net and prune a linear time algorithm for euclidean distance problems we provide a general framework for getting expected linear time constant factor approximations and in many cases fptas s to several well known problems in computational geometry such as k center clustering and farthest nearest neighbor the new approach is robust to variations in the input problem and yet it is simple elegant and practical in particular many of these well studied problems which fit easily into our framework either previously had no linear time approximation algorithm or required rather involved algorithms and analysis a short list of the problems we consider include farthest nearest neighbor k center clustering smallest disk enclosing k points kth largest distance kth smallest m nearest neighbor distance kth heaviest edge in the mst and other spanning forest type problems problems involving upward closed set systems and more finally we show how to extend our framework such that the linear running time bound holds with high probability

construction of wavelets and prewavelets over triangulations constructions of wavelets and prewavelets over triangulations with an emphasis of the continuous piecewise polynomial setting are discussed some recent results on piecewise linear prewavelets and orthogonal wavelets are presented

some multilevel methods on graded meshes we consider yserentant s hierarchical basis method and multilevel diagonal scaling method on a class of refined meshes used in the numerical approximation of boundary value problems on polygonal domains in the presence of singularities we show as in the uniform case that the stiffness matrix of the first method has a condition number bounded by ln 1 h 2 where h is the meshsize of the triangulation for the second method we show that the condition number of the iteration operator is bounded by ln 1 h which is worse than in the uniform case but better than the hierarchical basis method as usual we deduce that the condition number of the bpx iteration operator is bounded by ln 1 h finally graded meshes fulfilling the general conditions are presented and numerical tests are given which confirm the theoretical bounds

locally supported rational spline wavelets on a sphere in this paper we construct certain continuous piecewise rational wavelets on arbitrary spherical triangulations giving explicit expressions of these wavelets our wavelets have small support a fact which is very important in working with large amounts of data since the algorithms for decomposition compression and reconstruction deal with sparse matrices we also give a quasi interpolant associated to a given triangulation and study the approximation error some numerical examples are given to illustrate the efficiency of our wavelets

swapping edges of arbitrary triangulations to achieve the optimal order of approximation in the representation of scattered data by smooth pp piecewise polynomial functions perhaps the most important problem is to find an optimal triangulation of the given sample sites called vertices of course the notion of optimality depends on the desirable properties in the approximation or modeling problems in this paper we are concerned with optimal approximation order with respect to the given order r of smoothness and degree k of the polynomial pieces of the smooth pp functions we will only consider c1 pp approximation with r 1 and k 4 the main result in this paper is an efficient method for triangulating any finitely many arbitrarily scattered sample sites such that these sample sites are the only vertices of the triangulation and that for any discrete data given at these sample sites there is a c1 piecewise quartic polynomial on this triangulation that interpolates the given data with the fifth order of approximation

finite element wavelets with improved quantitative properties in w dahmen r stevenson element by element construction of wavelets satisfying stability and moment conditions siam j numer anal 37 1 1999 319 352 electronic finite element wavelets were constructed on polygonal domains or lipschitz manifolds that are piecewise parametrized by mappings with constant jacobian determinants the wavelets could be arranged to have any desired order of cancellation properties and they generated stable bases for the sobolev spaces h s for s 32 or s 1 on manifolds unfortunately it appears that the quantitative properties of these wavelets are rather disappointing in this paper we modify the construction from the above mentioned work to obtain finite element wavelets which are much better conditioned

incomplete iterations in multistep backward difference methods for parabolic problems with smooth and nonsmooth data backward difference methods for the discretization of parabolic boundary value problems are considered in this paper in particular we analyze the case when the backward difference equations are only solved approximately by a preconditioned iteration we provide an analysis which shows that these methods remain stable and accurate if a suitable number of iterations often independent of the spatial discretization and time step size are used results are provided for the smooth as well as nonsmooth initial data cases finally the results of numerical experiments illustrating the algorithms performance on model problems are given

