the primal dual active set strategy as a semismooth newton method this paper addresses complementarity problems motivated by constrained optimal control problems it is shown that the primal dual active set strategy which is known to be extremely efficient for this class of problems and a specific semismooth newton method lead to identical algorithms the notion of slant differentiability is recalled and it is argued that the max function is slantly differentiable in lp spaces when appropriately combined with a two norm concept this leads to new local convergence results of the primal dual active set strategy global unconditional convergence results are obtained by means of appropriate merit functions

a posteriori error control for distributed elliptic optimal control problems with control constraints discretized by hp finite elements abstract a distributed elliptic control problem with control constraints is considered which is formulated as a three field problem and consists of two variational equations for the state and the co state variables as well as of a variational inequality for the control variable two discretization approaches with h p finite elements are discussed in the first discretization approach all variables state co state and control are discretized a semi smooth newton method is introduced for solving the resulting algebraic system in the second discretization approach only the state and the co state variables are discretized whereas the control is determined by projection a simple fixed point scheme is presented for the iterative solution of this approach the main focus of the paper is on the derivation of reliable and efficient a posteriori error estimates which enables h p adaptive mesh refinements in particular the estimates can be applied to the iteration solutions so that they can be used as a stopping criterion of the iterative solution schemes in several numerical experiments the order of convergence of the adaptive discretization approaches and the efficiency as well as the reliability of the a posteriori error estimates are studied

interacting pairs of periodic solutions lead to tori in lasers subject to delayed feedback models of class b lasers subject to either an optoelectronic or an optical feedback are investigated analytically and numerically we derive slow time amplitude equations from the laser delay differential equations and find multiple bifurcating and isolated branches of periodic solutions we then show that secondary bifurcations to tori result from the interaction of pairs of hopf bifurcations the branches emerging from these bifurcations are followed numerically using a continuation method developed for delay differential equations

on h p adaptive bem for frictional contact problems in linear elasticity a mixed formulation for a tresca frictional contact problem in linear elasticity is considered in the context of boundary integral equations which is later extended to coulomb friction the discrete lagrange multiplier an approximation of the surface traction on the contact boundary part is a linear combination of biorthogonal basis functions in case of curved isoparametric elements these are the solutions of local problems in particular the biorthogonality allows to rewrite the variational inequality constraints as a simple set of complementarity problems thus enabling an efficient application of a semi smooth newton solver for the discrete mixed problem converging locally super linearly in the frictional case and quadratically in the frictionless case typically the solution of frictional contact problems is of reduced regularity at the interface between contact to non contact and from stick to slip to identify the a priori unknown locations of these interfaces two a posteriori error estimations are introduced in a first step the error is split into specific error contributions resulting from the contact and friction conditions and from the discretization error of a variational equation for the latter a residual and a bubble error estimation are considered explicitly the numerical experiments show the applicability of the derived error estimations and the superiority of h p adaptivity compared to low order uniform and adaptive approaches

computational survey on a posteriori error estimators for nonconforming finite element methods for the poisson problem this paper compares different a posteriori error estimators for nonconforming first order crouzeix raviart finite element methods for simple second order partial differential equations all suggested error estimators yield a guaranteed upper bound of the discrete energy error up to oscillation terms with explicit constants novel equilibration techniques and an improved interpolation operator for the design of conforming approximations of the discrete nonconforming finite element solution perform very well in an error estimator competition with six benchmark examples

comparison of mixed h p bem stabilized and non stabilized for frictional contact problems we present boundary integral equation procedures for contact problems with tresca or coulomb friction which are based on mixed formulations where besides the displacement also the traction on the contact boundary part appears as a lagrange multiplier this approach allows for an easy and efficient way to perform an h p be method by the use of biorthogonal basis functions this is especially suited for applying the semi smooth newton ssn method which is a very efficient solver superior to standard algorithm like uzawa with an adaptive algorithm we perform locally mesh refinements and increase of polynomial degrees for the be solution thus correctly representing the contact phenomena we also present as stabilized version of our mixed h p bem scheme with gauss lobatto lagrange basis which circumvents the discrete inf sup condition numerical results supporting our theory are reported

higher order fem for the obstacle problem of the p laplacian a variational inequality approach abstract we consider higher order finite element discretizations of a nonlinear variational inequality formulation arising from an obstacle problem with the p laplacian differential operator for p 1 we prove an a priori error estimate and convergence rates with respect to the mesh size h and in the polynomial degree q under assumed regularity moreover we derive a general a posteriori error estimate which is valid for any uniformly bounded sequence of finite element functions all our results contain the known results for the linear case of p 2 we present numerical results on the improved convergence rates of adaptive schemes mesh size adaptivity with and without polynomial degree adaptation for the singular case of p 1 5 and for the degenerated case of p 3

