastic permeability field and furthermore the solution is projected onto the finite dimensional space spanned by these basis functions we employ multiscale methods using limited global information since the permeability fields do not have apparent scale separation the proposed approach does not require any interpolation in stochastic space and can easily be coupled with interpolation based approaches to predict the solution on the coarse grid numerical results are presented for permeability fields with normal and exponential variograms

global local nonlinear model reduction for flows in heterogeneous porous media abstract in this paper we combine discrete empirical interpolation techniques global mode decomposition methods and local multiscale methods such as the generalized multiscale finite element method gmsfem to reduce the computational complexity associated with nonlinear flows in highly heterogeneous porous media to solve the nonlinear governing equations we employ the gmsfem to represent the solution on a coarse grid with multiscale basis functions and apply proper orthogonal decomposition on a coarse grid computing the gmsfem solution involves calculating the residual and the jacobian on a fine grid as such we use local and global empirical interpolation concepts to circumvent performing these computations on the fine grid the resulting reduced order approach significantly reduces the flow problem size while accurately capturing the behavior of fully resolved solutions we consider several numerical examples of nonlinear multiscale partial differential equations that are numerically integrated using fully implicit time marching schemes to demonstrate the capability of the proposed model reduction approach to speed up simulations of nonlinear flows in high contrast porous media

multiscale model reduction for transport and flow problems in perforated domains abstract convection dominated transport phenomenon is important for many applications in these applications the transport velocity is often a solution of heterogeneous flow problems which results to a coupled flow and transport phenomena in this paper we consider a coupled flow stokes problem and transport unsteady convection diffusion problem in perforated domains perforated domains see fig 1 represent void space outside hard inclusions as in porous media filters and so on we construct a coarse scale solver based on generalized multiscale finite element method gmsfem for a coupled flow and transport the main idea of the gmsfem is to develop a systematic approach for computing multiscale basis functions we use a mixed formulation and appropriate multiscale basis functions for both flow and transport to guarantee a mass conservation for the transport problem we use petrov galerkin mixed formulation which provides a stability as a first approach we use the multiscale flow solution in constructing the basis functions for the transport equation in the second approach we construct multiscale basis functions for coupled flow and transport without solving global flow problem the novelty of this approach is to construct a coupled multiscale basis function numerical results are presented for computations using offline basis we also present an algorithm for adaptively adding online multiscale basis functions which are computed using the residual information numerical examples using online gmsfem show the speed up of convergence

multiscale finite volume method for parabolic problems arising from compressible multiphase flow in porous media the multiscale finite volume msfv method was originally developed for the solution of heterogeneous elliptic problems with reduced computational cost recently some extensions of this method for parabolic problems have been proposed these extensions proved effective for many cases however they are neither general nor completely satisfactory for instance they are not suitable for correctly capturing the transient behavior described by the parabolic pressure equation in this paper we present a general multiscale finite volume method for parabolic problems arising for example from compressible multiphase flow in porous media opposed to previous methods here the basis and correction functions are solutions of full parabolic governing equations in localized domains at the same time to enhance the computational efficiency of the scheme the basis functions are kept pressure independent and do not have to be recalculated as pressure evolves this general approach requires no additional assumptions and its good efficiency and high accuracy is demonstrated for various challenging test cases finally to improve the quality of the results and also to extend the scheme for highly anisotropic heterogeneous problems it is combined with the iterative msfv i msfv method for parabolic problems as one iterates the i msfv solutions of compressible multiphase problems parabolic problems converge to the corresponding fine scale reference solutions in the same way as demonstrated recently for incompressible cases elliptic problems therefore the proposed msfv method can also be regarded as an efficient linear solver for parabolic problems and studies of its efficiency are presented for many test cases

convergence of a nonconforming multiscale finite element method the multiscale finite element method msfem t y hou x h wu and z cai math comp 1998 to appear t y hou and x h wu j comput phys 134 1997 pp 169 189 has been introduced to capture the large scale solutions of elliptic equations with highly oscillatory coefficients this is accomplished by constructing the multiscale base functions from the local solutions of the elliptic operator our previous study reveals that the leading order error in this approach is caused by the resonant sampling which leads to large error when the mesh size is close to the small scale of the continuous problem similar difficulty also arises in numerical upscaling methods an oversampling technique has been introduced to alleviate this difficulty t y hou and x h wu j comput phys 134 1997 pp 169 189 a consequence of the oversampling method is that the resulting finite element method is no longer conforming here we give a detailed analysis of the nonconforming error our analysis also reveals a new cell resonance error which is caused by the mismatch between the mesh size and the wavelength of the small scale we show that the cell resonance error is of lower order our numerical experiments demonstrate that the cell resonance error is generically small and is difficult to observe in practice

