regularity of solutions of elliptic problems with a curved fracture abstract we consider the laplace equation with a right hand side concentrated on a curved fracture of class c m 2 for some nonnegative integer m i e a sort of dirac mass we show that the solution belongs to a weighted sobolev space of order m the weight being the distance to this fracture our proof relies on a priori estimates in a dihedron or a cone with singularities for elliptic operators with variable coefficients in both cases such an estimate is obtained using a dyadic covering of the domain

optimal design of magnitude responses of rational infinite impulse response filters this correspondence considers a design of magnitude responses of optimal rational infinite impulse response iir filters the design problem is formulated as an optimization problem in which a total weighted absolute error in the passband and stopband of the filters the error function reflects a ripple square magnitude is minimized subject to the specification on this weighted absolute error function defined in the corresponding passband and stopband as well as the stability condition since the cost function is nonsmooth and nonconvex while the constraints are continuous this kind of optimization problem is a nonsmooth nonconvex continuous functional constrained problem to address this issue our previous proposed constraint transcription method is applied to transform the continuous functional constraints to equality constraints then the nonsmooth problem is approximated by a sequence of smooth problems and solved via a hybrid global optimization method the solutions obtained from these smooth problems converge to the global optimal solution of the original optimization problem hence small transition bandwidth filters can be obtained

optimal control of an eddy current problem with a dipole source abstract this paper is concerned with the analysis of a class of optimal control problems governed by a time harmonic eddy current system with a dipole source which is taken as the control variable a mathematical model is set up for the state equation where the dipole source takes the form of a dirac mass located in the interior of the conducting domain a non standard approach featuring the fundamental solution of a curl curl id operator is proposed to address the well posedness of the state problem leading to a split structure of the state field as the sum of a singular part and a regular part the aim of the control is the best approximation of desired electric and magnetic fields via a suitable l 2 quadratic tracking cost functional here special attention is devoted to establishing an adjoint calculus which is consistent with the form of the state variable and in this way first order optimality conditions are eventually derived

a trilinear immersed finite element method for solving the electroencephalography forward problem this article describes a trilinear immersed finite element method for solving the electroencephalography forward problem which is a three dimensional elliptic interface problem in the head geometry the method uses hexahedral cartesian meshes independent of the interfaces between head tissues thus avoiding the sometimes difficult task of generating geometry fitting meshes the interfaces are represented with level sets and the finite element space is locally modified to better approximate the discontinuities of the solution numerical results show that this method achieves the same accuracy as the standard linear finite element method with geometry fitting mesh

pointwise a posteriori error estimates for elliptic problems on highly graded meshes pointwise a posteriori error estimates are derived for linear second order elliptic problems over general polygonal domains in 2d the analysis carries over regardless of convexity accounting even for slit domains and applies to highly graded unstructured meshes as well a key ingredient is a new asymptotic a priori estimate for regularized green s functions the estimators lead always to upper bounds for the error in the maximum norm along with lower bounds under very mild regularity and nondegeneracy assumptions the effect of both point and line singularities is examined three popular local estimators for the energy norm are shown to be equivalent when suitably interpreted to those introduced here

hierarchical bayesian inference for the eeg inverse problem using realistic fe head models depth localization and source separation for focal primary currents the estimation of the activity related ion currents by measuring the induced electromagnetic fields at the head surface is a challenging and severely ill posed inverse problem this is especially true in the recovery of brain networks involving deep lying sources by means of eeg meg recordings which is still a challenging task for any inverse method recently hierarchical bayesian modeling hbm emerged as a unifying framework for current density reconstruction cdr approaches comprising most established methods as well as offering promising new methods our work examines the performance of fully bayesian inference methods for hbm for source configurations consisting of few focal sources when used with realistic high resolution finite element fe head models the main foci of interest are the correct depth localization a well known source of systematic error of many cdr methods and the separation of single sources in multiple source scenarios both aspects are very important in the analysis of neurophysiological data and in clinical applications for these tasks hbm provides a promising framework and is able to improve upon established cdr methods such as minimum norm estimation mne or sloreta in many aspects for challenging multiple source scenarios where the established methods show crucial errors promising results are attained additionally we introduce wasserstein distances as performance measures for the validation of inverse methods in complex source scenarios