global and superlinear convergence of the smoothing newton method and its application to general box constrained variational inequalities the smoothing newton method for solving a system of nonsmooth equations f x 0 which may arise from the nonlinear complementarity problem the variational inequality problem or other problems can be regarded as a variant of the smoothing method at the kth step the nonsmooth function f is approximated by a smooth function f and the derivative of f at x k is used as the newton iterative matrix the merits of smoothing methods and smoothing newton methods are global convergence and convenience in handling in this paper we show that the smoothing newton method is also superlinearly convergent if f is semismooth at the solution and f satisfies a jacobian consistency property we show that most common smooth functions such as the gabriel more function have this property as an application we show that for box constrained variational inequalities if the involved function is p uniform the iteration sequence generated by the smoothing newton method will converge to the unique solution of the problem globally and superlinearly quadratically

recursive probability density estimation for weakly dependent stationary processes recursive estimation of the univariate probability density function f x for stationary processes x j is considered quadratic mean convergence and asymptotic normality for density estimators f n x are established for strong mixing and for asymptotically uncorrelated processes x j recent results for nonrecursive density estimators are extended to the recursive case

an efficient method for computing invariant manifolds of planar maps abstract an efficient method for computing invariant manifolds of mappings in the plane is described and compared to a simple method in common use the emphasis is on numerically producing smoothly resolved segments of invariant manifolds of arbitrary length while requiring a minimum number of calls to the map itself a brief estimate of errors is given and applications are discussed qualitative and quantitative comparisons between the two methods are made using an example problem the present method is applied to computing unstable manifolds for the well know periodically forced duffing system and the henon map for which the manifold is a strange attractor

a primal dual active set algorithm for three dimensional contact problems with coulomb friction in this paper efficient algorithms for contact problems with tresca and coulomb friction in three dimensions are presented and analyzed the numerical approximation is based on mortar methods for nonconforming meshes with dual lagrange multipliers using a nonsmooth complementarity function for the three dimensional friction conditions a primal dual active set algorithm is derived the method determines active contact and friction nodes and at the same time resolves the additional nonlinearity originating from sliding nodes no regularization and no penalization are applied and superlinear convergence can be observed locally in combination with a multigrid method it defines a robust and fast strategy for contact problems with tresca or coulomb friction the efficiency and flexibility of the method is illustrated by several numerical examples

polynomial interpolation results in sobolev spaces abstract in this paper we derive new results related to the lagrange polynomial interpolation on the nodes of gauss type quadrature formulas in particular we obtain asymptotic error estimates of the same order as the distance to the best fit in the polynomial space

polynomial degree robust a posteriori estimates in a unified setting for conforming nonconforming discontinuous galerkin and mixed discretizations we present equilibrated flux a posteriori error estimates in a unified setting for conforming nonconforming discontinuous galerkin and mixed finite element discretizations of the two dimensional poisson problem relying on the equilibration by the mixed finite element solution of patchwise neumann problems the estimates are guaranteed locally computable locally efficient and robust with respect to polynomial degree maximal local overestimation is guaranteed as well numerical experiments suggest asymptotic exactness for the incomplete interior penalty discontinuous galerkin scheme

a note on the design of hp adaptive finite element methods for elliptic partial differential equations we introduce an hp adaptive finite element algorithm based on a combination of reliable and efficient residual error indicators and a new hp extension control technique which assesses the local regularity of the underlying analytical solution on the basis of its local legendre series expansion numerical experiments confirm the robustness and reliability of the proposed algorithm

goal oriented error estimation based on equilibrated flux reconstruction for finite element approximations of elliptic problems abstract we propose an approach for goal oriented error estimation in finite element approximations of second order elliptic problems that combines the dual weighted residual method and equilibrated flux reconstruction methods for the primal and dual problems the objective is to be able to consider discretization schemes for the dual solution that may be different from those used for the primal solution it is only assumed here that the discretization methods come with a priori error estimates and an equilibrated flux reconstruction algorithm a high order discontinuous galerkin dg method is actually the preferred choice for the approximation of the dual solution thanks to its flexibility and straightforward construction of equilibrated fluxes one contribution of the paper is to show how the order of the dg method for asymptotic exactness of the proposed estimator can be chosen in the cases where a conforming finite element method a dg method or a mixed raviart thomas method is used for the solution of the primal problem numerical experiments are also presented to illustrate the performance and convergence of the error estimates in quantities of interest with respect to the mesh size

solving linear variational inequality problems by a self adaptive projection method we propose a self adaptive projection method for solving linear variational inequality problems and show its global convergence under mild conditions some numerical results are also addressed which indicate that the method is quite robust and efficient

non uniform order mixed fem approximation implementation post processing computable error bound and adaptivity the present work provides a straightforward and focused set of tools and corresponding theoretical support for the implementation of an adaptive high order finite element code with guaranteed error control for the approximation of elliptic problems in mixed form the work contains details of the discretisation using non uniform order mixed finite elements of arbitrarily high order a new local post processing scheme for the primary variable the use of the post processing scheme in the derivation of new fully computable bounds for the error in the flux variable and an hp adaptive refinement strategy based on the a posteriori error estimator numerical examples are presented illustrating the results obtained when the procedure is applied to a challenging problem involving a ten pole electric motor with singularities arising from both geometric features and discontinuities in material properties the procedure is shown to be capable of producing high accuracy numerical approximations with relatively modest numbers of unknowns