soft decision decoding using punctured codes let a q ary linear n k code be used over a memoryless channel we design a soft decision decoding algorithm that tries to locate a few most probable error patterns on a shorter length s spl isin k n first we take s cyclically consecutive positions starting from any initial point then we cut the subinterval of length s into two parts and examine t most plausible error patterns on either part to obtain codewords of a punctured s k code we try to match the syndromes of both parts finally the designed codewords of an s k code are re encoded to find the most probable codeword on the full length n for any long linear code the decoding error probability of this algorithm can be made arbitrarily close to the probability of its maximum likelihood ml decoding given sufficiently large t by optimizing s we prove that this near ml decoding can be achieved by using only t spl ap q sup n k k n k error patterns for most long linear codes this optimization also gives about t re encoded codewords as a result we obtain the lowest complexity order of q sup n k k n k known to date for near ml decoding for codes of rate 1 2 the new bound grows as a cubic root of the general trellis complexity q sup min n k k for short blocks of length 63 the algorithm reduces the complexity of the trellis design by a few decimal orders

multiscale empirical interpolation for solving nonlinear pdes in this paper we propose a multiscale empirical interpolation method for solving nonlinear multiscale partial differential equations the proposed method combines empirical interpolation techniques and local multiscale methods such as the generalized multiscale finite element method gmsfem to solve nonlinear equations the gmsfem is used to represent the solution on a coarse grid with multiscale basis functions computed offline computing the gmsfem solution involves calculating the system residuals and jacobians on the fine grid we use empirical interpolation concepts to evaluate these residuals and jacobians of the multiscale system with a computational cost which is proportional to the size of the coarse scale problem rather than the fully resolved fine scale one the empirical interpolation method uses basis functions which are built by sampling the nonlinear function we want to approximate a limited number of times the coefficients needed for this approximation are computed in the offline stage by inverting an inexpensive linear system the proposed multiscale empirical interpolation techniques 1 divide computing the nonlinear function into coarse regions 2 evaluate contributions of nonlinear functions in each coarse region taking advantage of a reduced order representation of the solution and 3 introduce multiscale proper orthogonal decomposition techniques to find appropriate interpolation vectors we demonstrate the effectiveness of the proposed methods on several nonlinear multiscale pdes that are solved with newton s methods and fully implicit time marching schemes our numerical results show that the proposed methods provide a robust framework for solving nonlinear multiscale pdes on a coarse grid with bounded error and significant computational cost reduction

area weighted moments of convex polygons on the square lattice when constructing series expansions for the zero field partition function of the q state potts model on the square lattice using the finite lattice method the series can be extended by one term by adding a correction of q 1 times the number of convex polygons of an appropriate order by virtue of duality such corrections apply for both high and low temperature series for low temperature series corrections to the field dependent partition function are of the same order in the temperature variable as the zero field correction the general corrections are given by area weighted moments of the number of convex polygons the authors obtain conjectured closed form expressions for the two generating functions that give corrections to the first two field derivatives of the low temperature partition function

the numbers of line defect self avoiding polygons and related subgraphs of the square lattice abstract those self avoiding polygons of the square lattice which have a perimeter larger by 2d than the perimeter of their bounding rectangle and for which this is due to a line defect are defined also classes of spiked convex self avoiding polygons are introduced a large number of generating functions for these types of graphs can be calculated exactly all imply an asymptotic behaviour for the number of such graphs with perimeter p as p 2 2 p for p

multiscale mortar expanded mixed discretization of nonlinear elliptic problems abstract we consider the discretization of nonlinear second order elliptic partial differential equations by multiscale mortar expanded mixed method this is a domain decomposition method in which the model problem is restricted to the small pieces by dividing the computational domain into the non overlapping subdomains an unknown lagrange multiplier is introduced on the interfaces which serve as pressure dirichlet boundary condition for local subdomain problems a finite element space is defined on interface to approximate the pressure boundary such that the normal fluxes match weakly on interface we demonstrated solvability of discretization and established a priori l2 error estimates for both vector and scalar approximations we proved the optimal order convergence rates by an appropriate choice of mortar space and polynomial degree of approximation the uniqueness of discrete problem is shown for sufficiently small values of mesh size an error estimate for the mortar pressure is derived via linear interface formulation with pressure dependent coefficient we also present the analysis of the linear second order elliptic problem and prove the similar results the computational experiments are presented to validate theory