linear independence and stability of piecewise linear prewavelets on arbitrary triangulations in this paper we establish linear independence and stability of certain piecewise linear prewavelets over arbitrary bounded triangulations these prewavelets are natural generalizations of the locally supported element constructed by kotyczka and oswald for an infinite three directional mesh

uniqueness and estimation of three dimensional motion parameters of rigid objects with curved surfaces two main results are established in this paper first we show that seven point correspondences are sufficient to uniquely determine from two perspective views the three dimensional motion parameters within a scale factor for the translations of a rigid object with curved surfaces the seven points should not be traversed by two planes with one plane containing the origin nor by a cone containing the origin second a set of essential parameters are introduced which uniquely determine the motion parameters up to a scale factor for the translations and can be estimated by solving a set of linear equations which are derived from the correspondences of eight image points the actual motion parameters can subsequently be determined by computing the singular value decomposition svd of a 3 3 matrix containing the essential parameters

c 1 spline wavelets on triangulations in this paper we investigate spline wavelets on general triangulations in particular we are interested in c 1 wavelets generated from piecewise quadratic polynomials by using the powell sabin elements we set up a nested family of spaces of c 1 quadratic splines which are suitable for multiresolution analysis of besov spaces consequently we construct c 1 wavelet bases on general triangulations and give explicit expressions for the wavelets on the three direction mesh a general theory is developed so as to verify the global stability of these wavelets in besov spaces the wavelet bases constructed in this paper will be useful for numerical solutions of partial differential equations

filter bank algorithms for piecewise linear prewavelets on arbitrary triangulations abstract this paper studies algorithms for decomposition reconstruction and approximation based on piecewise linear prewavelets on bounded triangulations of arbitrary topology our key mathematical result is showing that the schur complement of the associated two scale matrix is symmetric positive definite and well conditioned numerical examples suggest that thresholding based on prewavelets yields a smaller approximation error than when based on the simple faber decomposition scheme

nonlinear galerkin methods the finite elements case with the increase in the computing power and the advent of supercomputers the approximation of evolution equations on large intervals of time is emerging as a new type of numerical problem in this article we consider the approximation of evolution equations on large intervals of time when the space discretization is accomplished by finite elements the algorithm that we propose called the nonlinear galerkin method stems from the theory of dynamical systems and amounts to some approximation of the attractor in the discrete finite elements space essential here is the utilization of incremental unknown which is accomplished in finite elements by using hierarchical bases beside a detailed description of the algorithm the article includes some technical results on finite elements spaces and a full study of the stability and convergence of the method

a class of iterative methods for solving saddle point problems we consider the numerical solution of indefinite systems of linear equations arising in the calculation of saddle points we are mainly concerned with sparse systems of this type resulting from certain discretizations of partial differential equations we present an iterative method involving two levels of iteration similar in some respects to the uzawa algorithm we relate the rates of convergence of the outer and inner iterations proving that under natural hypotheses the outer iteration achieves the rate of convergence of the inner iteration the technique is applied to finite element approximations of the stokes equations

element by element construction of wavelets satisfying stability and moment conditions in this paper we construct a class of locally supported wavelet bases for c0 lagrange finite element spaces on possibly nonuniform meshes on n dimensional domains or manifolds the wavelet bases are stable in the sobolev spaces mathcal h s for s frac 3 2 s leq 1 on lipschitz manifolds and the wavelets can in principle be arranged to have any desired order of vanishing moments as a consequence these bases can be used e g for constructing an optimal solver of discretized mathcal h s elliptic problems for s in above ranges the construction of the wavelets consists of two parts an implicit part involves some computations on a reference element which for each type of finite element space have to be performed only once in addition there is an explicit part which takes care of the necessary adaptations of the wavelets to the actual mesh the only condition we need for this construction to work is that the refinements of initial elements are uniform we will show that the wavelet bases can be implemented efficiently