error estimators for a mixed method we present two a posteriori error estimators for elliptic partial differential equations when we use a mixed method one is based on a adequate evaluation of the residual of the finite element solution and the other on the solution of a local problem we prove their equivalence with the norm of the error when the data is locally smooth

recursive density estimation under dependence recursive estimators of the density of weakly dependent random variables are studied under certain absolute regularity and strong mixing conditions uniform strong consistency of the density estimators is established and their rates of convergence are obtained this study is concerned more with the almost sure uniform consistency of a sequence and its rate of convergence than with pointwise convergence since parameter estimation in time series analysis is often carried out under the gaussian assumption it is useful to check whether or not the density of a time series is gaussian or nearly so the method of proof used here is based on approximations of absolutely regular and strong mixing random variables rvs by independent rvs

averaging techniques yield reliable a posteriori finite element error control for obstacle problems the reliability of frequently applied averaging techniques for a posteriori error control has recently been established for a series of finite element methods in the context of second order partial differential equations this paper establishes related reliable and efficient a posteriori error estimates for the energy norm error of an obstacle problem on unstructured grids as a model example for variational inequalities the surprising main result asserts that the distance of the piecewise constant discrete gradient to any continuous piecewise affine approximation is a reliable upper error bound up to known higher order terms consistency terms and a multiplicative constant

one dimensional unstable eigenfunction and manifold computations in delay differential equations in this paper we present a new numerical technique for computing the unstable eigenfunctions of a saddle periodic orbit in a delay differential equation this is used to obtain the necessary starting data for an established algorithm for computing one dimensional 1d unstable manifolds of an associated saddle fixed point of a suitable poincare map to illustrate our method we investigate an intermittent transition to chaos in a delay system describing a semiconductor laser subject to phase conjugate feedback

growing 1d and quasi 2d unstable manifolds of maps we present a new 1d algorithm for computing the global one dimensional unstable manifold of a saddle point of a map our method can be generalized to compute two dimensional unstable manifolds of maps with three dimensional state spaces this is shown here with a quasi 2d q2d algorithm for the special case of a quasiperiodically forced map which allows for a substantial simplification of the general case described in krauskopf and osinga 1998 the key idea is to grow the manifold in steps which consist of finding a new point on the manifold at a prescribed distance from the last point the speed of growth is determined only by the curvature of the manifold and not by the dynamics the performance of the 1d algorithm is demonstrated with a constructed test example and it is then used to compute one dimensional manifolds of a map modeling mixing in a stirring tank with the q2d algorithm we compute two dimensional unstable manifolds in the quasiperiodically forced henon map

recursive kernel density estimation for time series we consider the recursive estimation of the probability density function of continuous random variables from a strongly mixing random sample we revisit here earlier research on this subject by considering a more general class of recursive estimators including the usual ones we derive the quadratic mean error of the considered class of estimators moreover we establish a central limit theorem by using lindeberg s method resulting in a simplification of the existing assumptions on the sequence of smooth parameters and the mixing coefficient this is the main contribution of this paper finally the feasibility of the proposed estimator is illustrated throughout an empirical study

hp interpolation of nonsmooth functions and an application to hp a posteriori error estimation the quasi interpolation operators of clement and scott zhang type are generalized to the setting of the hp fem finite element method new polynomial lifting and inverse estimates are presented the classical residual based a posteriori error estimator is generalized to the hp fem

robust equilibrated residual error estimator for diffusion problems conforming elements this paper analyzes an equilibrated residual a posteriori error estimator for the diffusion problem the estimator which is a modification of those in d braess and j schoberl math comput 77 2008 pp 651 672 r verfurth siam j numer anal 47 2009 pp 3180 3194 is based on the prager synge identity and on a local recovery of an equilibrated flux numerical results for an interface test problem show that the modification is necessary for the robustness of the estimator when the distribution of diffusion coefficients is local quasi monotone it is shown theoretically that the estimator is robust with respect to the size of jumps

an adaptive wavelet collocation method for solving optimal control of elliptic variational inequalities of the obstacle type abstract this paper presents a fast computational technique based on the wavelet collocation method for the numerical solution of an optimal control problem governed by elliptic variational inequalities of obstacle type in this problem the solution divides the domain into contact and noncontact sets the boundary between the contact and noncontact sets is a free boundary which is not known a priori and the solution is not smooth on it accordingly a very fine grid is needed in order to obtain a solution with a reasonable accuracy in this paper our aim is to propose an adaptive scheme in order to generate an appropriate and economic irregular dyadic mesh for finding the optimal control and state functions the irregular mesh will be generated such that its density around the free boundary is higher than in other places and high resolution computations are focused on these zones to this aim we use an adaptive wavelet collocation method and take advantage of the fast wavelet transform of compact supported interpolating wavelets to develop a multi level algorithm which generates an adaptive computational grid using this adaptive grid takes less cpu time than using a full regular mesh at each step of the algorithm the active set method is used for solving the optimality system of the obstacle problem on the adapted mesh finally the numerical examples are presented to show the validity and efficiency of the technique