multiscale direction splitting algorithms for parabolic equations with highly heterogeneous coefficients in this paper we discuss two methods for upscaling of highly heterogeneous data for parabolic problems in the context of a direction splitting time approximation the first method is a direct application of the idea of jenny et al 2003 in the context of the direction splitting approach the second method devises the approximation from the schur complement corresponding to the interface unknowns of the coarse grid by applying a proper l 2 projection operator to it the spatial discretization employed in this paper is based on a mac finite volume stencil but the same approach can be implemented within a proper finite element discretization a key feature of the present approach is that it can extend to 3d problems with very little computational overhead the properties of the resulting approximations are demonstrated numerically on some benchmark coefficient data available in the literature

beast decoding of block codes obtained via convolutional codes beast is a bidirectional efficient algorithm for searching trees in this correspondence beast is extended to maximum likelihood ml decoding of block codes obtained via convolutional codes first it is shown by simulations that the decoding complexity of beast is significantly less than that of the viterbi algorithm then asymptotic upper bounds on the beast decoding complexity for three important ensembles of codes are derived they verify beast s high efficiency compared to other algorithms for high rates the new asymptotic bound for the best ensemble is in fact better than previously known bounds

exact solution of the convex polygon perimeter and area generating function an explicit expression is derived for the three variable generating function p x y z sigma m or 1 sigma n or 1 sigma r or 1 x2ny2mzrcn m r where cn m r is the number of convex polygons with horizontal width n vertical height m and area r

global estimates for mixed methods for second order elliptic equations global error estimates in l2 92 l 9 and h s q 2 in r2 or a re derived for a mixed finite element method for the dirichlet problem for the elliptic operator lp div a grad p bp cp based on the raviart thomas nedelec space vh x wh c h div 92 x l2 g2 optimal order estimates are obtained for the approximation of p and the associated velocity field u a grad p bp in l2 g2 and h s 92 0 6 s k 1 and if q c r2 forp in lv2

special finite element methods for a class of second order elliptic problems with rough coefficients in this paper the approximate solution of a class of second order elliptic equations with rough coefficients is considered problems of the type considered arise in the analysis of unidirectional composites where the coefficients represent the properties of the material several methods for this class of problems are presented and it is shown that they have the same accuracy as usual methods have for problems with smooth coefficients the methods are referred to as special finite element methods because they are of finite element type but employ special shape functions chosen to accurately model the unknown solution

self avoiding walks neighbour avoiding walks and trails on semiregular lattices we study self avoiding and neighbour avoiding walks and lattice trails on two semiregular lattices the lattice and the lattice for the lattice we find the exact connective constant for both self avoiding walks neighbour avoiding walks and trails for the lattice we generate long series which permit the accurate estimation of the connective constant for self avoiding walks and trails

self avoiding polygons on the square lattice we have developed an improved algorithm that allows us to enumerate the number of self avoiding polygons on the square lattice to perimeter length 90 analysis of the resulting series yields very accurate estimates of the connective constant 2 638 158 529 27 1 biased and the critical exponent 0 500 0005 10 unbiased the critical point is indistinguishable from a root of the polynomial 581x4 7x2 13 0 an asymptotic expansion for the coefficients is given for all n there is strong evidence for the absence of any non analytic correction to scaling exponent

generating functions for classes of self avoiding polygons on the square lattice abstract a transfer matrix technique is presented which allows the explicit calculation of the generating functions for those self avoiding polygons on the square lattice which have a perimeter equal to or two larger than the perimeter of their bounding rectangles

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

generalized finite element methods their performance and their relation to mixed methods the notion of a generalized finite element method is introduced this class of methods is analyzed and their relation to mixed methods is discussed the class of generalized finite element methods offers a wide variety of computational procedures from which particular procedures can be selected for particular problems a particular generalized finite element method which is very effective for problems with rough coefficients is discussed in detail

a beast for prowling in trees when searching for convolutional codes and tailbiting codes of high complexity it is of vital importance to use fast algorithms for computing their weight spectra which corresponds to finding low weight paths in their code trellises this can be efficiently done by a combined search in both forward and backward code trees a bidirectional efficient algorithm for searching such code trees beast is presented for large encoder memories it is shown that beast is significantly more efficient than comparable algorithms beast made it possible to find new convolutional and tailbiting codes that have larger free minimum distances than the previously best known codes with the same parameters tables of such codes are presented