nonlinear galerkin method in the finite difference case and wavelet like incremental unknowns the img algorithm inertial manifold multigrid algorithm which uses the first order incremental unknowns was introduced in 20 the img algorithm is aimed at numerically implementing inertial manifolds see e g 19 when finite difference discretizations are used for that purpose it is necessary to decompose the unknown function into its long wavelength and its short wavelength components first order incremental unknowns iu were proposed in 20 as a means to realize this decomposition our aim in the present article is to propose and study other forms of incremental unknowns in particular the wavelet like incremental unknowns wiu so called because of their oscillatory nature in this report we first extend the general convergence results in 20 by proving them under slightly weaker conditions we then present three sets of incremental unknowns i e the first order as in 20 the second order and wavelet like incremental unknowns we show that these incremental unknown can be used to construct convergent img algorithms special stress is put on the wavelet like incremental unknowns since this set of unknowns has thel 2 orthogonality property between different levels of unknowns and this should make them particularly appropriate for the approximation of evolution equations by inertial algorithms

approximate halfspace range counting we present a simple scheme extending the shallow partitioning data structures of matousek which supports efficient approximate halfspace range counting queries in mathbb r d with relative error varepsilon specifically the problem is given a set p of n points in mathbb r d to preprocess them into a data structure that returns for a query halfspace h a number t so that 1 varepsilon h cap p leq t leq 1 varepsilon h cap p one of our data structures requires linear storage and o n 1 delta preprocessing time for any delta 0 and answers a query in time o varepsilon gamma n 1 1 lfloor d 2 rfloor 2 b log ast n for any gamma 2 lfloor d 2 rfloor the choice of gamma and delta affects b and the implied constants several variants and extensions are also discussed as presented the construction of the structure is mostly deterministic except for one critical randomized step and so are the query storage and preprocessing costs the quality of approximation for every query is guaranteed with high probability the construction can also be fully derandomized at the expense of increasing preprocessing time

multiresolution analysis over triangles based on quadratic hermite interpolation given a triangulation t of r 2 a recipe to build a spline space s t over this triangulation and a recipe to rene the triangulation t into a triangulation t 0 the question arises whether s t s t 0 i e whether any spline surface over the original triangulation t can also be represented as a spline surface over the rened triangulation t 0 in this paper we will discuss how to construct such a nested sequence of spaces based on powell sabin 6 splits for a regular triangulation the resulting spline space consists of piecewise c 1 quadratics and renement is obtained by subdividing every triangle into four subtriangles at the edge midpoints we develop explicit formulas for wavelet transformations based on quadratic hermite interpolation and give a stability result with respect to a natural norm c 2000 elsevier science b v all rights reserved msc 41a63 65d07 65d17 65t60

localization of elliptic multiscale problems this paper constructs a local generalized finite element basis for elliptic problems with heterogeneous and highly varying coefficients the basis functions are solutions of local problems on vertex patches the error of the corresponding generalized finite element method decays exponentially with respect to the number of layers of elements in the patches hence on a uniform mesh of size h patches of diameter h log 1 h are sufficient to preserve a linear rate of convergence in h without pre asymptotic or resonance effects the analysis does not rely on regularity of the solution or scale separation in the coefficient this result motivates new and justifies old classes of variational multiscale methods see more at http www ams org journals mcom 2014 83 290 s0025 5718 2014 02868 8 sthash z2ccfxig dpuf

piecewise linear prewavelets on arbitrary triangulations this paper studies locally supported piecewise linear prewavelets on bounded triangulations of arbitrary topology it is shown that a concrete choice of prewavelets form a basis of the wavelet space when the degree of the vertices in the triangulation is not too high