a posteriori error control and adaptivity of hp finite elements for mixed and mixed hybrid methods abstract in this paper mixed and mixed hybrid methods for h and h p adaptive finite elements on quadrilateral meshes are discussed for variational equations and in particular for variational inequalities the main result is the derivation of reliable error estimates for mixed methods for the obstacle problem the estimates rely on the use of a post processing of the potential in h 1 and on the introduction of a certain lagrange multiplier which is associated with the obstacle constraints the error estimates consist of the dual norm of the residual which is defined by an appropriate approximation of the lagrange multiplier plus some computable remainder terms in numerical experiments the applicability of the post processing procedure on quadrilateral meshes with multilevel hanging nodes is verified and the use of the estimates in h and h p adaptive schemes is demonstrated by means of convergence rates and effectivity indices

a posteriori error estimate for the mixed finite element method a computable error bound for mixed finite element methods is established in the model case of the poisson problem to control the error in the h div x l 2 norm the reliable and efficient a posteriori error estimate applies e g to raviart thomas brezzi douglas marini and brezzi douglas fortin marini elements

on density estimation from ergodic processes we consider the problem of lp consistent density estimation from the initial segments of strongly dependent processes it is shown that no procedure can consistently estimate the one dimensional marginal density of every stationary ergodic process for which such a density exists a similar result is established for the problem of estimating the support of the marginal distribution of an ergodic process

biorthogonal bases with local support and approximation properties we construct locally supported basis functions which are biorthogonal to conforming nodal finite element basis functions of degree p in one dimension in contrast to earlier approaches these basis functions have the same support as the nodal finite element basis functions and reproduce the conforming finite element space of degree p 1 working with gaus lobatto nodes we find an interesting connection between biorthogonality and quadrature formulas one important application of these newly constructed biorthogonal basis functions are two dimensional mortar finite elements the weak continuity condition of the constrained mortar space is realized in terms of our new dual bases as a result local static condensation can be applied which is very attractive from the numerical point of view numerical results are presented for cubic mortar finite elements

hp fem for a stabilized three field formulation of the biharmonic problem abstract in this paper a three field formulation for the biharmonic equation is considered that consists of the potential u the flux and a lagrange multiplier the use of stabilization techniques circumvents the discrete inf sup condition and thus enables the application of arbitrary conforming finite element spaces for the three fields a priori error estimates as well as reliable and efficient a posteriori error estimates are derived the proof of the latter is based on an implicit never to be computed h 2 reconstruction of the discrete potential several numerical experiments underline the theoretical results and show the existence of discretization spaces for which the a posteriori error estimate is p robust with an efficiency index tending towards the optimal value of one the numerical experiments indicate that the proposed method is applicable even if the theoretically required minimal regularity of the lagrange multiplier for the a priori error estimate is not fulfilled in particular h and h p adaptivity recover optimal algebraic and exponential converge rates respectively

on global properties of time delayed feedback control weakly nonlinear analysis we address the problem which type of initial state yields successful stabilisation when time delayed feedback control is applied using well known concepts from bifurcation theory we provide a general mechanism that limits the corresponding basins of attraction application of normal form analysis to general delay differential equations proves the universality of our results the empirical criteria which determine the prospective size of the basin are easily accessible and a demonstration in terms of a simple numerical simulation is provided

a framework for obtaining guaranteed error bounds for finite element approximations we give an overview of our recent progress in developing a framework for the derivation of fully computable guaranteed posteriori error bounds for finite element approximation including conforming non conforming mixed and discontinuous finite element schemes whilst the details of the actual estimator are rather different for each particular scheme there is nonetheless a common underlying structure at work in all cases we aim to illustrate this structure by treating conforming non conforming and discontinuous finite element schemes in a single framework in taking a rather general viewpoint some of the finer details of the analysis that rely on the specific properties of each particular scheme are obscured but in return we hope to allow the reader to see the wood despite the trees

estimating a function from ergodic samples with additive noise we study the problem of estimating an unknown function from ergodic samples corrupted by additive noise it is shown that one can consistently recover an unknown measurable function in this setting if the one dimensional 1 d distribution of the samples is comparable to a known reference distribution and the noise is independent of the samples and has known mixing rates the estimates are applied to deterministic sampling schemes in which successive samples are obtained by repeatedly applying a fixed map to a given initial vector and it is then shown how the estimates can be used to reconstruct an ergodic transformation from one of its trajectories

mixed finite element methods of higher order for model contact problems this paper presents mixed finite element methods of higher order for a simplified signorini problem and an idealized frictional problem the discretization is based on a mixed variational formulation proposed by haslinger et al which is extended to higher order finite elements to guarantee the unique existence of the solution of the mixed method a discrete inf sup condition is proven approximation results of the p method of finite elements and some inverse estimates for higher order polynomials are applied numerical results confirm the theoretical findings