generalized multiscale finite element methods for space time heterogeneous parabolic equations abstract in this paper we consider local multiscale model reduction for problems with multiple scales in space and time we developed our approaches within the framework of the generalized multiscale finite element method gmsfem using space time coarse cells the main idea of gmsfem is to construct a local snapshot space and a local spectral decomposition in the snapshot space previous research in developing multiscale spaces within gmsfem focused on constructing multiscale spaces and relevant ingredients in space only in this paper our main objective is to develop a multiscale model reduction framework within gmsfem that uses space time coarse cells we construct space time snapshot and offline spaces we compute these snapshot solutions by solving local problems a complete snapshot space will use all possible boundary conditions however this can be very expensive we propose using randomized boundary conditions and oversampling cf calo et al 2016 we construct the local spectral decomposition based on our analysis as presented in the paper we present numerical results to confirm our theoretical findings and to show that using our proposed approaches we can obtain an accurate solution with low dimensional coarse spaces we discuss using online basis functions constructed in the online stage and using the residual information online basis functions use global information via the residual and provide fast convergence to the exact solution provided a sufficient number of offline basis functions we present numerical studies for our proposed online procedures we remark that the proposed method is a significant extension compared to existing methods which use coarse cells in space only because of 1 the parabolic nature of cell solutions 2 extra degrees of freedom associated with space time cells and 3 local boundary conditions in space time cells

lower bounds on trellis complexity of block codes the trellis state complexity s of a linear block code is defined as the logarithm of the maximum number of states in its minimal trellis we present a new lower bound on the state complexity of linear codes which includes most of the existing bounds as special cases the new bound is obtained by dividing the time axis for the code into several sections of varying lengths as opposed to the division into two sections the past and the future employed in the well known dlp bounds for a large number of codes this results in a considerable improvement upon the dlp bound moreover we generalize the new bound to nonlinear codes and introduce several alternative techniques for lower bounding the trellis complexity based on the distance spectrum and other combinatorial properties of the code we also show how the general ideas developed in this paper may be employed to lower bound the maximum and the total number of branches in the trellis leading to considerably tighter bounds on these quantities furthermore the asymptotic behavior of the new bounds is investigated and shown to improve upon the previously known asymptotic estimates of trellis state complexity

flux formulation of parabolic equations with highly heterogeneous coefficients abstract in this paper we study the flux formulation of unsteady diffusion equations with highly heterogeneous permeability coefficients and their discretization in the proposed approach first an equation governing the flux of the unknown scalar quantity is solved and then the scalar is recovered from its flux the problem for the flux is further discretized by splitting schemes that yield locally one dimensional problems and therefore the resulting linear systems are tridiagonal if the spatial discretization uses cartesian grids a first and a formally second order time discretization splitting scheme have been implemented in both two and three dimensions and we present results for a few model problems using a challenging benchmark dataset

generalized multiscale finite element method symmetric interior penalty coupling motivated by applications to numerical simulations of flows in highly heterogeneous porous media we develop multiscale finite element methods for second order elliptic equations we discuss a multiscale model reduction technique in the framework of the discontinuous galerkin finite element method we propose two different finite element spaces on the coarse mesh the first space is based on a local eigenvalue problem that uses an interior weighted l 2 norm and a boundary weighted l 2 norm for computing the mass matrix the second choice is based on generation of a snapshot space and subsequent selection of a subspace of a reduced dimension the approximation with these multiscale spaces is based on the discontinuous galerkin finite element method framework we investigate the stability and derive error estimates for the methods and further experimentally study their performance on a representative number of numerical examples

multiscale finite volume method for compressible multiphase flow in porous media the multiscale finite volume msfv method has been recently developed and tested for multiphase flow problems with simplified physics i e incompressible flow without gravity and capillary effects and proved robust accurate and efficient however applications to practical problems necessitate extensions that enable the method to deal with more complex processes in this paper we present a modified version of the msfv algorithm that provides a suitable and natural framework to include additional physics the algorithm consists of four main steps computation of the local basis functions which are used to extract the coarse scale effective transmissibilities solution of the coarse scale pressure equation reconstruction of conservative fine scale fluxes and solution of the transport equations within this framework we develop a msfv method for compressible multiphase flow the basic idea is to compute the basis functions as in the case of incompressible flow such that they remain independent of the pressure the effects of compressibility are taken into account in the solution of the coarse scale pressure equation and if necessary in the reconstruction of the fine scale fluxes we consider three models with an increasing level of complexity in the flux reconstruction and test them for highly compressible flows tracer transport in gas flow imbibition and drainage of partially saturated reservoirs depletion of gas water reservoirs and flooding of oil gas reservoirs we demonstrate that the msfv method provides accurate solutions for compressible multiphase flow problems whereas slightly compressible flows can be treated with a very simple model a more sophisticate flux reconstruction is needed to obtain accurate fine scale saturation fields in highly compressible flows