rational sampling filter banks based on iir filters the problem of splitting the spectrum of a digital signal by using nonuniform infinite impulse response iir filter banks is addressed near perfect reconstruction npr is considered the method uses the modulation of different iir prototypes the cancellation of the main aliasing components constrains the prototypes to be dependent on each other by using this approach linear phase prototypes are needed and noncausal filtering is required numerical examples of filter bank design are given and the computational complexity is compared with the finite impulse response fir case

l2 estimates for the finite element method for the dirichlet problem with singular data in this paper we consider the approximation by the finite element method of second order elliptic problems on convex domains and homogeneous dirichlet condition on the boundary in these problems the data are borel measures using a quasiuniform mesh of finite elements and polynomials of degree 1 we prove that in two dimensions the convergence is of orderh inl 2 and in three dimensions of orderh 1 2

adaptive galerkin methods with error control for a dynamical ginzburg landau model in superconductivity the time dependent ginzburg landau model which describes the phase transitions taking place in superconductors is a coupled system of nonlinear parabolic equations it is discretized semi implicitly in time and in space via continuous piecewise linear finite elements a posteriori error estimates are derived for the l infty l 2 norm by studying a dual problem of the linearization of the original system other than the dual of error equations numerical simulations are included which illustrate the reliability of the estimators and the flexibility of the proposed adaptive method

a posteriori error estimates for elliptic problems with dirac delta source terms the aim of this paper is to introduce residual type a posteriori error estimators for a poisson problem with a dirac delta source term in l p norm and w1 p seminorm the estimators are proved to yield global upper and local lower bounds for the corresponding norms of the error they are used to guide adaptive procedures which are experimentally shown to lead to optimal orders of convergence

maximum norm a posteriori error estimates for singularly perturbed reaction diffusion problems on anisotropic meshes residual type a posteriori error estimates in the maximum norm are given for singularly perturbed semilinear reaction diffusion equations posed in polygonal domains linear finite elements are considered on anisotropic triangulations the error constants are independent of the diameters and the aspect ratios of mesh elements and of the small perturbation parameter

a review of a posteriori error estimation techniques for elasticity problems within the framework of linearized elasticity we give an overview of the most popular a posteriori error estimators we discuss their common features the main differences and the underlying principles this will allow us to judge the quality of the different approaches and their suitability for nonlinear problems we also briefly describe a general framework which allows the construction of a posteriori error estimators for nonlinear elasticity

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

sharply local pointwise a posteriori error estimates for parabolic problems we prove pointwise a posteriori error estimates for semi and fully discrete finite element methods for approximating the solution u to a parabolic model problem our estimates may be used to bound the finite element error u u h l d where d is an arbitrary subset of the space time domain of the definition of the given pde in contrast to standard global error estimates these estimators de emphasize spatial error contributions from space time regions removed from d our results are valid on arbitrary shape regular simplicial meshes which may change in time and also provide insight into the contribution of mesh change to local errors when implemented in an adaptive method these estimates require only enough spatial mesh refinement away from d in order to ensure that local solution quality is not polluted by global effects

design of 1 d and 2 d iir eigenfilters an effective method is proposed for the design of recursive digital filters in the time spatial domain the method is based on the computation of an eigenvalue and its corresponding eigenvector of an appropriate real symmetric and positive definite matrix derived from the objective error function of the impulse response in the least squares sense the method can be used to design 1d iir digital filters as well as 2d iir digital filters the method is easy and the performance is comparable to those of the existing methods one of the main advantages of the proposed method is that the solution is obtained directly without the need for iteration

a posteriori error estimation and dynamic adaptivity for symmetric discontinuous galerkin approximations of reactive transport problems a symmetric discontinuous galerkin scheme is applied for solving reactive transport problems an explicit a posteriori error estimator in l 2 l 2 is derived for the semi discrete scheme applied to transport with general kinetic reactions explicit a posteriori error estimators in terms of linear functionals and negative norms are also obtained for transport problems with simple reactions an adaptive approach based on the a posteriori error estimators is proposed to modify the mesh dynamically numerical examples demonstrate the efficiency and effectivity of the estimators moreover it is shown in this study that the proposed adaptivity eliminates the need of slope limiters 2005 elsevier b v all rights reserved