universal denoising of discrete time continuous amplitude signals we consider the problem of reconstructing a discrete time signal sequence with continuous valued components corrupted by a known memoryless channel when performance is measured using a per symbol loss function satisfying mild regularity conditions we develop a sequence of denoisers that although independent of the distribution of the underlying ldquocleanrdquo sequence is universally optimal in the limit of large sequence length this sequence of denoisers is universal in the sense of performing as well as any sliding window denoising scheme which may be optimized for the underlying clean signal our results are initially developed in a ldquosemi stochasticrdquo setting where the noiseless signal is an unknown individual sequence and the only source of randomness is due to the channel noise it is subsequently shown that in the fully stochastic setting where the noiseless sequence is a stationary stochastic process our schemes universally attain optimum performance the proposed schemes draw from nonparametric density estimation techniques and are practically implementable we demonstrate efficacy of the proposed schemes in denoising gray scale images in the conventional additive white gaussian noise awgn setting with additional promising results for less conventional noise distributions

a behavioral approach to delay differential systems we will study linear time invariant delay differential systems from the behavioral point of view as it was introduced for dynamical systems by willems dynam report 2 1989 pp 171 269 a ring cal h which lies between bbb r s z z 1 and bbb r s z z 1 will be presented whose elements can be interpreted as a generalized version of delay differential operators on cal c infty bbb r bbb r in this framework a behavior is the kernel of such an operator using the ring cal h an algebraic characterization of inclusion respectively equality of the behaviors under consideration is given finally controllability of the behaviors is characterized in terms of the rank of the associated matrices in the case of time delay state space systems this criterion becomes the known hautus criterion for spectral controllability

a posteriori error estimates for the primary and dual variables for the div first order least squares finite element method abstract we propose a posteriori error estimators for first order div least squares ls finite element method for linear elasticity stokes equations and general second order scalar elliptic problems our main interest is obtaining a posteriori error estimators for the dual variables fluxes strains stress etc which are main quantity of interest in many applications we also provide a posteriori error estimators for the primary variable these estimators are obtained from the local least squares functional by assigning weight coefficients scaling the respective residuals the weight coefficients are given in terms of local meshsize h k we establish the global upper bounds and local lower bounds for the estimators the estimators can be easily computed from the finite element solution together with the given problem data and provide basis for mesh refinement criteria for efficient computation of finite element solution the indicators and estimators are identical numerical experiments show a superior performance of our a posteriori estimators for user specific norm over the standard ls functional

the l sub 1 and l sub 2 strong consistency of recursive kernel density estimation from dependent samples the l sub 1 and l sub 2 strong consistency of recursive kernel density estimators is established for mixing and ergodic stationary processes sharp rates of almost sure convergence are obtained in the l sub 2 case for mixing processes in addition the concept and properties of hilbert space valued mixingales are developed and strong laws of large numbers are given

some basic remarks on eigenmode expansions of time delay dynamics abstract we describe in elementary terms how eigenmode expansions can be used to deal with differential difference equations as particular applications we present the full analytical solution of linear stochastic time delay systems and the weakly nonlinear analysis of nonlinear differential difference equations in the limit of large time delay our exposition is essentially based on an explicit analytical expression for the linear spectrum in terms of the lambert w function and on the explicit formula for the eigenfunctions of the adjoint equation

first order least squares method for the obstacle problem we define and analyse a least squares finite element method for a first order reformulation of the obstacle problem moreover we derive variational inequalities that are based on similar but non symmetric bilinear forms a priori error estimates including the case of non conforming convex sets are given and optimal convergence rates are shown for the lowest order case we provide a posteriori bounds that can be used as error indicators in an adaptive algorithm numerical studies are presented

recursive nonparametric estimation for time series this paper considers online kernel estimation for both short and long range dependent time series data utilizing the predictive dependence measure of wu we carefully study the asymptotic properties of recursive kernel density and regression estimators for a general class of stationary processes in particular we prove that the proposed estimators have the asymptotic normality and the corresponding central limit theorems are provided in addition we establish the sharp laws of the iterated logarithms that precisely characterize the asymptotic almost sure behavior of the proposed estimators

semiconductor laser with phase conjugate feedback dynamics and bifurcations this paper presents the dynamics and bifurcations of a semiconductor laser subject to instantaneous phase conjugate feedback recently the behavior of such a laser has been explored by means of bifurcation diagrams however the exact nature of the involved dynamics and bifurcations remained unclear here we present a detailed study of the changes of the dynamics as the feedback strength is varied most prominent are symmetry breaking and restoring bifurcations tori and their bifurcations and a sudden transition between chaos and a stable limit cycle due to a saddle node bifurcation of limit cycles

