discontinuous galerkin methods for friedrichs systems i general theory this paper presents a unified analysis of discontinuous galerkin methods to approximate friedrichs systems an abstract set of conditions is identified at the continuous level to guarantee existence and uniqueness of the solution in a subspace of the graph of the differential operator then a general discontinuous galerkin method that weakly enforces boundary conditions and mildly penalizes interface jumps is proposed all the design constraints of the method are fully stated and an abstract error analysis in the spirit of strang s second lemma is presented finally the method is formulated locally using element fluxes and links with other formulations are discussed details are given for three examples namely advection reaction equations advection diffusion reaction equations and the maxwell equations in the so called elliptic regime

topological properties of concept spaces full version based on the observation that the category of concept spaces with the positive information topology is equivalent to the category of countably based t 0 topological spaces we investigate further connections between the learning in the limit model of inductive inference and topology in particular we show that the texts or positive presentations of concepts in inductive inference can be viewed as special cases of the admissible representations of computable analysis we also show that several structural properties of concept spaces have well known topological equivalents in addition to topological methods we use algebraic closure operators to analyze the structure of concept spaces and we show the connection between these two approaches the goal of this paper is not only to introduce new perspectives to learning theorists but also to present the field of inductive inference in a way more accessible to domain theorists and topologists

fundamental properties of infinite trees infinite trees naturally arise in the formalization and the study of the semantics of programming languages this paper investigates some of their combinatorial and algebraic properties that are especially relevant to semantics

a posteriori error estimators for the upstream weighting mixed methods for convection diffusion problems abstract in this paper we design a posteriori error estimators for the mixed finite element approximation of the convection diffusion problems with dominant convection which exhibit solutions with internal layers and boundary layers a discontinuous upstream weighting scheme in association with the mixed method is used for the convection term the a posteriori error estimator yields global upper and local lower bounds on the error measured in the l2 norm we present several numerical results to show the performance of the error estimator we find that our estimator is not only reliable and efficient but also robust for various numerical examples

newton method for reactive solute transport with equilibrium sorption in porous media we present a mass conservative numerical scheme for reactive solute transport in porous media the transport is modeled by a convection diffusion reaction equation including equilibrium sorption the scheme is based on the mixed finite element method mfem more precisely the lowest order raviart thomas elements and one step euler implicit the underlying fluid flow is described by the richards equation a possibly degenerate parabolic equation which is also discretized by mfem this work is a continuation of radu et al 2008 and radu et al 2009 1 2 where the algorithmic aspects of the scheme and the analysis of the discretization method are presented respectively here we consider the newton method for solving the fully discrete nonlinear systems arising on each time step after discretization the convergence of the scheme is analyzed in the case when the solute undergoes equilibrium sorption of freundlich type the problem becomes degenerate and a regularization step is necessary we derive sufficient conditions for the quadratic convergence of the newton scheme

ockham s razor empirical complexity and truth finding efficiency the nature of empirical simplicity and its relationship to scientific truth are long standing puzzles in this paper empirical simplicity is explicated in terms of empirical effects which are defined in terms of the structure of the inference problem addressed problem instances are classified according to the number of empirical effects they present simple answers are satisfied by simple worlds an efficient solution achieves the optimum worst case cost over each complexity class with respect to such costs as the number of retractions or errors prior to convergence and elapsed time to convergence it is shown that always choosing the simplest theory compatible with experience and hanging on to it while it remains the simplest is both necessary and sufficient for efficiency

large time asymptotics in contaminant transport in porous media in this paper we derive large time solutions of the partial differential equations modelling contaminant transport in porous media for initial data with bounded support while the main emphasis is on two space dimensions for the sake of completeness we give a brief summary of the corresponding results for one space dimension the philosophy behind the paper is to compare the results of a formal asymptotic analysis of the governing equations as t to infty with numerical solutions of the complete initial value problem the analytic results are obtained using the method of dominant balance which identifies the dominant terms in the model equations determining the behaviour of the solution in the large time limit these are found in terms of time scaled space similarity variables and the procedure results in a reduction of the number of independent variables in the original partial differential equation this generates what we call a reduced equation the solution of which depends crucially on the value