matching 3 d anatomical surfaces with non rigid deformations using octree splines presents a new method for determining the minimal nonrigid deformation between two 3d surfaces such as those which describe anatomical structures in 3d medical images although the authors match surfaces they represent the deformation as a volumetric transformation their method performs a least squares minimization of the distance between the two surfaces of interest to quickly and accurately compute distances between points on the two surfaces the authors use a precomputed distance map represented using an octree spline whose resolution increases near the surface to quickly and robustly compute the deformation the authors use a second octree spline to model the deformation function the coarsest level of the deformation encodes the global e g affine transformation between the two surfaces while finer levels encode smooth local displacements which bring the two surfaces into closer registration the authors present experimental results on both synthetic and real 3d surfaces

an adaptive wavelet viscosity method for systems of hyperbolic conservation laws for the burgers equation as an example of a hyperbolic conservation law we have considered in our previous paper castano diez et al 2008 8 a weak formulation with a stabilization for handling discontinuities commonly called a viscosity approach numerically this was realized by locally introducing degrees of freedom around the discontinuities by means of an adaptive wavelet method in an a posteriori fashion in the present paper we apply this method to systems of conservation laws specifically euler s equations for gas dynamics moreover as the viscosity stabilization produces some gibbs phenomena we discuss different post processing techniques known from data and image processing together with a number of numerical comparisons

on approximating the depth and related problems we study the question of finding a deepest point in an arrangement of regions and provide a fast algorithm for this problem using random sampling showing it sufficient to solve this problem when the deepest point is shallow this implies among other results a fast algorithm for approximately solving linear programming problems with violations we also use this technique to approximate the disk covering the largest number of red points while avoiding all the blue points given two such sets in the plane using similar techniques implies that approximate range counting queries have roughly the same time and space complexity as emptiness range queries

a robust hierarchical basis preconditioner on general meshes in this paper we introduce a multi level direct sum space decomposition of general possibly locally refined linear or multi linear finite element spaces the resulting additive schwarz preconditioner is optimal for symmetric second order elliptic problems moreover it turns out to be robust with respect to coefficient jumps over edges in the coarsest mesh perturbations with positive zeroth order terms and after a further decomposition of the spaces also with respect to anisotropy along the grid lines important for an efficient implementation is that stable bases of the subspaces defining our decomposition consisting of functions having small supports can be easily constructed

semialgebraic range reporting and emptiness searching with applications in a typical range emptiness searching resp reporting problem we are given a set p of n points in mathbb r d and we wish to preprocess it into a data structure that supports efficient range emptiness resp reporting queries in which we specify a range sigma which in general is a semialgebraic set in mathbb r d of constant description complexity and we wish to determine whether p cap sigma emptyset or to report all the points in p cap sigma range emptiness searching and reporting arise in many applications and have been treated by matousek comput geom theory appl 2 1992 pp 169 186 in the special case where the ranges are half spaces bounded by hyperplanes as shown in matousek s work the two problems are closely related and they have solutions for the case of half spaces with similar performance bounds in this paper we extend the analysis to general semialgebraic ranges and show how to adapt matousek s technique without the need to linearize the ranges into a higher dimensional space this yields more efficient solutions to several useful problems and we demonstrate the new technique in four applications with the following results i an algorithm for ray shooting amid balls in mathbb r 3 which uses o n storage and o n preprocessing we use the notation o n gamma to mean an upper bound of the form c varepsilon n gamma varepsilon which holds for any varepsilon 0 where c varepsilon is a constant that depends on varepsilon and answers a query in o n 2 3 time improving the previous bound of o n 3 4 ii an algorithm that preprocesses in o n time a set p of n points in mathbb r 3 into a data structure with o n storage so that for any query line ell or for that matter any simply shaped convex set the point of p farthest from ell can be computed in o n 1 2 time this in turn yields an algorithm that computes the largest area triangle spanned by p in time o n 26 11 as well as nontrivial algorithms for computing the largest perimeter or largest height triangle spanned by p iii an algorithm that preprocesses in o n time a set p of n points in mathbb r 2 into a data structure with o n storage so that for any query alpha fat triangle delta we can determine in o 1 time whether delta cap p is empty alternatively we can report in o 1 o k time the points of delta cap p where k delta cap p iv an algorithm that preprocesses in o n time a set p of n points in mathbb r 2 into a data structure with o n storage so that given any query semidisk c or a circular cap larger than a semidisk we can determine in o 1 time whether c cap p is empty or report the k points in c cap p in o 1 o k time adapting the recent techniques of b aronov and s har peled siam j comput 38 2008 pp 899 921 b aronov s har peled and m sharir on approximate halfspace range counting and relative epsilon approximations in proceedings of the 23rd acm symposium comput geom 2007 pp 327 336 b aronov and m sharir siam j comput 39 2010 pp 2704 2725 we can turn our solutions into efficient algorithms for approximate range counting with small relative error for the cases mentioned above our technique is closely related to the notions of nearest or farthest neighbor generalized voronoi diagrams and of the union or intersection of geometric objects where sharper bounds on the combinatorial complexity of decompositions of complements of these structures yield faster range emptiness searching or reporting algorithms