eigenfilter design of higher order digital differentiators an effective approach is proposed for designing higher order differentiators by the eigenfilter method by minimizing a quadratic measure of the error in the frequency band an eigenvector of an appropriate matrix is computed to get the filter coefficients this method is not only simple and fast but also optimal in the least squares sense comparison to the mcclellan parks algorithm for minimax equiripple filters shows that both are optimal in the sense of different minimum norms of the error function but much better performance is obtained with the proposed approach in most of the frequency band except in the narrowband region near the cutoff edge

a posteriori pointwise error computation for 2 d transport equations based on the variational multiscale method abstract this article presents a general framework to estimate the pointwise error of linear partial differential equations the error estimator is based on the variational multiscale theory in which the error is decomposed in two components according to the nature of the residuals element interior residuals and inter element jumps the relationship between the residuals coarse scales and the error components fine scales is established yielding to a very simple model in particular the pointwise error is modeled as a linear combination of bubble functions and green s functions if residual free bubbles and the classical green s function are employed the technology leads to an exact explicit method for the pointwise error if bubble functions and free space green s functions are employed then a local projection problem must be solved within each element and a global boundary integral equation must be solved on the domain boundary as a consequence this gives a model for the so called fine scale green s functions the numerical error is studied for the standard galerkin and supg methods with application to the heat equation the reaction diffusion equation and the convection diffusion equation numerical results show that stabilized methods minimize the propagation of pollution errors which stay mostly locally

adaptive finite element method for parabolic equations with dirac measure abstract in this paper we study the adaptive finite element method for parabolic equations with dirac measure two kinds of problems with separate measure data in time and measure data in space are considered it is well known that the solutions of such kind of problems may exhibit lower regularity due to the existence of the dirac measure and thus fit to adaptive fem for space discretization and variable time steps for time discretization for both cases we use piecewise linear and continuous finite elements for the space discretization and backward euler scheme or equivalently piecewise constant discontinuous galerkin method for the time discretization the a posteriori error estimates based on energy and l 2 norms for the fully discrete problems are then derived to guide the adaptive procedure numerical results are provided at the end of the paper to support our theoretical findings

parallel algebraic multigrid methods on distributed memory computers algebraic multigrid methods are well suited as preconditioners for iterative solvers we consider linear systems of equations which are sparse and symmetric positive definite and which stem from a finite element discretization of a second order self adjoint elliptic partial differential equation or a system of them since preconditioners based on algebraic multigrid are very efficient additional speedup can only be achieved by parallelization in this paper we propose a general parallel algebraic multigrid algorithm for finite element discretizations based on domain decomposition ideas which is well suited for distributed memory computers this paper pays special attention to the coarsening strategy which has to be adapted in the parallel case moreover a general framework of data distribution gives rise to a construction scheme for the prolongation operators results of numerical studies on parallel computers with distributed memory are presented which show the high efficiency of the approach

expected performance of m solution backtracking this paper derives upper bounds on the expected number of search tree nodes visited during an m solution backtracking search a search which terminates after some preselected number m problem solutions are found the search behavior is assumed to have a general probabilistic structure our results are stated in terms of node expansion and contraction a visited search tree node is said to be expanding if the mean number of its children visited by the search exceeds 1 and is contracting otherwise we show that if every node expands or if every node contracts then the number of search tree nodes visited by a search has an upper bound which is linear in the depth of the tree in the mean number of children a node has and in the number of solutions sought bounds linear in the depth of the tree are derived for the case where the upper portion of the tree contracts while the lower portion expands we derive linear bounds for some special cases of an expanding upper portion and contracting lower portion ho

optimal a priori error estimates for an elliptic problem with dirac right hand side it is well known that finite element solutions for elliptic pdes with dirac measures as source terms converge due to the fact that the solution is not in h 1 suboptimal in classical norms a standard remedy is to use graded meshes then quasioptimality i e optimal up to a log factor for low order finite elements can be recovered e g in the l 2 norm here we show for the lowest order case quasioptimality and for higher order finite elements optimal order a priori estimates on a family of quasi uniform meshes in an l 2 seminorm the seminorm is defined as an l 2 norm on a fixed subdomain which excludes the locations of the delta source terms our motivation in the use of such a norm results from the observation that in many applications the error at the singularity is dominated by the model error e g in dimension reduced settings or is not the quantity of interest e g in optimal control problems the quasi optimal and optimal order a priori bounds are obtained recursively by using aub