numerical bifurcation analysis of the traveling wave model of multisection semiconductor lasers abstract the traveling wave model consisting of a hyperbolic system of linear first order pde s nonlinearly coupled with a system of ode s is used to model the dynamics of multisection semiconductor lasers it is shown how low dimensional systems of ode s based on the projection of the original system into the subspace spanned only by a few instantaneously changing spectral elements allow a precise approximation of the original model the constructed low dimensional mode approximation systems are accessible to classical bifurcation analysis and numerical tools the presented method is applied to characterise quasi periodic orbits arising in a three section distributed feedback laser the obtained results are highly important for tailoring such pulsating lasers to be applied in optical communication systems

a posteriori error control of h p finite elements for variational inequalities of the first and second kind in this paper residual based a posteriori error estimates for variational inequalities including those of the second kind are proposed the variational formulation and its discretizations are considered in an abstract framework of general banach spaces the main idea is to express the residual in terms of some lagrange multipliers which can be obtained for instance by some post processing the residual itself is defined for an arbitrary element of the trial space the error of this element and of the solution of the variational inequality is then estimated by the dual norm of the residual plus some computable remainder terms this concept is applied to a variety of frictional contact problems such as signorini and obstacle problems as well as to the bingham fluid problem the applicability of the estimates is confirmed by several numerical experiments in particular the general framework allows for the discretization with h p adaptivity which leads to nearly exponential convergence rates in most cases

hp adaptive ipdg tdg fem for parabolic obstacle problems for a parabolic obstacle problem two equivalent hp fem discretization methods based on interior penalty discontinuous galerkin in space and discontinuous galerkin in time are presented the first approach is based on a variational inequality vi formulation and the second approach on a mixed method in which the non penetration condition is resolved by a lagrange multiplier the discrete lagrange multiplier is a linear combination of biorthogonal basis functions allowing to write the discrete vi constraints as a set of complementarity problems employing a penalized fischer burmeister non linear complementarity function the discrete mixed problem can be solved by a locally q quadratic converging semi smooth newton ssn method the hierarchical a posteriori error estimator for the vi formulation which under the saturation assumption is both efficient and reliable allows hp adaptivity the numerical experiments show improved convergence compared to uniform and h adaptive meshes furthermore an a priori error estimate is given for the vi formulation

guaranteed a posteriori error estimator for mixed finite element methods of elliptic problems abstract in this work we analyze a posteriori error estimator for a general class of mixed finite element methods of elliptic problems which guarantees an upper bound on the vector error thus extending the recent result obtained for the lowest order triangular raviart thomas mixed finite element method in m ainsworth a posteriori error estimation for lowest order raviart thomas mixed finite elements siam j sci comput 30 2007 08 189 204 the error estimator is constructed through a variant of stenberg s postprocessing procedure and the guaranteed upper bound is readily established by making use of the argument similar to the hypercircle method however the proof of the lower bound given in the above reference does not seem to apply to other kinds of mixed elements so we employ a different technique using the discrete friedrichs inequality to establish the lower bound of the error estimator

distribution free consistency of a nonparametric kernel regression estimate and classification it is shown that the kernel estimate of the regression e y x x is weakly or strongly consistent for almost all x mu where mu is the probability measure of x the result is valid for any distribution of x the asymptotical optimality of classification rules derived from the estimate is examined the optimality is independent of class distributions i e it is distribution free

a posteriori error estimation for lowest order raviart thomas mixed finite elements for the lowest order raviart thomas mixed finite element we derive an a posteriori error estimator that provides actual guaranteed computable upper bounds on the error in the flux variable regardless of jumps in the material coefficients across interfaces moreover the estimator is efficient in that it provides a local lower bound on the error up to a constant that is independent of the solution and the local mesh size the estimator may be evaluated at virtually no additional cost compared to the evaluation of the finite element approximation itself

biorthogonal basis functions in hp adaptive fem for elliptic obstacle problems abstract in this paper the discretization of a non symmetric elliptic obstacle problem with h p adaptive h 1 conforming finite elements is discussed for this purpose a higher order mixed finite element discretization is introduced where the dual space is discretized via biorthogonal basis functions the h p adaptivity is realized via automatic adaptive mesh refinement based on a residual a posteriori error estimation which is also derived in this paper the use of biorthogonal basis functions leads to unilateral box constraints and componentwise complementarity conditions enabling the highly efficient application of a quadratically converging semi smooth newton scheme which can be modified to ensure global convergence h p adaptivity usually implies meshes with hanging nodes and varying polynomial degrees which have to be handled appropriately within the h 1 conforming finite element discretization this is typically done by using so called connectivity matrices in this paper a procedure is proposed which efficiently computes these matrices for biorthogonal basis functions finally the applicability of the theoretical findings is demonstrated with several numerical experiments

numerical bifurcation analysis of delay differential equations numerical methods for the bifurcation analysis of delay differential equations ddes have only recently received much attention partially because the theory of ddes smoothness boundedness stability of solutions is more complicated and less established than the corresponding theory of ordinary differential equations as a consequence no established software packages exist at present for the bifurcation analysis of ddes we outline existing numerical methods for the computation and stability analysis of steady state solutions and periodic solutions of systems of ddes with several constant delays