a matrix analysis of operator based upscaling for the wave equation scientists and engineers who wish to understand the earth s subsurface are faced with a daunting challenge features of interest range from the microscale centimeters to the macroscale hundreds of kilometers it is unlikely that computational power limitations will ever allow modeling of this level of detail numerical upscaling is one technique intended to reduce this computational burden the operator based algorithm developed originally for elliptic flow problems is modified for the acoustic wave equation with the wave equation written as a first order system in space we solve for pressure and its gradient acceleration the upscaling technique relies on decomposing the solution space into coarse and fine components operator based upscaling applied to the acoustic wave equation proceeds in two steps step one involves solving for fine grid features internal to coarse blocks this stage can be solved quickly via a well chosen set of coarse grid boundary conditions each coarse problem is solved independently of its neighbors in step two we augment the coarse scale problem via this internal subgrid information unfortunately the complexity of the numerical upscaling algorithm has always obscured the physical meaning of the resulting solution via a detailed matrix analysis the coarse scale acceleration is shown to be the solution of the original constitutive equation with input density field corresponding to an averaged density along coarse block edges the pressure equation corresponds to the standard acoustic wave equation at nodes internal to coarse blocks however along coarse cell boundaries the upscaled solution solves a modified wave equation which includes a mixed second derivative term

analysis of a two scale locally conservative subgrid upscaling for elliptic problems we present a two scale theoretical framework for approximating the solution of a second order elliptic problem the elliptic coefficient is assumed to vary on a scale that can be resolved on a fine numerical grid but limits on computational power require that computations be performed on a coarse grid we consider the elliptic problem in mixed variational form over w times bf v subset l 2 times h rm div we base our scale expansion on local mass conservation over the coarse grid it is used to define a direct sum decomposition of w times bf v into coarse and subgrid subspaces w c times bf v c and delta w times delta bf v such that 1 nabla cdot bf v c w c and nabla cdot delta bf v delta w and 2 the space delta bf v is locally supported over the coarse mesh we then explicitly decompose the variational problem into coarse and subgrid scale problems the subgrid problem gives a well defined operator taking w c times bf v c to delta w times delta bf v which is localized in space and it is used to upscale that is to remove the subgrid from the coarse scale problem using standard mixed finite element spaces two scale mixed spaces are defined a mixed approximation is defined which can be viewed as a type of variational multiscale method or a residual free bubble technique a numerical green s function approach is used to make the approximation to the subgrid operator efficient to compute a mixed method pi operator is defined for the two scale approximation spaces and used to show optimal order error estimates

a weak galerkin generalized multiscale finite element method in this paper we propose a general framework for weak galerkin generalized multiscale wg gms finite element method for the elliptic problems with rapidly oscillating or high contrast coefficients this general wg gms method features in high order accuracy on general meshes and can work with multiscale basis derived by different numerical schemes a special case is studied under this wg gms framework in which the multiscale basis functions are obtained by solving local problem with the weak galerkin finite element method convergence analysis and numerical experiments are obtained for the special case

analysis of multiscale mortar mixed approximation of nonlinear elliptic equations abstract a multiscale mortar mixed finite element method is established to approximate non linear second order elliptic equations the method is based on non overlapping domain decomposition and mortar finite element methods the existence and uniqueness of the approximation are demonstrated and a priori l 2 error estimates for the velocity and pressure are derived an error bound for mortar pressure is proved convergence estimates of the mortar pressure are based on a linear interface formulation having the discrete pressure dependent coefficient optimal order convergence is achieved on the fine scale by a proper choice of mortar space and polynomial degree of approximation the quadratic convergence of the newton raphson method is proved for the nonlinear algebraic system arising from the mortar mixed formulation of the problem numerical experiments are performed to support theoretic results