iterative learning of simple external contextual languages it is investigated for which choice of a parameter q denoting the number of contexts the class of simple external contextual languages is iteratively learnable on the one hand the class admits for all values of q polynomial time learnability provided an adequate choice of the hypothesis space is given on the other hand additional constraints like consistency and conservativeness or the use of a one one hypothesis space changes the picture iterative learning limits the long term memory of the learner to the current hypothesis and these constraints further hinder storage of information via padding of this hypothesis it is shown that if q 3 then simple external contextual languages are not iteratively learnable using a class preserving one one hypothesis space while for q 1 it is iteratively learnable even in polynomial time it is also investigated for which choice of the parameters the simple external contextual languages can be learnt by a consistent and conservative iterative learner

classes with easily learnable subclasses in this paper we study the question of whether identifiable classes have subclasses which are identifiable under a more restrictive criterion the chosen framework is inductive inference in particular the criterion of explanatory learning ex of recursive functions as introduced by gold inform comput 10 1967 447 among the more restrictive criteria is finite learning where the learner outputs on every function to be learned exactly one hypothesis which has to be correct the topic of the present paper are the natural variants a and b below of the classical question whether a given learning criterion like finite learning is more restrictive than ex learning a does every infinite ex identifiable class have an infinite finitely identifiable subclass b if an infinite ex identifiable class s has an infinite finitely identifiable subclass does it necessarily follow that some appropriate learner ex identifies s as well as finitely identifies an infinite subclass of s these questions are also treated in the context of ordinal mind change bounds

a finite volume method on general meshes for a degenerate parabolic convection reaction diffusion equation we propose a finite volume method on general meshes for the discretization of a degenerate parabolic convection reaction diffusion equation equations of this type arise in many contexts such as for example the modeling of contaminant transport in porous media the diffusion term which can be anisotropic and heterogeneous is discretized using a recently developed hybrid mimetic mixed framework we construct a family of discretizations for the convection term which uses the hybrid interface unknowns we consider a wide range of unstructured possibly nonmatching polyhedral meshes in arbitrary space dimension the scheme is fully implicit in time it is locally conservative and robust with respect to the peclet number we obtain a convergence result based upon a priori estimates and the frechet kolmogorov compactness theorem we implement the scheme both in two and three space dimensions and compare the numerical results obtained with the upwind and the centered discretizations of the convection term numerically

analysis of an upwind mixed finite element method for nonlinear contaminant transport equations in this paper the numerical approximation of a nonlinear diffusion equation arising in contaminant transport is studied the equation is characterized by advection diffusion and adsorption assuming the adsorption term is modeled by a freundlich isotherm it can be nonlinear in concentration and nondifferentiable as the concentration approaches zero we consider the approximation of this equation by a method which upwinds the advection and incorporates diffusion using a mixed finite element method error estimates for a semidiscrete formulation are derived and numerical results for a fully discrete formulation are given

inductive inference of approximations in this paper we investigate inductive inference identification criteria which permit infinitely many errors in explanations but which require that the density of these errors be no more than a certain prespectified amount we introduce three hierarchies of such criteria each of which has the same order type as the real unit interval these three hierarchies are progressively more strict in the way they measure density of errors of explanations the strictest of the three turns out to have all of its members save one incomparable to the identification criterion which permits finitely many errors in explanations

finite element approximation of the transport of reactive solutes in porous media part 1 error estimates for nonequilibrium adsorption processes in this paper we analyze a fully practical piecewise linear finite element approximation involving numerical integration backward euler time discretization and possibly regularization of the following degenerate parabolic system arising in a model of reactive solute transport in porous media find u x t v x t such that begin eqnarray partial t u partial t v delta u f quad mbox in omega times 0 t qquad u 0 quad mbox on partial omega times 0 t partial tv k varphi u v quad mbox in omega times 0 t u cdot 0 g 1 cdot quad v cdot 0 g 2 cdot quad mbox in omega subset real d quad 1 le d le 3 end eqnarray for given data k in real f g 1 g 2 and a monotonically increasing varphi in c 0 real cap c 1 infty 0 cup 0 infty satisfying varphi 0 0 which is only locally holder continuous with exponent p in 0 1 at the origin e g varphi s equiv s p this lack of lipschitz continuity at the origin limits the regularity of the unique solution u v and leads to difficulties in the finite element error analysis