a posteriori error estimators for the raviart thomas element when error estimators for the rte are developed two difficulties prevent the success of the straightforward application of frequently used arguments the hdiv norm is an anisotropic norm i e it refers to differential operators of different orders moreover the traces of hdiv functions are only in h 1 2 therefore one does not obtain optimal a posteriori error estimates when using natural norms this drawback is overcome by using mesh dependent norms

stochastic bifurcation in a driven laser system experiment and theory we analyze the effects of stochastic perturbations in a physical example occurring as a higher dimensional dynamical system the physical model is that of a class b laser which is perturbed stochastically with finite noise the effect of the noise perturbations on the dynamics is shown to change the qualitative nature of the dynamics experimentally from a stochastic periodic attractor to one of chaoslike behavior or noise induced chaos to analyze the qualitative change we apply the technique of the stochastic frobenius perron operator l billings et al phys rev lett 88 234101 2002 to a model of the experimental system our main result is the identification of a global mechanism to induce chaoslike behavior by adding stochastic perturbations in a realistic model system of an optics experiment in quantifying the stochastic bifurcation we have computed a transition matrix describing the probability of transport from one region of phase space to another which approximates the stochastic frobenius perron operator this mechanism depends on both the standard deviation of the noise and the global topology of the system our result pinpoints regions of stochastic transport whereby topological deterministic dynamics subjected to sufficient noise results in noise induced chaos in both theory and experiment

limits to consistent on line forecasting for ergodic time series this article concerns problems of time series forecasting under the weakest of assumptions related results are surveyed and are points of departure for the developments here some of which are new and others are new derivations of previous findings the contributions in this study are all negative showing that various plausible prediction problems are unsolvable or in other cases are not solvable by predictors which are known to be consistent when mixing conditions hold

an adaptive stabilized finite element method for the generalized stokes problem in this work we present an adaptive strategy based on an a posteriori error estimator for a stabilized finite element method for the stokes problem with and without a reaction term the hierarchical type estimator is based on the solution of local problems posed on appropriate finite dimensional spaces of bubble like functions an equivalence result between the norm of the finite element error and the estimator is given where the dependence of the constants on the physics of the problem is explicited several numerical results confirming both the theoretical results and the good performance of the estimator are given

hp fem convergence for unilateral contact problems with tresca friction in plane linear elastostatics abstract this paper is concerned with the convergence of the h p version of the finite element method h p fem for some nonsmooth unilateral problems in linear elastostatics we consider in particular the deformation of an elastic body unilaterally supported by a rigid foundation admitting tresca friction given friction along the rigid foundation solely subjected to body forces and surface tractions without being fixed along some part of its boundary for the discretization of the unilateral constraint and the nonsmooth friction functional we employ gauss lobatto quadrature we show convergence of the h p fem approximations for mechanically definite problems without imposing any regularity assumption moreover we treat the coercive case when the body is fixed along some part of the boundary based on an abstract cea falk estimate and operator interpolation arguments we establish an a priori error estimate in the energy norm under a reasonable regularity assumption

an infeasible active set method for quadratic problems with simple bounds a primal dual active set method for quadratic problems with bound constraints is presented based on a guess on the active set a primal dual pair x s is computed that satisfies the first order optimality condition and the complementarity condition if x s is not feasible a new active set is determined and the process is iterated sufficient conditions for the iterations to stop in a finite number of steps with an optimal solution are provided computational experience indicates that this approach often requires only a few less than 10 iterations to find the optimal solution

a mortar finite element method using dual spaces for the lagrange multiplier the mortar finite element method allows the coupling of different discretization schemes and triangulations across subregion boundaries in the original mortar approach the matching at the interface is realized by enforcing an orthogonality relation between the jump and a modified trace space which serves as a space of lagrange multipliers in this paper this lagrange multiplier space is replaced by a dual space without losing the optimality of the method the advantage of this new approach is that the matching condition is much easier to realize in particular all the basis functions of the new method are supported in a few elements the mortar map can be represented by a diagonal matrix in the standard mortar method a linear system of equations must be solved the problem is considered in a positive definite nonconforming variational as well as an equivalent saddle point formulation

the norm of a discretized gradient in h div for a posteriori finite element error analysis this paper characterizes the norm of the residual of mixed schemes in their natural functional framework with fluxes or stresses in h mathrm div h div and displacements in l 2 l2 under some natural conditions on an associated fortin interpolation operator reliable and efficient error estimates are introduced that circumvent the duality technique and so do not suffer from reduced elliptic regularity for non convex domains for the laplace stokes and lame equations this generalizes known estimators to non convex domains and introduces new a posteriori error estimators

a posteriori error estimators for obstacle problems another look we show that a posteriori estimators for the obstacle problem are easily obtained from the theory for linear equations the theory would be even simpler if the lagrange multiplier does not have a nonconforming contribution as it has in actual finite element computations