exact solution of the row convex polygon perimeter generating function an explicit expression is obtained for the perimeter generating function g y sigma n or 2 any2n for row convex polygons on the square lattice where an is the number of 2n step row convex polygons an asymptotic expression for an approximately a mu nn 3 2 is obtained where mu 3 22 and a are given the authors also show that the generating function is an algebraic function and that it satisfies an inhomogeneous linear differential equation of degree three

flux norm approach to finite dimensional homogenization approximations with non separated scales and high contrast we consider linear divergence form scalar elliptic equations and vectorial equations for elasticity with rough l r d coefficients a x that in particular model media with non separated scales and high contrast in material properties while the homogenization of pdes with periodic or ergodic coefficients and well separated scales is now well understood we consider here the most general case of arbitrary bounded coefficients for such problems we introduce explicit and optimal finite dimensional approximations of solutions that can be viewed as a theoretical galerkin method with controlled error estimates analogous to classical homogenization approximations in particular this approach allows one to analyze a given medium directly without introducing the mathematical concept of an e family of media as in classical homogenization we define the flux norm as the l 2 norm of the potential part of the fluxes of solutions which is equivalent to the usual h 1 norm we show that in the flux norm the error associated with approximating in a properly defined finite dimensional space the set of solutions of the aforementioned pdes with rough coefficients is equal to the error associated with approximating the set of solutions of the same type of pdes with smooth coefficients in a standard space for example piecewise polynomial we refer to this property as the transfer property a simple application of this property is the construction of finite dimensional approximation spaces with errors independent of the regularity and contrast of the coefficients and with optimal and explicit convergence rates this transfer property also provides an alternative to the global harmonic change of coordinates for the homogenization of elliptic operators that can be extended to elasticity equations the proofs of these homogenization results are based on a new class of elliptic inequalities these inequalities play the same role in our approach as the div curl lemma in classical homogenization

high temperature expansions for classical spin models abstract it is shown that the transitivity of the symmetry group of a classical spin model leads to a high temperature expansion of the free energy of such a model in terms of multiply connected graphs for models with a completely permissible symmetry group e g for ising and potts models these graphs fall into equivalence classes such classes are classified according to their cyclomatic index several different methods for obtaining the high temperature series in a practical way are also discussed the extension of these results to models with an infinite symmetry group is indicated

differential porosimetry and permeametry for random porous media accurate determination of geometrical and physical properties of natural porous materials is notoriously difficult continuum multiscale modeling has provided carefully calibrated realistic microstructure models of reservoir rocks with floating point accuracy previous measurements using synthetic microcomputed tomography ct were based on extrapolation of resolution dependent properties for discrete digitized approximations of the continuum microstructure this paper reports continuum measurements of volume and specific surface with full floating point precision it also corrects an incomplete description of rotations in earlier publications more importantly the methods of differential permeametry and differential porosimetry are introduced as precision tools the continuum microstructure chosen to exemplify the methods is a homogeneous carefully calibrated and characterized model for fontainebleau sandstone the sample has been publicly available since 2010 on the worldwide web as a benchmark for methodical studies of correlated random media high precision porosimetry gives the volume and internal surface area of the sample with floating point accuracy continuum results with floating point precision are compared to discrete approximations differential porosities and differential surface area densities allow geometrical fluctuations to be discriminated from discretization effects and numerical noise differential porosimetry and fourier analysis reveal subtle periodic correlations the findings uncover small oscillatory correlations with a period of roughly 850 m thus implying that the sample is not strictly stationary the correlations are attributed to the deposition algorithm that was used to ensure the grain overlap constraint differential permeabilities are introduced and studied differential porosities and permeabilities provide scale dependent information on geometry fluctuations thereby allowing quantitative error estimates

a new multiscale finite element method for high contrast elliptic interface problems we introduce a new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes and without resolving the interfaces a typical application would be the modelling of flow in a porous medium containing a number of inclusions of low or high permeability embedded in a matrix of high respectively low permeability our method is h 1 conforming with degrees of freedom at the nodes of a triangular mesh and requiring the solution of subgrid problems for the basis functions on elements which straddle the coefficient interface but which use standard linear approximation otherwise a key point is the introduction of novel coefficientdependent boundary conditions for the subgrid problems under moderate assumptions we prove that our methods have optimal convergence rate of o h in the energy norm and o h 2 in the l 2 norm where h is the coarse mesh diameter and the hidden constants in these estimates are independent of the contrast i e ratio of largest to smallest value of the pde coefficient for standard elements the best estimate in the energy norm would be o h 1 2 e with a hidden constant which in general depends on the contrast the new interior boundary conditions depend not only on the contrast of the coefficients but also on the angles of intersection of the interface with the element edges