interior maximum norm estimates for finite element methods part ii we consider bilinear forms a connected with second order elliptic problems and assume that for u h in a finite element space s h we have a u u h x f x for x in s h with local compact support we give local estimates for u u h in l and w 1 of the type local best approximation plus weak outside influences plus the local size of f

analysis of the efficiency of an a posteriori error estimator for linear triangular finite elements this paper adresses the problem of determining upper and lower bounds for the effectivity index on the a posteriori estimate of the error in the finite element method these bounds are given explicitly for a certain concrete estimator for linear elements and unstructured triangular meshes they depend strongly on the geometry of the triangles and relatively weakly on the smoothness of the solution an example shows that the bounds are not over pessimistic in babuska plank and rodriguez 4 detailed numerical experimentation is given

an adaptive finite element method for two phase stefan problems in two space dimensions i stability and error estimates a simple and efficient adaptive local mesh refinement algorithm is devised and analyzed for two phase stefan problems in 2d a typical triangula tion is coarse away from the discrete interface where discretization parameters satisfy a parabolic relation whereas it is locally refined in the vicinity of the dis crete interface so that the relation becomes hyperbolic several numerical tests are performed on the computed temperature to extract information about its first and second derivatives as well as to predict discrete free boundary locations mesh selection is based upon equidistributing pointwise interpolation errors be tween consecutive meshes and imposing that discrete interfaces belong to the so called refined region consecutive meshes are not compatible in that they are not produced by enrichment or coarsening procedures but rather regenerated a general theory for interpolation between noncompatible meshes is set up in lp based norms the resulting scheme is stable in various sobolev norms and necessitates fewer spatial degrees of freedom than previous practical methods 3 2 2 on quasi uniform meshes namely 0 r as opposed to 0 x to achieve the same global asymptotic accuracy here r 0 is the uniform time step 112 a rate of convergence of essentially 0 x is derived in the natural energy spaces provided the total number of mesh changes is restricted to 0 x which in turn is compatible with the mesh selection procedure an auxiliary quasi optimal pointwise error estimate for the laplace operator is proved as well numerical results illustrate the scheme s efficiency in approximating both solutions and interfaces

pointwise a posteriori error estimates for monotone semi linear equations we derive upper and lower a posteriori estimates for the maximum norm error in finite element solutions of monotone semi linear equations the estimates hold for lagrange elements of any fixed order non smooth nonlinearities and take numerical integration into account the proof hinges on constructing continuous barrier functions by correcting the discrete solution appropriately and then applying the continuous maximum principle no geometric mesh constraints are thus required numerical experiments illustrate reliability and efficiency properties of the corresponding estimators and investigate the performance of the resulting adaptive algorithms in terms of the polynomial order and quadrature

a priori mesh grading for an elliptic problem with dirac right hand side the green function of the poisson equation in two dimensions is not contained in the sobolev space h 1 omega such that finite element error estimates for the discretization of a problem with the dirac measure on the right hand side are nonstandard and quasi uniform meshes are inappropriate by using graded meshes l 2 error estimates of almost optimal order are shown as a byproduct we show for the poisson equation with a right hand side in l 2 that appropriate mesh refinement near some interior point diminishes the error at this point by nearly one order

adaptive finite element methods for parabolic problems i a linear model problem this paper is the first part in a series of papers on adaptive finite element methods for parabolic problems in this paper an adaptive algorithm is presented and analyzed for choosing the space a

a posteriori error estimators for the stokes equations we present two a posteriori error estimators for the mini element discretization of the stokes equations one is based on a suitable evaluation of the residual of the finite element solution the other one is based on the solution of suitable local stokes problems involving the residual of the finite element solution both estimators are globally upper and locally lower bounds for the error of the finite element discretization numerical examples show their efficiency both in estimating the error and in controlling an automatic self adaptive mesh refinement process the methods presented here can easily be generalized to the navier stokes equations and to other discretization schemes