unified primal formulation based a priori and a posteriori error analysis of mixed finite element methods we derive in this paper a unified framework for a priori and a posteriori error analysis of mixed finite element discretizations of second order elliptic problems it is based on the classical primal weak formulation the postprocessing of the potential proposed in t arbogast and z chen on the implementation of mixed methods as nonconforming methods for second order elliptic problems math comp 64 1995 943 972 and the discrete friedrichs inequality our analysis in particular avoids any explicit use of the uniform discrete inf sup condition and in a straightforward manner and under minimal necessary assumptions known convergence and superconvergence results are recovered the same framework then turns out to lead to optimal a posteriori energy error bounds in particular estimators for all families and orders of mixed finite element methods on grids consisting of simplices or rectangular parallelepipeds are derived they give a guaranteed and fully computable upper bound on the energy error represent error local lower bounds and are robust under some conditions on the diffusion dispersion tensor they are thus suitable for both overall error control and adaptive mesh refinement moreover the developed abstract framework and a posteriori error estimates are quite general and apply to any locally conservative method we finally prove that in parallel and simultaneously in converse to galerkin finite element methods under some circumstances the weak solution is the orthogonal projection of the postprocessed mixed finite element approximation onto the h 1 0 space and also establish several links between mixed finite element approximations and some generalized weak solutions

a posteriori error estimates of higher order finite elements for frictional contact problems abstract in this paper a posteriori estimates are derived for higher order finite element methods and frictional contact problems the discretization is based on a mixed approach where the geometrical and frictional constraints are captured by lagrange multipliers the use of higher order polynomials leads to a certain non conformity in the discretization which requires special attention in the error analysis as a main result an error estimation is proposed which consists of the dual norm of a residual plus some computable remainder terms the residual is estimated by well known a posteriori error estimates for variational equations the remainder terms represent typical sources resulting from the non conforming mixed discretization numerical experiments confirm the applicability of the a posteriori estimates to adaptive mesh refinements

a p version finite element method for nonlinear elliptic variational inequalities in 2d this article introduces and analyzes a p version fem for variational inequalities resulting from obstacle problems for some quasi linear elliptic partial differential operators we approximate the solution by controlling the obstacle condition in images of the gauss lobatto points we show existence and uniqueness for the discrete solution up from the p version for the obstacle problem we prove the convergence of up towards the solution with respect to the energy norm and assuming some additional regularity for the solution we derive an a priori error estimate in numerical experiments the p version turns out to be superior to the h version concerning the convergence rate and the number of unknowns needed to achieve a certain exactness of the approximation

a penalized fischer burmeister ncp function we introduce a new ncp function in order to reformulate the nonlinear complementarity problem as a nonsmooth system of equations this new ncp function turns out to have stronger theoretical properties than the widely used fischer burmeister function and other ncp functions suggested previously moreover numerical experience indicates that a semismooth newton method based on this new ncp function performs considerably better than the corresponding method based on the fischer burmeister function

aspects of an adaptive hp finite element method adaptive strategy conforming approximation and efficient solvers the main components needed for an adaptive hp version finite element algorithm are discussed an adaptive hp refinement strategy effective methods for constructing conforming hp approximations and efficient solvers for the large ill conditioned systems of linear equations together these provide the methodology for an effective adaptive hp version algorithm the presentation emphasizes the links between the differing components showing how the algorithms may be implemented efficiently in practice the main principles are illustrated by use of concrete examples so that a non expert may develop their own adaptive hp code the performance of the whole algorithm is illustrated for some representative problems taken from linear elasticity

the rates of convergence of kernel regression estimates and classification rules both nonrecursive and recursive nonparametric regression estimates are studied the rates of weak and strong convergence of kernel estimates as well as corresponding multiple classification errors are derived without assuming the existence of the density of the measurements an application of the obtained results to a nonparametric bayes predication is presented

collocation methods for the computation of periodic solutions of delay differential equations in this paper we investigate collocation methods for the computation of periodic solutions of autonomous delay differential equations ddes periodic solutions are found by solving a periodic two point boundary value problem which is an infinite dimensional problem for ddes in contrast to the case of ordinary differential equations we investigate three collocation methods based on piecewise polynomials we discuss computational issues and show numerical orders of convergence using an extensive number of tests we compare our numerical results with known theoretical convergence results for initial value problems for ddes in particular we show how superconvergence at the mesh points can be lost or recovered depending on the dde model under consideration and on the choice of collocation discretization we end with a brief discussion of adaptive mesh selection

computing unstable manifolds of periodic orbits in delay differential equations we present the first algorithm for computing unstable manifolds of saddle type periodic orbits with one unstable floquet multiplier in systems of autonomous delay differential equations ddes with one fixed delay specifically we grow the one dimensional unstable manifold wu q of an associated saddle fixed point q of a poincare map defined by a suitable poincare section starting close to q along the linear approximation to wu q given by the associated eigenfunction our algorithm grows the manifold as a sequence of points where the distance between points is governed by the curvature of the one dimensional intersection curve wu q of wu q with our algorithm makes it possible to study global bifurcations in ddes we illustrate this with the break up of an invariant torus and a subsequent crisis bifurcation to chaos in a dde model of a semiconductor laser with phase conjugate feedback