recovery of the three dimensional shape of an object from a single view given a single picture which is a projection of a three dimensional scene onto the two dimensional picture plane we usually have definite ideas about the 3 d shapes of objects to do this we need to use assumptions about the world and the image formation process since there exist a large number of shapes which can produce the same picture the purpose of this paper is to identify some of these assumptions mostly geometrical ones by demonstrating how the theory and techniques which exploit such assumptions can provide a systematic shape recovery method the method consists of two parts the first is the application of the origami theory which models the world as a collection of plane surfaces and recovers the possible shapes qualitatively the second is the technique of mapping image regularities into shape constraints for recovering the probable shapes quantitatively actual shape recovery from a single view is demonstrated for the scenes of an object such as a box and a chair given a single image the method recovers the 3 d shapes of an object in it and generates images of the same object as we would see it from other directions

interpretation of straight line correspondences using angular relations abstract formal analysis of the structure from motion problem has produced computational schemes that are often unstable and sensitive to image measurement errors this is in part due to the fact that the formal analysis is mainly concerned as it should be with the most general cases treating special cases with marginal interest however special cases abound in man made environments their occurrence if ascertained simplifies drastically the task of 3d interpretation after examining the problem of interpreting general configurations of lines in space this paper proposes a number of simple and efficient computational units that are tuned to the occurrence of special line configurations the process of propagation which allows interpretation to spread over the image is also advocated such computational units and processes can be organized in a rule based framework where the occurrence of special configurations can either be tested directly or hypothesized and used until an inconsistency is demonstrated

algebraic multilevel preconditioning methods ii a recursive way of constructing preconditioning matrices for the stiffness matrix in the discretization of selfadjoint second order elliptic boundary value problems is proposed it is based on a sequence of nested finite element spaces with the usual nodal basis functions using a nodeordering corresponding to the nested meshes the finite element stiffness matrix is recursively split up into two level block structures and is factored approximately in such a way that any successive schur complement is replaced approximated by a matrix defined recursively and thereform only implicitely given to solve a system with this matrix we need to perform a fixed number v of iterations on the preceding level using as an iteration matrix the preconditioning matrix already defined on that level it is shown that by a proper choice of iteration parameters it suffices to use v left 1 gamma 2 right tfrac 1 2 iterations for the so constructedv foldv cycle wherev 2 corresponds to aw cycle preconditioning matrices to be spectrally equivalent to the stiffness matrix the conditions involve only the constant in the strengthened c b s inequality for the corresponding two level hierarchical basis function spaces and are therefore independent of the regularity of the solution for instance if we use successive uniform refinements of the meshes the method is of optimal order of computational complexity if gamma 2 tfrac 8 9 under reasonable assumptions of the finite element mesh the condition numbers turn out to be so small that there are in practice few reasons to use an accelerated iterative method like the conjugate gradient method for instance