surface crouzeix raviart element for the laplace beltrami equation in this paper we are concerned with the nonconforming finite element discretization of geometric partial differential equations we construct a surface crouzeix raviart element on the linear approximated surface analogous to a flat surface the optimal convergence theory for the new nonconforming surface finite element method is developed even though the geometric error exists by taking an intrinsic viewpoint of manifolds we introduce a new superconvergent gradient recovery method for the surface crouzeix raviart element using only the information of discretized surface the potential of serving as an asymptotically exact a posteriori error estimator is also exploited a series of benchmark numerical examples are presented to validate the theoretical results and numerically demonstrate the superconvergence of the gradient recovery method

decompositions into codimension two manifolds let m denote an orientable n 2 manifold and let g denote an upper semicontinuous decomposition of m into continua having the shape of closed orientable n manifolds the main result establishes that the decomposition space m g is a 2 manifold

a controlled plus construction for crumpled laminations given a closed n manifold m n 4 and a finitely generated perfect subgroup p of it1 m we previously developed a controlled version of quillen s plus construction namely a cobordism w m n with the inclusion j n f w a homotopy equivalence and kernel of i i1 m f it1 w equalling the smallest normal subgroup of xl1 m containing p together with a closed map p w 0 1 such that p l t is a closed n manifold for every t e 0 1 and in particular m p 1 o and n p l l we accomplished this by constructing an acyclic map of manifolds f m f n having the right fundamental groups and w arose as the mapping cylinder of f with a collar attached along n the main result here presents a condition under which the desired controlled plus construction can still be accomplished in many cases even when it1 m contains no finitely generated perfect subgroups by products of these results include a new method for constructing wild embeddings of codimension one manifolds and a better understanding of perfect subgroups of finitely presented groups

geometrically consistent mesh modification a new paradigm of adaptivity is to execute refinement coarsening and smoothing of meshes on manifolds with incomplete information about their geometry and yet preserve position and curvature accuracy we refer to this collectively as geometrically consistent gc mesh modification we discuss the concept of discrete gc show the failure of naive approaches and propose and analyze a simple algorithm that is gc and accuracy preserving

higher order finite element methods and pointwise error estimates for elliptic problems on surfaces we define higher order analogues to the piecewise linear surface finite element method studied in g dziuk finite elements for the beltrami operator on arbitrary surfaces in partial differential equations and calculus of variations springer verlag berlin 1988 pp 142 155 and prove error estimates in both pointwise and l 2 based norms using the laplace beltrami problem on an implicitly defined surface gamma as a model pde we define lagrange finite element methods of arbitrary degree on polynomial approximations to gamma which likewise are of arbitrary degree then we prove a priori error estimates in the l 2 h 1 and corresponding pointwise norms that demonstrate the interaction between the pde error that arises from employing a finite dimensional finite element space and the geometric error that results from approximating gamma we also consider parametric finite element approximations that are defined on gamma and thus induce no geometric error computational examples confirm the sharpness of our error estimates

reliable and robust a posteriori error estimation for singularly perturbed reaction diffusion problems problems with singular perturbations exhibit solutions with strong boundary layers and other types of local behavior such features lend themselves to adaptive solution methods the quality of any adaptive algorithm ultimately rests on the reliability and robustness of the a posteriori error control an estimator that has proved to be one of the most reliable is the equilibrated residual method the main property of the estimator is that it bounds the true error from above however the method is not robust in the singularly perturbed limit the current work generalizes the error estimator based on the equilibrated residuals and coincides with the standard method in the unperturbed limit it is shown that the new method is robust in the singularly perturbed limit while maintaining reliability yielding a guaranteed upper bound on the true error finally the application of the estimator to the problem of controlling the spatial error in rothe s method for the time discretization of a simple parabolic problem is included

an adaptive strategy for elliptic problems including a posteriori controlled boundary approximation we derive a posteriori error estimates for the approximation of linear elliptic problems on domains with piecewise smooth boundary the numerical solution is assumed to be defined on a finite element mesh whose boundary vertices are located on the boundary of the continuous problem no assumption is made on a geometrically fitting shape a posteriori error estimates are given in the energy norm and the l 2 norm and efficiency of the adaptive algorithm is proved in the case of a saturated boundary approximation furthermore a strategy is presented to compute the effect of the non discretized part of the domain on the error starting from a coarse mesh this especially implies that parts of the domain where the measured error is small stay non discretized the presented algorithm includes a stable path following to supply a sufficient polygonal approximation of the boundary the reliable computation of the a posteriori estimates and a mesh adaptation based on delaunay techniques numerical examples illustrate that errors outside the initial discretization will be detected