godunov mixed methods for advection diffusion equations in multidimensions time split methods for multidimensional advection diffusion equations are considered in these methods advection is approximated by a godunov type procedure and diffusion is approximated by a low

characteristic galerkin methods for contaminant transport with nonequilibrium adsorption kinetics a procedure based on combining the method of characteristics with a galerkin finite element method is analyzed for approximating reactive transport in groundwater in particular equations modeling

a combined finite volume nonconforming finite element scheme for compressible two phase flow in porous media we propose and analyze a combined finite volume nonconforming finite element scheme on general meshes to simulate the two compressible phase flow in porous media the diffusion term which can be anisotropic and heterogeneous is discretized by piecewise linear nonconforming triangular finite elements the other terms are discretized by means of a cell centered finite volume scheme on a dual mesh where the dual volumes are constructed around the sides of the original mesh the relative permeability of each phase is decentred according the sign of the velocity at the dual interface this technique also ensures the validity of the discrete maximum principle for the saturation under a non restrictive shape regularity of the space mesh and the positiveness of all transmissibilities next a priori estimates on the pressures and a function of the saturation that denote capillary terms are established these stabilities results lead to some compactness arguments based on the use of the kolmogorov compactness theorem and allow us to derive the convergence of a subsequence of the sequence of approximate solutions to a weak solution of the continuous equations provided the mesh size tends to zero the proof is given for the complete system when the density of the each phase depends on its own pressure

on the role of procrastination in machine learning the traditional model of inductive inference is enhanced to allow learning machines to procrastinate about how many trials they will need to complete an inference or about how accurate their solution will be hierarchies of classes of learnable phenomena represented by sets of recursive functions isomorphic to the structure of the constructive ordinals are revealed the existence of such hierarchies indicates a potential advantage of procrastination as a learning technique tradeoffs between the two types of procrastination are shown not to exist in general one of the new classes of sets of inferrible functions introduced in this paper turns out to have the somewhat unusual property of being closed under finite unions a new technique based on a hardest to lear set relative to a class is used

a priori and a posteriori error analysis of a staggered discontinuous galerkin method for convection dominant diffusion equations abstract an upwind staggered discontinuous galerkin upwind sdg method for convection dominant diffusion problems is developed optimal a priori error estimates can be achieved for both the scalar and vector functions approximated by the method to efficiently capture the layer problems we propose a robust a posteriori error estimator for upwind sdg method measured in a natural norm and a semi norm associated with the convective derivative the semi norm can be neglected when the mesh peclet number is sufficiently small the key is to bound the conforming contribution by the dual norm of the residual and the robustness arises from the incorporation of the semi norm associated with the convective derivative finally various numerical examples are tested to illustrate the performance of the robust a posteriori error estimator the results indicate that optimal convergence rates can be achieved and the singularity can be well captured by the proposed error estimator

on the inference of approximate programs abstract a more powerful criterion of successful inductive inference is studied this criterion considers the inference successful when the result is a program which computes the function to be inferred everywhere except on a sparse but infinite subset of its domain in previous work sparse was taken to mean finite relationships with previously defined criteria are examined royer 19 considered the same issues and independently obtained some of our results

a combined finite volume nonconforming mixed hybrid finite element scheme for degenerate parabolic problems we propose and analyze a numerical scheme for nonlinear degenerate parabolic convection diffusion reaction equations in two or three space dimensions we discretize the diffusion term which generally involves an inhomogeneous and anisotropic diffusion tensor over an unstructured simplicial mesh of the space domain by means of the piecewise linear nonconforming crouzeix raviart finite element method or using the stiffness matrix of the hybridization of the lowest order raviart thomas mixed finite element method the other terms are discretized by means of a cell centered finite volume scheme on a dual mesh where the dual volumes are constructed around the sides of the original mesh checking the local peclet number we set up the exact necessary amount of upstream weighting to avoid spurious oscillations in the convection dominated case this technique also ensures the validity of the discrete maximum principle under some conditions on the mesh and the diffusion tensor we prove the convergence of the scheme only supposing the shape regularity condition for the original mesh we use a priori estimates and the kolmogorov relative compactness theorem for this purpose the proposed scheme is robust only 5 point 7 point in space dimension three locally conservative efficient and stable which is confirmed by numerical experiments

