composite geometric concepts and polynomial predictability abstract we study the predictability of geometric concepts in particular those defined by boolean combinations of simple geometric objects first we give a negative results showing that the problem of predicting the class of convex polytopes encoded by listing their vertices is prediction complete for p thus an efficient solution to this prediction problem implies the existence of efficient solutions to all prediction problems whose associated evaluation problems are in p assuming the existence of a one way function that is hard on iterates there are such prediction problems which do not admit efficient solutions thus we show under minimal cryptographic assumptions that the class of convex polytopes encoded by listing their vertices is not predictable as a side effect we show that determining membership in the convex hull of a given set of points is complete for p with respect to log space reductions next we establish the predictability of the class consisting of unions of a fixed number of flats by reducing its prediction problem to that of the class of flats which has previously been shown to be predictable finally we give an occam algorithm for predicting fixed finite unions of boxes both constructive results for flats and boxes hold if the dimension is variable

least squares mixed finite element methods for non selfadjoint elliptic problems i error estimates a least squares mixed finite element method for general second order non selfadjoint elliptic problems in two and three dimensional domains is formulated and analyzed the finite element spaces for the primary solution approximation u h and the flux approximation sigma h consist of piecewise polynomials of degree k and r respectively the method is mildly nonconforming on the boundary the cases k r and k 1 r are studied it is proved that the method is not subject to the lbb condition optimal l 2 and h 1 error estimates are derived for regular finite element partitions numerical experiments confirming the theoretical rates of convergence are presented

least squares solutions as solutions of a perturbation form of the galerkin methods interior pointwise error estimates and pollution effect first order div least squares ls finite element methods for second order elliptic partial differential equations are considered the main idea of this paper is a consideration of the ls solution u h for the primary function u as the solution of a perturbation form of the galerkin methods consequently the results concerning the perturbation form of the galerkin methods can be applied for the ls methods here we use the pointwise error estimates for the perturbation form provided by schatz a h schatz pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part ii interior estimates siam j numer anal 38 2000 1269 1293 to obtain the pointwise estimates for the ls solution the estimates consist of the local truncation error and pollution terms we provide estimates for pollution terms our process for arriving at the estimates involves obtaining the norm of a linear functional which accounts for the perturbation our estimates are valid for both smooth and nonsmooth problems for the smooth problems the pollution is weaker than the local truncation error for the nonsmooth problems we provide a criterion for determining the strength of the pollution in particular we identify superconvergent points for nonconvex polygonal domains

noise tolerant distribution free learning of general geometric concepts we present an efficient algorithm for pac learning a very general class of geometric concepts over r d for fixed d more specifically let t be any set of s halfspaces let x x 1 x d be an arbitrary point in r d with each t t we associate a boolean indicator function i t x which is 1 if and only if x is in the halfspace t the concept class c d s that we study consists of all concepts formed by any boolean function over i t1 i ts for t i t this class is much more general than any geometric concept class known to be pac learnable our results can be extended easily to learn efficiently any boolean combination of a polynomial number of concepts selected from any concept class c over r given that the vc dimension of c has dependence only on d and there is a polynomial time algorithm to determine if there is a concept from c consistent with a given set of labeled examples we also present a statistical query version of our algorithm that can tolerate random classification noise finally we present a generalization of the standard e net result of haussler and welzl 1987 and apply it to give an alternative noise tolerant algorithm for d 2 based on geometric subdivisions

the l 2 norm error estimates for the div least squares method this paper studies l 2 norm error estimates for the div least squares method for which the associated homogeneous least squares functional is equivalent to the h mbox div times h 1 norm for the respective dual and primal variables least squares of this type for the second order elliptic equations elasticity and the stokes equations are an active area of research and error estimates in the h mbox div times h 1 norm were previously established in this paper we establish optimal l 2 norm error estimates for the primal variable under the minimum regularity requirement through a refined duality argument

local error estimates for least squares finite element methods for first order system we present local energy type error estimates for first order system div least squares ls finite element methods the estimate shows that the local energy norm error is bounded by the local best approximation and weaker norms which account for the pollution the estimate is similar to the one for the standard galerkin methods however our estimate needs to consider the effect of error of dual flux variables since ls methods approximate the primary and dual variables simultaneously the effect of error of the dual variables is shown to be of higher order moreover our estimate shows the convergence behavior when locally enriched approximation spaces are used in the area of interest as an elementary consequence of the estimate asymptotically exact a posteriori error estimator is constructed for the local area of interest under mild assumptions