a posteriori error estimates for the problem of electrostatics with a dipole source abstract electroencephalography is a non invasive technique for detecting brain activity from the measurement of the electric potential on the head surface in mathematical terms it reduces to an inverse problem in which the goal is to determine the source that has generated the electric field from measurements of boundary values of the electric potential since for reasonable models the time variation of the electric and magnetic fields can be disregarded the mathematical modeling of the corresponding forward problem leads to an electrostatics problem with a current dipole source this is a singular problem since the current dipole model involves first order derivatives of a dirac delta measure its solution lies in l p for 1 p 3 2 in three dimensional domains and 1 p 2 in the two dimensional case we consider the numerical approximation of the forward problem by means of standard piecewise linear continuous finite elements we prove a priori error estimates in the l p norm then we propose a residual type a posteriori error estimator we prove that it is reliable and efficient namely it yields global upper and local lower bounds for the corresponding norms of the error finally we use this estimator to guide an adaptive procedure which is experimentally shown to lead to an optimal order of convergence

algebraic approach to two dimensional recursive digital filter synthesis an effective algorithm is developed for synthesizing two dimensional recursive digital filters which approximate prescribed ideal frequency response specifications the algorithm is based on an algebraic approach that uses the eigenvalue eigenvector decomposition of the ideal filter s excitation response matrix in conjunction with a recently developed signal enhancement method this results in a recursive filter which has a unit impulse that closely approximates the ideal filter s unit impulse response illustrative examples and comparisons to an existing technique are included

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

local error analysis for the stokes equations with a punctual source term the solution of the stokes problem with a punctual force in source term is not h 1 times mathbb l 2 and therefore the approximation by a finite element method is suboptimal in the case of poisson problem with a dirac mass in the right hand side an optimal convergence for the lagrange finite elements has been shown on a subdomain which excludes the singularity in mathbb l 2 norm by koppl and wohlmuth here we show a quasi optimal local convergence in h 1 times mathbb l 2 norm for a mathbb p k mathbb p k 1 finite element method k geqslant 2 and for the mathbb p 1 mathrm b mathbb p 1 the error is still analysed on a subdomain which does not contain the singularity the proof is based on local arnold and liu error estimates a weak version of aubin nitsche duality lemma applied to the stokes problem and discrete inf sup conditions these theoretical results are generalized to a wide class of finite element methods before being illustrated by numerical simulations

a discontinuous galerkin method to solve the eeg forward problem using the subtraction approach in order to perform electroencephalography eeg source reconstruction i e to localize the sources underlying a measured eeg the electric potential distribution at the electrodes generated by a dipolar current source in the brain has to be simulated which is the so called eeg forward problem to solve it accurately it is necessary to apply numerical methods that are able to take the individual geometry and conductivity distribution of the subject s head into account in this context the finite element fe method fem has shown high numerical accuracy with the possibility to model complex geometries and conductive features e g white matter conductivity anisotropy in this article we introduce and analyze the application of a discontinuous galerkin dg method an fem that includes features of the finite volume framework to the eeg forward problem the dg fem approach fulfills the conservation property of electric charge also in the discrete case making it attractive for a variety of applicat

robust error estimates for the finite element approximation of elliptic optimal control problems in this paper we consider the finite element approximation of an elliptic optimal control problem based on an assumption on the adjoint state of the continuous problem with a small parameter which represents a regularization of the bang bang type control problem we derive robust a priori error estimates for optimal control and state and a posteriori error estimate is also presented numerical experiments confirm our theoretical results

numerical mathematics of the subtraction method for the modeling of a current dipole in eeg source reconstruction using finite element head models in electroencephalography eeg source analysis a dipole is widely used as the model of the current source the dipole introduces a singularity on the right hand side of the governing poisson type differential equation that has to be treated specifically when solving the equation toward the electric potential in this paper we give a proof for existence and uniqueness of the weak solution in the function space of zero mean potential functions using a subtraction approach the method divides the total potential into a singularity and a correction potential the singularity potential is due to a dipole in an infinite region of homogeneous conductivity we then state convergence properties of the finite element fe method for the numerical solution to the correction potential we validate our approach using tetrahedra and regular and geometry conforming node shifted hexahedra elements in an isotropic three layer sphere model and a model with anisotropic middle compartment validation is carried out using sophisticated visualization techniques correlation coefficient cc and magnification factor mag for a comparison of the numerical results with analytical series expansion formulas at the surface and within the volume conductor for the subtraction approach with regard to the accuracy in the anisotropic three layer sphere model cc of 0 998 or better and mag of 4 3 or better over the whole range of realistic eccentricities and to the computational complexity 2mm node shifted hexahedra achieve the best results a relative fe solver accuracy of 10 4 is sufficient for the used algebraic multigrid preconditioned conjugate gradient approach finally we visualize the computed potentials of the subtraction method in realistically shaped fe head volume conductor models with anisotropic skull compartments