solution of nonlinear convection diffusion problems by a conservative galerkin characteristics method we consider a scheme for nonlinear degenerate convection dominant diffusion problems that arise in contaminant transport in porous media with equilibrium adsorption isotherm this scheme is based on a regularization relaxation scheme that has been introduced by jager and kacur numer math 60 407 427 1991 m2an math model numer anal 29 n5 605 627 1995 with a type of numerical integration by bermejo siam j numer anal 32 425 455 1995 to the modified method of characteristics with adjusted advection mmocaa that was recently developed by douglas et al numer math 83 3 353 369 1999 comput geosci 1 155 190 1997 we present another variant of adjusting advection method the convergence of the scheme is proved an error estimate of the approximated scheme is derived computational experiments are carried out to illustrate the capability of the scheme to conserve the mass

infinite labeled trees from rational to sturmian trees this paper studies infinite unordered d ary trees with nodes labeled by 0 1 we introduce the notions of rational and sturmian trees along with the definitions of strongly balanced trees and mechanical trees and study the relations among them in particular we show that strongly balanced trees exist and coincide with mechanical trees in the irrational case providing an effective construction such trees also have a minimal factor complexity hence are sturmian we also give several examples illustrating the inclusion relations between these classes of trees

comparison of identification criteria for machine inductive inference abstract a natural pl 1 hierarchy of successively more general criteria of success for inductive inference machines is described based on the size of sets of anomalies in programs synthesized by such machines these criteria are compared to others in the literature some of our results are interpreted as tradeoff results or as showing the inherent relative computational complexity of certain processes and others are interpreted from a positivistic mechanistic philosophical stance as theorems in philosophy of science the techniques of recursive function theory are employed including ordinary and infinitary recursion theorems

tree shift topological entropy abstract we give a definition of topological entropy for tree shifts prove that the limit in the definition exists and show that it dominates the topological entropy of the associated one dimensional shift of finite type when the labeling of the tree shares the same restrictions

a posteriori error estimates for lowest order mixed finite element discretizations of convection diffusion reaction equations we establish residual a posteriori error estimates for lowest order raviart thomas mixed finite element discretizations of convection diffusion reaction equations on simplicial meshes in two or three space dimensions the upwind mixed scheme is considered as well and the emphasis is put on the presence of an inhomogeneous and anisotropic diffusion dispersion tensor and on a possible convection dominance global upper bounds for the approximation error in the energy norm are derived where in particular all constants are evaluated explicitly so that the estimators are fully computable our estimators give local lower bounds for the error as well and they hold from the cases where convection or reaction are not present to convection or reaction dominated problems we prove that their local efficiency depends only on local variations in the coefficients and on the local peclet number moreover the developed general framework allows for asymptotic exactness and full robustness with respect to inhomogeneities and anisotropies the main idea of the proof is a construction of a locally postprocessed approximate solution using the mean value and the flux in each element known in the mixed finite element method and a subsequent use of the abstract framework arising from the primal weak formulation of the continuous problem numerical experiments confirm the guaranteed upper bound and excellent efficiency and robustness of the derived estimators

solute transport in porous media with equilibrium and non equilibrium multiple site adsorption travelling waves we study travelling wave solutions of a model describing solute transport through porous media where the chemical species undergoes absorption i e a retention release reaction with the surface of the porous sceleton processes of this type are of fundamental importance in different fields we do not consider the process at the microscopic scale of single grains and pores but at the microscopic scale appropriate for the measurement of the phenomena described above

extremal splittings of point processes the sequence with nth term defined by n 1 p np is an extremal zero one valued sequence of asymptotic mean p in the following sense for example if a fraction p of customers from a point process with iid interarrival times is sent to an exponential server queue according to a prespecified splitting sequence then the long term average queue size is minimized when the above sequence is used the proof involves consideration of the lower convex envelope j which is a function on rm of a function j on zm an explicit representation is given for j in terms of j for j in a broad class of functions which we call multimodular the expected queue size just before an arrival considered as a function of the zero one splitting sequence is shown to belong to this class