algebraic languages and polyominoes enumeration abstract in this paper the use of algebraic languages theory in solving an open problem in combinatorics is shown by constructing a bijection between convex polyominoes and words of an algebraic language and by solving the corresponding algebraic system we prove that the number of convex polyominoes with perimeter 2n 8 is 2n 11 4 n 4 2n 1 2n n

suboptimal decoding of linear codes partition technique general symmetric channels are introduced and near maximum likelihood decoding in these channels is studied first we define a class of suboptimal decoding algorithms based on an incomplete search through the code trellis it is proved that the decoding error probability of suboptimal decoding is bounded above for any q ary code of length n and code rate r by twice the error probability of its maximum likelihood decoding and tends to the latter as n grows second we design a suboptimal trellis like algorithm which reduces the known decoding complexity of the order of q sup n min r 1 r operations to that of q sup nr i r operations for all cyclic codes and virtually all long linear codes we also consider the corresponding bounds for concatenated codes an important corollary is that this suboptimal decoding can provide complexity below the lower bounds on trellis complexity at a negligible expense in terms of decoding error probability

a multiscale restriction smoothed basis method for high contrast porous media represented on unstructured grids a wide variety of multiscale methods have been proposed in the literature to reduce runtime and provide better scaling for the solution of poisson type equations modeling flow in porous media we present a new multiscale restricted smoothed basis msrsb method that is designed to be applicable to both rectilinear grids and unstructured grids like many other multiscale methods msrsb relies on a coarse partition of the underlying fine grid and a set of local prolongation operators multiscale basis functions that map unknowns associated with the fine grid cells to unknowns associated with blocks in the coarse partition these mappings are constructed by restricted smoothing starting from a constant a localized iterative scheme is applied directly to the fine scale discretization to compute prolongation operators that are consistent with the local properties of the differential operators the resulting method has three main advantages first of all both the coarse and the fine grid can have general polyhedral geometry and unstructured topology this means that partitions and good prolongation operators can easily be constructed for complex models involving high media contrasts and unstructured cell connections introduced by faults pinch outs erosion local grid refinement etc in particular the coarse partition can be adapted to geological or flow field properties represented on cells or faces to improve accuracy secondly the method is accurate and robust when compared to existing multiscale methods and does not need expensive recomputation of local basis functions to account for transient behavior dynamic mobility changes are incorporated by continuing to iterate a few extra steps on existing basis functions this way the cost of updating the prolongation operators becomes proportional to the amount of change in fluid mobility and one reduces the need for expensive tolerance based updates finally since the msrsb method is formulated on top of a cell centered conservative finite volume method it is applicable to any flow model in which one can isolate a pressure equation herein we only discuss single and two phase incompressible models compressible flow e g as modeled by the black oil equations is discussed in a separate paper

multi scale finite volume method for elliptic problems in subsurface flow simulation in this paper we present a multi scale finite volume msfv method to solve elliptic problems with many spatial scales arising from flow in porous media the method efficiently captures the effects of small scales on a coarse grid is conservative and treats tensor permeabilities correctly the underlying idea is to construct transmissibilities that capture the local properties of the differential operator this leads to a multi point discretization scheme for the finite volume solution algorithm transmissibilities for the msfv have to be constructed only once as a preprocessing step and can be computed locally therefore this step is perfectly suited for massively parallel computers furthermore a conservative fine scale velocity field can be constructed from the coarse scale pressure solution two sets of locally computed basis functions are employed the first set of basis functions captures the small scale heterogeneity of the underlying permeability field and it is computed in order to construct the effective coarse scale transmissibilities a second set of basis functions is required to construct a conservative fine scale velocity field the accuracy and efficiency of our method is demonstrated by various numerical experiments

a family of multiscale hybrid mixed finite element methods for the darcy equation with rough coefficients abstract we aim at proposing novel stable finite element methods for the mixed darcy equation with heterogeneous coefficients within a space splitting framework we start from the primal hybrid formulation of the elliptic model for the pressure localization of this infinite dimensional problem leads to element level boundary value problems which embed multiscale and high contrast features in a natural way with neumann boundary conditions driven by the lagrange multipliers such a procedure leads to methods involving the space of piecewise constants for the pressure together with a discretization of the fluxes choosing arbitrarily polynomial interpolations the lowest order raviart thomas element as well as some recent multiscale methods are recovered in addition the methods assure local mass conservation and can be interpreted as stabilized primal hybrid methods extensive numerical validation attests to the accuracy of the new methods on academic and more realistic problems with rough coefficients