reasoning about noisy sensors and effectors in the situation calculus abstract agents interacting with an incompletely known world need to be able to reason about the effects of their actions and to gain further information about that world they need to use sensors of some sort unfortunately both the effects of actions and the information returned from sensors are subject to error to cope with such uncertainties the agent can maintain probabilistic beliefs about the state of the world with probabilistic beliefs the agent will be able to quantify the likelihood of the various outcomes of its actions and is better able to utilize the information gathered from its error prone actions and sensors in this paper we present a model in which we can reason about an agent s probabilistic degrees of belief and the manner in which these beliefs change as various actions are executed we build on a general logical theory of action developed by reiter and others formalized in the situation calculus we propose a simple axiomatization that captures an agent s state of belief and the manner in which these beliefs change when actions are executed our model displays a number of intuitively reasonable properties

convergence of adaptive finite element methods for general second order linear elliptic pdes we prove convergence of adaptive finite element methods afems for general nonsymmetric second order linear elliptic pdes thereby extending the result of morin nochetto and siebert it siam j numer anal 38 2000 pp 466 488 it siam rev 44 2002 pp 631 658 the proof relies on quasi orthogonality which accounts for the bilinear form not being a scalar product together with novel error and oscillation reduction estimates which now do not decouple we show that afems are a contraction for the sum of energy error plus oscillation numerical experiments including oscillatory coefficients and both coercive and noncoercive convection diffusion pde illustrate the theory and yield optimal meshes

isogeometric analysis of high order partial differential equations on surfaces we consider the numerical approximation of high order partial differential equations pdes defined on surfaces in the three dimensional space with particular emphasis on closed surfaces we consider computational domains that can be represented by b splines or nurbs as for example the sphere and we spatially discretize the pdes by means of nurbs based isogeometric analysis in the framework of the standard galerkin method we numerically solve benchmark laplace beltrami problems of the fourth and sixth order as well as the corresponding eigenvalue problems with the goal of analyzing the role of the continuity of the nurbs basis functions on closed surfaces in this respect we show that the use of globally high order continuous basis functions as allowed by the construction of periodic nurbs leads to the efficient solution of the high order pdes finally we consider the numerical solution of high order phase field problems on closed surfaces namely the cahn hilliard and crystal equations

a priori anisotropic mesh adaptation on implicitly defined surfaces mesh adaptation on surfaces demands particular care due to the important role played by the surface fitting we propose an adaptive procedure based on a new error analysis which combines a rigorous anisotropic estimator for the l 1 norm of the interpolation error with an anisotropic heuristic control of the geometric error we resort to a metric based adaptive algorithm which employs local operations to modify the initial mesh according to the information provided by the error analysis an extensive numerical validation corroborates the robustness of the error analysis as well as of the adaptive procedure

an adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems an efficient and reliable a posteriori error estimate is derived for linear parabolic equations which does not depend on any regularity assumption on the underlying elliptic operator an adaptive algorithm with variable time step sizes and space meshes is proposed and studied which at each time step delays the mesh coarsening until the final iteration of the adaptive procedure allowing only mesh and time step size refinements before it is proved that at each time step the adaptive algorithm is able to reduce the error indicators and thus the error below any given tolerance within a finite number of iteration steps the key ingredient in the analysis is a new coarsening strategy numerical results are presented to show the competitive behavior of the proposed adaptive algorithm

parametric fem for geometric biomembranes we consider geometric biomembranes governed by an l 2 gradient flow for bending energy subject to area and volume constraints helfrich model we give a concise derivation of a novel vector formulation based on shape differential calculus and corresponding discretization via parametric fem using quadratic isoparametric elements and a semi implicit euler method we document the performance of the new parametric fem with a number of simulations leading to dumbbell red blood cell and toroidal equilibrium shapes while exhibiting large deformations