on the complexity of inductive inference the notion of the complexity of performing an inductive inference is defined some examples of the tradeoffs between the complexity of performing an inference and the accuracy of the inferred result are presented an axiomatization of the notion of the complexity of inductive inference is developed and several results are presented which both resemble and contrast with results obtainable for the axiomatic computational complexity of recursive functions

convergence analysis of mixed numerical schemes for reactive flow in a porous medium this paper deals with the numerical analysis of an upscaled model describing the reactive flow in a porous medium the solutes are transported by advection and diffusion and undergo precipitation and dissolution the reaction term and in particular the dissolution term have a particular multivalued character which leads to stiff dissolution fronts we consider the euler implicit method for the temporal discretization and the mixed finite element for the discretization in space more precisely we use the lowest order raviart thomas elements as an intermediate step we consider also a semidiscrete mixed variational formulation continuous in space we analyze the numerical schemes and prove the convergence to the continuous formulation apart from the proof for the convergence this also yields an existence proof for the solution of the model in mixed variational formulation numerical experiments are performed to study the convergence behavior

generalized notions of mind change complexity gold introduced the notion of learning in the limit where a class s is learnable iff there is a recursive machine m which reads the course of values of a function f and converges to a program for f whenever f is in s an important measure for the speed of convergence in this model is the quantity of mind changes before the onset of convergence the oldest model is to consider a constant bound on the number of mind changes m makes on any input function such a bound is referred here as type 1 later this was generalized to a bound of type 2 where a counter ranges over constructive ordinals and is counted down at every mind change although ordinal bounds permit the inference of richer concept classes than constant bounds they still are a severe restriction therefore the present work introduces two more general approaches to bounding mind changes these are based on counting by going down in a linearly ordered set type 3 and on counting by going down in a partially ordered set type 4 in both cases the set must not contain infinite descending recursive sequences these four types of mind changes yield a hierarchy and there are identifiable classes that cannot be learned with the most general mind change bound of type 4 it is shown that existence of type 2 bound is equivalent to the existence of a learning algorithm which converges on every also nonrecursive input function and the existence of type 4 is shown to be equivalent to the existence of a learning algorithm which converges on every recursive function a partial characterization of type 3 yields a result of independent interest in recursion theory the interplay between mind change complexity and choice of hypothesis space is investigated it is established that for certain concept classes a more expressive hypothesis space can sometimes reduce mind change complexity of learning these classes the notion of mind change bound for behaviourally correct learning is indirectly addressed by employing the above four types to restrict the number of predictive errors of commission in finite error next value learning nv a model equivalent to behaviourally correct learning again natural characterizations for type 2 and type 4 bounds are derived their naturalness is further illustrated by characterizing them in terms of branches of uniformly recursive families of binary trees

mind change complexity of inferring unbounded unions of restricted pattern languages from positive data this paper shows that the mind change complexity of inferring from positive data the class of unbounded unions of languages of regular patterns with constant segment length bound is of the form w w a b assuming that the patterns are defined over a finite alphabet containing at least two elements here a and b are natural numbers and we give tight bounds on their values based on the length of the constant segments and the size of the alphabet of the pattern languages this is to the authors knowledge the first time a natural class of languages has been shown to be inferable with mind change complexity above w w the proof uses the notion of closure operators on a class of languages and also uses the order type of well partial orderings to obtain a mind change bound the inference algorithm presented can be easily applied to a wide range of classes of languages finally we show an interesting connection between proof theory and mind change complexity