performance of linear space search algorithms abstract search algorithms that use space linear in the search depth are widely employed in practice to solve difficult problems optimally such as planning and scheduling in this paper we study the average case performance of linear space search algorithms including depth first branch and bound dfbnb iterative deepening id and recursive best first search rbfs to facilitate our analyses we use a random tree t b d that has mean branching factor b depth d and node costs that are the sum of the costs of the edges from the root to the nodes we prove that the expected number of nodes expanded by dfbnb on a random tree is no more than bd times the expected number of nodes expanded by best first search bfs on the same tree which usually requires space that is exponential in depth d we also show that dfbnb is asymptotically optimal when bfs runs in exponential time and id and rbfs are asymptotically optimal when the edge costs of t b d are integers if bp 0 is the expected number of children of a node whose costs are the same as that of their parent then the expected number of nodes expanded by these three linear space algorithms is exponential when bp o d 4 when bp 0 1 and at most quadratic when bp 0 1 in addition we study the heuristic branching factor of t b d and the effective branching factor of bfs dfbnb id and rbfs on t b d furthermore we use our analytic results to explain a surprising anomaly in the performance of these algorithms and to predict the existence of a complexity transition in the asymmetric traveling salesman problem

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

adaptive hpq finite element methods for the analysis of 3d based models of complex structures part 2 a posteriori error estimation abstract this is the second paper out of a series of papers devoted to model and hpq adaptive finite element methods assigned for the modeling and analysis of elastic structures of complex mechanical description in our previous publication we investigated the issue of hierarchical models and approximations of such structures we applied 3d or 3d based mechanical models hierarchical modeling and hierarchical approximations within the proposed finite element formulation furthermore we assumed that the mechanical model and discretization parameters such as the size h of the element and the longitudinal and transverse approximation orders p and q could vary locally i e they could be different in each finite element the a posteriori error estimation discussed in the present paper is based on the generalization of the residual equilibration method on the models with internal constraints the generalized method is applied to the assessment of the total and approximation errors while the modeling error is calculated as the difference between the former two errors the corresponding error controlled adaptive procedures are based on a three step strategy with possible iterations of h and p steps

adaptive finite element methods for an optimal control problem involving dirac measures the purpose of this work is the design and analysis of a reliable and efficient a posteriori error estimator for the so called pointwise tracking optimal control problem this linear quadratic optimal control problem entails the minimization of a cost functional that involves point evaluations of the state thus leading to an adjoint problem with dirac measures on the right hand side control constraints are also considered the proposed error estimator relies on a posteriori error estimates in the maximum norm for the state and in muckenhoupt weighted sobolev spaces for the adjoint state we present an analysis that is valid for two and three dimensional domains we conclude by presenting several numerical experiments which reveal the competitive performance of adaptive methods based on the devised error estimator

design of iir eigenfilters in the frequency domain the eigenfilter approach is an appealing way of designing digital filters mainly because of the simplicity of its implementation in this correspondence a new method of applying the eigenfilter approach to the design of infinite impulse response iir filters is described the procedure works in the frequency domain and yields the coefficients of a causal rational transfer function having an arbitrary number of poles and zeros some examples of filter design are given to show the effectiveness of the method presented

an a posteriori error analysis of an elliptic optimal control problem in measure space abstract we propose an a posteriori error estimator for an optimal control problem with sparsity promoting cost the control variable lies in the space of regular borel measures we consider a solution technique that relies on the discretization of the control variable as a linear combination of dirac measures the proposed a posteriori error estimator can be decomposed into the sum of two contributions an error estimator in the maximum norm for the discretization of the adjoint equation and an estimator in the l 2 norm that accounts for the approximation of the state equation we prove that the designed error estimator is locally efficient and we explore its reliability properties the analysis is valid for two and three dimensional domains we illustrate the theory with numerical examples