goal oriented local a posteriori error estimators for h div least squares finite element methods we propose a goal oriented local a posteriori error estimator for h div least squares ls finite element methods our main interest is to develop an a posteriori error estimator for the flux approximation in a preassigned region of interest d subset omega the estimator is obtained from the ls functional by scaling residuals with proper weight coefficients the weight coefficients are given in terms of local mesh size h t and a function omega d depending on the distance to d this new error estimator measures the pollution effect from the outside region of d and provides a basis for local refinement in order to efficiently approximate the solution in d numerical experiments show superior performances of our goal oriented a posteriori estimators over the standard ls functional and global error estimators

pointwise error estimates for first order div least squares finite element methods and applications to superconvergence and a posteriori error estimators first order div least squares finite element methods for second order elliptic partial differential equations are considered while there has been significant progress in error estimates for the methods to the best of our knowledge these estimates are based on the global norm of the error in this paper we provide highly localized pointwise error estimates for the primary variable u by establishing that the least squares solutions are locally higher order perturbations of the standard galerkin solutions as elementary consequences we identify the superconvergent points the set of superconvergent points for the primary function of the least squares solution is the same as that of the standard galerkin solution also we present a class of average type a posteriori error estimators for the method and conditions are given under which they are asymptotically exact or equivalent estimators in the maximum norm on each single element of the underlying mesh

multidimensional backward stochastic riccati equations and applications backward stochastic riccati differential equations bsrdes for short arise from the solution of general linear quadratic optimal stochastic control problems with random coefficients the existence and uniqueness question of the global adapted solutions has been open since bismut s pioneering research publication in 1978 seminaire de probabilites xii lecture notes in math 649 c dellacherie p a meyer and m weil eds springer verlag berlin 1978 pp 180 264 one distinguishing difficulty lies in the quadratic nonlinearity of the drift term in the second unknown component in a previous article stochastic process appl 97 2002 pp 255 288 the authors solved the one dimensional case driven by brownian motions in this paper the multidimensional case driven by brownian motions is studied a closeness property for solutions of bsrdes with respect to their coefficients is stated and is proved for general bsrdes which is used to obtain the existence of a global adapted solution to some bsrdes the global existence and uniqueness results are obtained for two classes of bsrdes whose generators contain a quadratic term of l the second unknown component more specifically the two classes of bsrdes are for the regular case n 0 email protected email protected email protected dk displaystyle a k ka q ld n d kd 1 d l dt l dw k t m end array right under the condition d 1 and for the singular case left begin array email protected email protected email protected dk displaystyle a k ka c kc q c l lc displaystyle kb c kd ld d kd 1 kb c kd ld dt displaystyle l dw k t m end array right under the condition d 1 and m n the arguments given in this paper are completely new and they consist of some simple techniques of algebraic matrix transformations and direct applications of the closeness property mentioned above we make full use of the special structure the nonnegativity of the quadratic term for example of the underlying riccati equation applications in optimal stochastic control are exposed

asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes part 1 a smooth problem and globally quasi uniform meshes a class of a posteriori estimators is studied for the error in the maximum norm of the gradient on single elements when the finite element method is used to approximate solutions of second order elliptic problems the meshes are unstructured and in particular it is not assumed that there are any known superconvergent points the estimators are based on averaging operators which are approximate gradients recovered gradients which are then compared to the actual gradient of the approximation on each element conditions are given under which they are asympotically exact or equivalent estimators on each single element of the underlying meshes asymptotic exactness is accomplished by letting the approximate gradient operator average over domains that are large in a controlled fashion to be detailed below compared to the size of the elements

a posteriori error estimators for the first order least squares finite element method in this paper we propose a posteriori error estimators for certain quantities of interest for a first order least squares finite element method in particular we propose an a posteriori error estimator for when one is interested in email protected s email protected 0 where email protected u our a posteriori error estimators are obtained by assigning proper weight in terms of local mesh size h t to the terms of the least squares functional an a posteriori error analysis yields reliable and efficient estimates based on residuals numerical examples are presented to show the effectivity of our error estimators

a random sampling based algorithm for learning intersections of halfspaces we give an algorithm to learn an intersection of k halfspaces in rn whose normals span an l dimensional subspace for any input distribution with a logconcave density such that the bounding hyperplanes of the k halfspaces pass through its mean the algorithm e learns with time and sample complexity bounded by nkl e o l log 1 e the hypothesis found is an intersection of o k log 1 e halfspaces this improves on blum and kannan s algorithm for the uniform distribution over a ball in the time and sample complexity previously doubly exponential and in the generality of the input distribution