mind change complexity of learning logic programs the present paper motivates the study of mind change complexity for learning minimal models of length bounded logic programs it establishes ordinal mind change complexity bounds for learnability of these classes both from positive facts and from positive and negative facts building on angluin s notion of finite thickness and wright s work on finite elasticity shinohara defined the property of bounded finite thickness to give a sufficient condition for learnability of indexed families of computable languages from positive data this paper shows that an effective version of shinohara s notion of bounded finite thickness gives sufficient conditions for learnability with ordinal mind change bound both in the context of learnability from positive data and for learnability from complete both positive and negative data let be a notation for the first limit ordinal then it is shown that if a language defining framework yields a uniformly decidable family of languages and has effective bounded finite thickness then for each natural number m 0 the class of languages defined by formal systems of length m is identifiable in the limit from positive data with a mind change bound of m is identifiable in the limit from both positive and negative data with an ordinal mind change bound of m the above sufficient conditions are employed to give an ordinal mind change bound for learnability of minimal models of various classes of length bounded prolog programs including shapiro s linear programs arimura and shinohara s depth bounded linearly covering programs and krishna rao s depth bounded linearly moded programs it is also noted that the bound for learning from positive data is tight for the example classes considered

learning by switching type of information the present work is dedicated to the study of modes of data presentation in the range between text and informant within the framework of inductive inference in this study the learner alternatingly requests sequences of positive and negative data we define various formalizations of valid data presentations in such a scenario we resolve the relationships between these different formalizations and show that one of these is equivalent to learning from informant we also show a hierarchy formed for each of the formalizations studied by considering the number of switches between requests for positive and negative data

on a generalized notion of mistake bounds this paper proposes the use of constructive ordinals as mistake bounds in the on line learning model this approach elegantly generalizes the applicability of the on line mistake bound model to learnability analysis of very expressive concept classes like pattern languages unions of pattern languages elementary formal systems and minimal models of logic programs the main result in the paper shows that the topological property of effective finite bounded thickness is a sufficient condition for on line learnability with a certain ordinal mistake bound an interesting characterization of the on line learning model is shown in terms of the identification in the limit framework it is established that the classes of languages learnable in the on line model with a mistake bound of are exactly the same as the classes of languages learnable in the limit from both positive and negative data by a popperian consistent learner with a mind change bound of this result nicely builds a bridge between the two models

godunov mixed methods for advective flow problems in one space dimension a time splitting method for solving advection dominated parabolic partial differential equations is presented in this method a higher order godunov procedure approximates advection and a mixed finite element procedure approximates diffusion several variations on the basic scheme are formulated for solving one dimensional quasilinear parabolic problems with dirichlet boundary conditions a maximum principle for one variant of the scheme is demonstrated and l infty l 2 and l 2 l 2 error estimates for the approximate solution and the diffusive flux respectively are derived these estimates indicate that one variant of the scheme is l infty stable in certain situations but possibly suboptimal in error while another variant is optimal and l 2 stable

on the intrinsic complexity of learning recursive functions the intrinsic complexity of learning compares the difficulty of learning classes of objects by using some reducibility notion for several types of learning recursive functions both natural complete classes are exhibited and necessary and sufficient conditions for completeness are derived informally a class is complete iff both its topological structure is highly complex while its algorithmic structure is easy some self describing classes turn out to be complete furthermore the structure of the intrinsic complexity is shown to be much richer than the structure of the mind change complexity though in general intrinsic complexity and mind change complexity can behave orthogonally

mind change efficient learning this paper studies efficient learning with respect to mind changes our starting point is the idea that a learner that is efficient with respect to mind changes minimizes mind changes not only globally in the entire learning problem but also locally in subproblems after receiving some evidence formalizing this idea leads to the notion of strong mind change optimality we characterize the structure of language classes that can be identified with at most mind changes by some learner not necessarily effective a language class l is identifiable with mind changes iff the accumulation order of l is at most accumulation order is a classic concept from point set topology we show that accumulation order is related to other established notions of structural complexity such as thickness and intrinsic complexity to aid the construction of learning algorithms we show that the characteristic property of strongly mind change optimal learners is that they output conjectures languages with maximal accumulation order we illustrate the theory by describing strongly mind change optimal learners for various problems such as identifying linear subspaces one variable patterns and fixed length patterns

convergence analysis for a conformal discretization of a model for precipitation and dissolution in porous media in this paper we discuss the numerical analysis of an upscaled core scale model describing the transport precipitation and dissolution of solutes in a porous medium the particularity lies in the modeling of the reaction term especially the dissolution term which has a multivalued character we consider the weak formulation for the upscaled equation and provide rigorous stability and convergence results for both the semi discrete time discretization and the fully discrete schemes in doing so compactness arguments are employed