an adaptive finite element method for the laplace beltrami operator on implicitly defined surfaces we present an adaptive finite element method for approximating solutions to the laplace beltrami equation on surfaces in mathbb r 3 which may be implicitly represented as level sets of smooth functions residual type a posteriori error bounds which show that the error may be split into a residual part and a geometric part are established in addition implementation issues are discussed and several computational examples are given

isogeometric analysis for surface pdes with extended loop subdivision abstract we investigate the isogeometric analysis for surface pdes based on the extended loop subdivision approach the basis functions consisting of quartic box splines corresponding to each subdivided control mesh are utilized to represent the geometry exactly and construct the solution space for dependent variables as well which is consistent with the concept of isogeometric analysis the subdivision process is equivalent to the h refinement of nurbs based isogeometric analysis the performance of the proposed method is evaluated by solving various surface pdes such as surface laplace beltrami harmonic biharmonic triharmonic equations which are defined on the limit surfaces of extended loop subdivision for different initial control meshes numerical experiments show that the proposed method has desirable performance in terms of the accuracy convergence and computational cost for solving the above surface pdes defined on both open and closed surfaces the proposed approach is proved to be second order accuracy in the sense of l 2 norm with theoretical and or numerical results which is also outperformed over the standard linear finite element by several numerical comparisons

convergence of adaptive finite element methods adaptive finite element methods fems have been widely used in applications for over 20 years now in practice they converge starting from coarse grids although no mathematical theory has been able to prove this assertion ensuring an error reduction rate based on a posteriori error estimators together with a reduction rate of data oscillation information missed by the underlying averaging process we construct a simple and efficient adaptive fem for elliptic partial differential equations we prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance extensions to higher order elements and applications to saddle point problems are discussed as well

semantical considerations on nonmonotonic logic abstract commonsense reasoning is nonmonotonic in the sense that we often draw on the basis of partial information conclusions that we later retract when we are given more complete information some of the most interesting products of recent attempts to formalize nonmonotonic reasoning are the nonmonotonic logics of mcdermott and doyle 1 2 these logics however all have peculiarities that suggest they do not quite succeed in capturing the intuitions that prompted their development in this paper we reconstruct nonmonotonic logic as a model of an ideally rational agent s reasoning about his own beliefs for the resulting system called autoepistemic logic we define an intuitively based semantics for which we can show autoepistemic logic to be both sound and complete we then compare autoepistemic logic with the approach of mcdermott and doyle showing how it avoids the peculiarities of their nonmonotonic logic

knowledge action and the frame problem this paper proposes a method for handling the frame problem for knowledge producing actions an example of a knowledge producing action is a sensing operation performed by a robot to determine whether or not there is an object of a particular shape within its grasp the work is an extension of reiter s approach to the frame problem for ordinary actions and moore s work on knowledge and action the properties of our specification are that knowledge producing actions do not affect fluents other than the knowledge fluent and actions that are not knowledge producing only affect the knowledge fluent as appropriate in addition memory emerges as a side effect if something is known in a certain situation it remains known at successor situations unless something relevant has changed also it will be shown that a form of regression examined by reiter for reducing reasoning about future situations to reasoning about the initial situation now also applies to knowledge producing actions

bounded situation calculus action theories in this paper 1 we investigate bounded action theories in the situation calculus a bounded action theory is one which entails that in every situation the number of object tuples in the extension of fluents is bounded by a given constant although such extensions are in general different across the infinitely many situations we argue that such theories are common in applications either because facts do not persist indefinitely or because the agent eventually forgets some facts as new ones are learned we discuss various classes of bounded action theories then we show that verification of a powerful first order variant of the calculus is decidable for such theories notably this variant supports a controlled form of quantification across situations we also show that through verification we can actually check whether an arbitrary action theory maintains boundedness

adaptive discontinuous galerkin methods on surfaces we present a dual weighted residual based a posteriori error estimate for a discontinuous galerkin approximation of a surface partial differential equation we restrict our analysis to a linear second order elliptic problem posed on hypersurfaces in mathbb r 3 r3 which are implicitly represented as level sets of smooth functions we show that the error in the energy norm may be split into a residual part and a higher order geometric part upper and lower bounds for the resulting a posteriori error estimator are proven and we consider a number of challenging test problems to demonstrate the reliability and efficiency of the estimator we also present a novel geometric driven refinement strategy for pdes on surfaces which considerably improves the performance of the method on complex surfaces