local enhancement of functional evaluation and adjoint error estimation for variational multiscale formulations abstract we develop superconvergent postprocessing techniques and accurate a posteriori error estimators for variational multiscale formulations vms of partial differential equations we show that including vms fine scale information in the quantity of interest qoi approximation can provide a superior rate of convergence including exact recovery for a non trivial class of qois further improved postprocessing is achieved by reinterpreting the standard vms fine scale approximation via a local bubble function projection decomposing the error in this modified qoi evaluation into an element local component which depends only on the stabilization and a non local dual weighted residual component allows us to obtain accurate qoi error estimates this analysis also leads to a new adjoint based expression for the classic stabilization parameter the intrinsic time numerical experiments on challenging boundary layer problems demonstrate the superior convergence of the modified qoi evaluation and the reliability of the adjoint error estimator for various qois we also provide computational confirmation of exact recovery of an average value qoi using our estimator with the canonical hughes 1995 as predicted by our analysis

trellis complexity of short linear codes an extended table of shuurman s bounds on the state complexity of short binary linear codes is presented some new lower and upper bounds are obtained most of the newly found codes are based on the so called double zero tail termination dzt construction

a stochastic multiscale framework for modeling flow through random heterogeneous porous media flow through porous media is ubiquitous occurring from large geological scales down to the microscopic scales several critical engineering phenomena like contaminant spread nuclear waste disposal and oil recovery rely on accurate analysis and prediction of these multiscale phenomena such analysis is complicated by inherent uncertainties as well as the limited information available to characterize the system any realistic modeling of these transport phenomena has to resolve two key issues i the multi length scale variations in permeability that these systems exhibit and ii the inherently limited information available to quantify these property variations that necessitates posing these phenomena as stochastic processes a stochastic variational multiscale formulation is developed to incorporate uncertain multiscale features a stochastic analogue to a mixed multiscale finite element framework is used to formulate the physical stochastic multiscale process recent developments in linear and non linear model reduction techniques are used to convert the limited information available about the permeability variation into a viable stochastic input model an adaptive sparse grid collocation strategy is used to efficiently solve the resulting stochastic partial differential equations spdes the framework is applied to analyze flow through random heterogeneous media when only limited statistics about the permeability variation are given

the size and number of rings on the square lattice the authors have calculated the number of self avoiding polygons on the square lattice to 56 steps and the caliper size to 54 steps analysis of the generating function permits estimates of the connective constant mu 2 638 1585 or 10 6 and the critical exponents alpha 0 500 06 or 0 000 06 and v 0 753 or 0 007 the singularity structure of the polygon generating function is found to be consistent with a correction to the scaling exponent delta 1 5 as predicted by nienhuis 1982 1984 the confluent part however maps into the additive analytic term due to the value of the exponent alpha

iterative galerkin enriched multiscale finite volume method the multiscale finite volume msfv method was designed to efficiently compute approximate numerical solutions of elliptic and parabolic problems with highly heterogeneous coefficients for a wide range of test cases it has been demonstrated that msfv solutions are in excellent agreement with reference data obtained with a standard fine scale finite volume method however for some problems involving large coherent structures with strong contrasts e g channels and shale layers the msfv method produces unsatisfactory results due to the dwindling validity of the localization assumption recently the iterative msfv i msfv method was introduced to iteratively improve the localization assumption and it was shown that the method converges to the fine scale reference solution in this work it is explained how the convergence rate of the i msfv method can be enhanced by consistent enrichment of the initial coarse space spanned by nodal basis functions a new set of enrichment basis functions associated with additional coarse scale degrees of freedom dof is introduced by construction the sum of all nodal and enrichment basis functions is one in the entire computational domain further a hybrid finite volume galerkin formulation for the coarse scale problem and a generic placement strategy for the additional dof are devised the resulting iterative galerkin enriched msfv i ge msfv method has all features of the i msfv method but is more robust and has improved convergence properties

exact solution of the staircase and row convex polygon perimeter and area generating function an explicit expression is obtained for the perimeter and area generating function g y z sigma n or 2 sigma m or 1 cn mynzm where cn m is the number of row convex polygons with area m and perimeter n a similar expression is obtained for the area perimeter generating function for staircase polygons both expressions contain q series