asymptotically exact a posteriori error estimators for first order div least squares methods in local and global l2 norm a new asymptotically exact a posteriori error estimator is developed for first order div least squares ls finite element methods let uh h be ls approximate solution for u au then e a1 2 h auh 0 is asymptotically exact a posteriori error estimator for a1 2 uuh 0 or a1 2 h 0 depending on the order of approximate spaces for and u for e to be asymptotically exact for a1 2 uuh 0 we require higher order approximation property for and vice versa when both au and are approximated in the same order of accuracy the estimator becomes an equivalent error estimator for both errors the underlying mesh is only required to be shape regular i e it does not require quasi uniform mesh nor any special structure for the underlying meshes confirming numerical results are provided and the performance of the estimator is explored for other choice of spaces for uh h

local energy estimates for the finite element method on sharply varying grids local energy error estimates for the nite element method for el liptic problems were originally proved in 1974 by nitsche and schatz these estimates show that the local energy error may be bounded by a local ap proximation term plus a global pollution term that measures the inuence of solution quality from outside the domain of interest and is heuristically of higher order however the original analysis of nitsche and schatz is restricted to quasi uniform grids we present local a priori energy estimates that are valid on shape regular grids an assumption which allows for highly graded meshes and which much more closely matches the typical practical situation our chief technical innovation is an improved superapproximation result

pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part ii interior estimates this part contains new interior pointwise error estimates for the finite element method for second order elliptic problems in mathbb r n global estimates were considered in part i in the sense to be discussed below these sharpen known interior quasi optimal l infty and w 1 infty estimates in that they indicate a more local dependence of the error at a point on the derivatives of the solution near the point the higher the order of the finite element the more local the behavior of the finite element approximation as a consequence of these estimates new types of local error expansions will be derived which are in the form of inequalities these expansion inequalities are valid for large classes of finite elements defined on irregular grids in mathbb r n and have applications to superconvergence extrapolation and a posteriori estimates for both smooth and nonsmooth problems

first order system least squares for second order partial differential equations part i this paper develops a least squares functional that arises from recasting general second order uniformly elliptic partial differential equations in n 2 or 3 dimensions as a system of first order equations in part i z cai r d lazarov t manteuffel and s mccormick siam j numer anal 31 1994 pp 1785 1799 a similar functional was developed and shown to be elliptic in the h divv times h 1 norm and to yield optimal convergence for finite element subspaces of h divv times h 1 in this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first order system the resulting functional is proved to be elliptic in the h 1 n 1 norm this immediately implies optimal error estimates for finite element approximation by standard subspaces of h 1 n 1 another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent as an alternative to perturbation based approaches the least squares approach developed here applies directly to convection diffusion reaction equations in a unified way and also admits a fast multigrid solver historically a missing ingredient in least squares methodology

least squares mixed finite elements for second order elliptic problems a theoretical analysis of a least squares mixed finite element method for second order elliptic problems in two and three dimensional domains is presented it is proved that the method is not subj

optimal error estimate for the div least squares method with data f inl 2 and application to nonlinear problems the div least squares methods have been studied by many researchers for the second order elliptic equations elasticity and the stokes equations and optimal error estimates have been obtained in the h mathrm div times h 1 norm however there is no known convergence rate when the given data f belongs only to l 2 space in this paper we will establish an optimal error estimate in the l 2 times h 1 norm with the given data f in l 2 and hence fill a theoretical gap of least squares methods as a consequence of this estimate we will provide a convergence analysis for the linearization process on solving navier stokes equations which uses the div least squares method for solving the corresponding stokes equations

sharp l_2 norm error estimates for first order div least squares methods a theoretical analysis of a first order div least squares finite element method is presented our main interest is providing l 2 norm error estimates for the primary function u and the flux mbox boldmath mathcal a nabla u while there has been significant progress in the error estimates for the flux in l 2 norm these estimates have drawbacks such as requiring smooth solutions or that the error estimate for mbox boldmath mbox boldmath h 0 is coupled with the error estimate for u u h 1 where u h mbox boldmath h is the least squares approximate solution for u mbox boldmath mathcal a nabla u in this paper with the minimum regularity assumption u in h 1 alpha alpha 1 2 we separate the error estimate for mbox boldmath mbox boldmath h 0 from u u h 1 and establish that mbox boldmath mbox boldmath h 0 is bounded by the best approximation for mbox boldmath in the finite element space and u u h 0 then we proceed to obtain two new estimates for u u h 0 the first estimate is useful to understand how the accuracy of u u h 0 affects the accuracy of mbox boldmath mbox boldmath h 0 and the other sheds new light on the convergence behavior of u u h 0 our analysis does not require smooth solutions and the domain is allowed to be nonconvex the resulting error estimates are valid for both smooth and low regularity solutions and show that optimal rates of convergence can be achieved by choosing proper finite element spaces such as the raviart thomas spaces for the flux mbox boldmath