quasi optimal convergence rate for an adaptive finite element method we analyze the simplest and most standard adaptive finite element method afem with any polynomial degree for general second order linear symmetric elliptic operators as is customary in practice the afem marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property we prove that the afem is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops this geometric decay is instrumental to derive the optimal cardinality of the afem we show that the afem yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity

convergence analysis of variational and non variational multigrid algorithms for the laplace beltrami operator we design and analyze variational and non variational multigrid algorithms for the laplace beltrami operator on a smooth and closed surface in both cases a uniform convergence for the v cycle algorithm is obtained provided the surface geometry is captured well enough by the coarsest grid the main argument hinges on a perturbation analysis from an auxiliary variational algorithm defined directly on the smooth surface in addition the vanishing mean value constraint is imposed on each level thereby avoiding singular quadratic forms without adding additional computational cost numerical results supporting our analysis are reported in particular the algorithms perform well even when applied to surfaces with a large aspect ratio

propositional knowledge base revision and minimal change abstract the semantics of revising knowledge bases represented by sets of propositional sentences is analyzed from a model theoretic point of view a characterization of all revision schemes that satisfy the gardenfors rationality postulates is given in terms of minimal change with respect to an ordering among interpretations revision methods proposed by various authors are surveyed and analyzed in this framework the correspondences between gardenfors like rationality postulates and minimal changes with respect to other orderings are also investigated

some a posteriori error estimators for elliptic partial differential equations we present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations the estimators are based on solving local neumann problems in each element the estimators differ in how they enforce consistency of the neumann problems we prove that as the mesh size decreases under suitable assumptions two of the error estimators approach upper bounds on the norm of the true error and all three error estimators are within multiplicative constants of the norm of the true error we present numerical results in which one of the error estimators appears to converge to the norm of the true error

iterated belief change in the situation calculus john mccarthy s situation calculus has left an enduring mark on artificial intelligence research this simple yet elegant formalism for modelling and reasoning about dynamic systems is still in common use more than forty years since it was first proposed the ability to reason about action and change has long been considered a necessary component for any intelligent system the situation calculus and its numerous extensions as well as the many competing proposals that it has inspired deal with this problem to some extent in this paper we offer a new approach to belief change associated with performing actions that addresses some of the shortcomings of these approaches in particular our approach is based on a well developed theory of action in the situation calculus extended to deal with belief moreover by augmenting this approach with a notion of plausibility over situations our account handles nested belief belief introspection mistaken belief and handles belief revision and belief update together with iterated belief change

on the logic of iterated belief revision abstract we show in this paper that the agm postulates are too weak to ensure the rational preservation of conditional beliefs during belief revision thus permitting improper responses to sequences of observations we remedy this weakness by proposing four additional postulates which are sound relative to a qualitative version of probabilistic conditioning contrary to the agm framework the proposed postulates characterize belief revision as a process which may depend on elements of an epistemic state that are not necessarily captured by a belief set we also show that a simple modification to the agm framework can allow belief revision to be a function of epistemic states we establish a model based representation theorem which characterizes the proposed postulates and constrains in turn the way in which entrenchment orderings may be transformed under iterated belief revision

guaranteed computable bounds for the exact error in the finite element solution part ii bounds for the energy norm of the error in two dimensions this paper addresses the computation of guaranteed upper and lower bounds for the energy norm of the exact error in the finite element solution these bounds are constructed in terms of the solutions of local residual problems with equilibrated residual loads and are rather sharp even for coarse meshes he sharpness of the bounds can be further improved by employing few iterations of a relatively inexpensive iterative scheme he main result is that the bounds are guaranteed for the nergy norm of the exact error unlike the bounds which ave been proposed in 13 14 which are guaranteed only for the nergy norm of the error with respect to an enriched truth esh finite element solution copyright 2000 john wiley sons ltd

some contributions to the metatheory of the situation calculus we focus on a rich axiomatization for actions in the situation calculus that includes among other features a solution to the frame problem for deterministic actions our work is foundational in nature directed at simplifying the entailment problem for these axioms specifically we make four contributions to the metatheory of situation calculus axiomatizations of dynamical systems 1 we prove that the above mentioned axiomatization for actions has a relative satisfiability property the full axiomatization is satisfiable iff the axioms for the initial state are 2 we define the concept of regression relative to these axioms and prove a soundness and completeness theorem for a regression based approach to the entailment problem for a wide class of queries 3 our formalization of the situation calculus requires certain foundational axioms specifying the domain of situations these include an induction axiom whose presence complicates human and automated reasoning in the situation calculus we characterize various classes of sentences whose proofs do not require induction and in some cases some of the other foundational axioms 4 we prove that the logic programming language golog never requires any of the foundational axioms for the evaluation of programs

data oscillation and convergence of adaptive fem data oscillation is intrinsic information missed by the averaging process associated with finite element methods fem regardless of quadrature ensuring a reduction rate of data oscillation together with an error reduction based on a posteriori error estimators we construct a simple and efficient adaptive fem for elliptic partial differential equations pdes with linear rate of convergence without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance

local problems on stars a posteriori error estimators convergence and performance a new computable a posteriori error estimator is introduced which relies on the solution of small discrete problems on stars it exhibits built in flux equilibration and is equivalent to the energy error up to data oscillation without any saturation assumption a simple adaptive strategy is designed which simultaneously reduces error and data oscillation and is shown to converge without mesh pre adaptation nor explicit knowledge of constants numerical experiments reveal a competitive performance show extremely good effectivity indices and yield quasi optimal meshes

afem for the laplace beltrami operator on graphs design and conditional contraction property we present an adaptive finite element method afem of any polynomial degree for the laplace beltrami operator on c 1 graphs in r d d 2 we first derive residual type a posteriori error estimates that account for the interaction of both the energy error in h 1 and the surface error in w 1 1 due to approximation of we devise a marking strategy to reduce the total error estimator namely a suitably scaled sum of the energy geometric and inconsistency error estimators we prove a conditional contraction property for the sum of the energy error and the total estimator the conditional statement encodes resolution of in w 1 1 we conclude with one numerical experiment that illustrates the theory

isogeometric analysis for second order partial differential equations on surfaces we consider the numerical solution of second order partial differential equations pdes on lower dimensional manifolds specifically on surfaces in three dimensional spaces for the spatial approximation we consider isogeometric analysis which facilitates the encapsulation of the exact geometrical description of the manifold in the analysis when this is represented by b splines or nurbs our analysis addresses linear nonlinear time dependent and eigenvalues problems involving the laplace beltrami operator on surfaces moreover we propose a priori error estimates under h refinement in the general case of second order pdes on the lower dimensional manifolds we highlight the accuracy and efficiency of isogeometric analysis with respect to the exactness of the geometrical representations of the surfaces

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

belief revision and projection in the epistemic situation calculus abstract this article considers defeasible beliefs in dynamic settings in particular we examine the belief projection problem what is believed after performing an action and or receiving new information the approach is based on an epistemic variant of reiter s situation calculus where actions not only have physical effects but may also provide new information to the agent the preferential belief structure is initially determined using conditional statements new information is then incorporated using two popular belief revision schemes namely natural and lexicographic revision the projection problem is solved twofold in this formalism by goal regression and by knowledge base progression

superconvergence and gradient recovery of linear finite elements for the laplace beltrami operator on general surfaces superconvergence results and several gradient recovery methods of finite element methods in flat spaces are generalized to the surface linear finite element method for the laplace beltrami equation on general surfaces with mildly structured triangular meshes for a large class of practically useful grids the surface linear finite element solution is proven to be superclose to an interpolant of the exact solution of the laplace beltrami equation and as a result various postprocessing gradient recovery including simple and weighted averaging local and global l 2 projections and zienkiewicz and zhu z z schemes are devised and proven to be a better approximation of the true gradient than the gradient of the finite element solution numerical experiments are presented to confirm the theoretical results

surface diffusion of graphs variational formulation error analysis and simulation surface diffusion is a fourth order highly nonlinear geometric driven motion of a surface with normal velocity proportional to the surface laplacian of mean curvature we present a novel variational formulation for graphs and derive a priori error estimates for a time continuous finite element discretization we also introduce a semi implicit time discretization and a schur complement approach to solve the resulting fully discrete linear systems after computational verification of the orders of convergence for polynomial degrees 1 and 2 we show several simulations in one dimension and two dimensions with and without forcing which explore the smoothing effect of surface diffusion as well as the onset of singularities in finite time such as infinite slopes and cracks