greenʼs function estimates for a singularly perturbed convection diffusion problem we consider a singularly perturbed convection diusion problem posed in the unit square with a horizontal convective direction its solutions exhibit parabolic and exponential boundary layers sharp estimates of the green s function and its first and second order derivatives are derived in the l1 norm the dependence of these estimates on the small diusion parameter is shown explicitly the obtained estimates will be used in a forthcoming numerical analysis of the considered problem

a full subtraction approach for finite element method based source analysis using constrained delaunay tetrahedralisation a mathematical dipole is widely used as a model for the primary current source in electroencephalography eeg source analysis in the governing poisson type differential equation the dipole leads to a singularity on the right hand side which has to be treated specifically in this paper we will present a full subtraction approach where the total potential is divided into a singularity and a correction potential the singularity potential is due to a dipole in an infinite region of homogeneous conductivity the correction potential is computed using the finite element fe method special care is taken in order to evaluate the right hand side integral appropriately with the objective of achieving highest possible convergence order for linear basis functions our new approach allows the construction of transfer matrices for fast computation of the inverse problem for anisotropic volume conductors a constrained delaunay tetrahedralisation cdt approach is used for the generation of high quality fe meshes we validate the new approach in a four layer sphere model with a highly conductive cerebrospinal fluid csf and an anisotropic skull compartment for radial and tangential sources with eccentricities up to 1 mm below the csf compartment we achieve a maximal relative error of 0 71 in a cdt fe model with 360 k nodes which is not locally refined around the source singularity and therefore useful for arbitrary dipole locations the combination of the full subtraction approach with the high quality cdt meshes leads to accuracies that to the best of the author s knowledge have not yet been presented before

a posteriori error analysis for a class of integral equations and variational inequalities we consider elliptic and parabolic variational equations and inequalities governed by integro differential operators of order 2s in 0 2 our main motivation is the pricing of european or american options under levy processes in particular pure jump processes or jump diffusion processes with tempered stable processes the problem is discretized using piecewise linear finite elements in space and the implicit euler method in time we construct a residual type a posteriori error estimator which gives a computable upper bound for the actual error in h s norm the estimator is localized in the sense that the residuals are restricted to the discrete non contact region numerical experiments illustrate the accuracy of the space and time estimators and show that they can be used to measure local errors and drive adaptive algorithms

a posteriori error estimates for nonlinear problems finite element discretizations of elliptic equations we give a general framework for deriving a posteriori error estimates for approximate solutions of nonlinear problems in a first step it is proven that the error of the approximate solution can be bounded from above and from below by an appropriate norm of its residual in a second step this norm of the residual is bounded from above and from below by a similar norm of a suitable finite dimensional approximation of the residual this quantity can easily be evaluated and for many practical applications sharp explicit upper and lower bounds are readily obtained the general results are then applied to finite element discretizations of scalar quasi linear elliptic partial differential equations of 2nd order the eigenvalue problem for scalar linear elliptic operators of 2nd order and the stationary incompressible navier stokes equations they immediately yield a posteriori error estimates which can easily be computed from the given data of the problem and the computed numerical solution and which give global upper and local lower bounds on the error of the numerical solution

a study of complexity transitions on the asymmetric traveling salesman problem abstract the traveling salesman problem tsp is one of the best known combinatorial optimization problems branch and bound bnb is the best method for finding an optimal solution of the tsp previous research has shown that there exists a transition in the average computational complexity of bnb on random trees we show experimentally that when the intercity distances of the asymmetric tsp are drawn uniformly from 0 1 2 r the complexity of bnb experiences an easy hard transition as r increases we also observe easy hard easy complexity transitions when asymmetric intercity distances are chosen from a log normal distribution this transition pattern is similar to one previously observed on the symmetric tsp we then explain these different transition patterns by showing that the control parameter that determines the complexity is the number of distinct intercity distances

the method of fundamental solutions in solving coupled boundary value problems for m eeg the estimation of neuronal activity in the human brain from electroencephalography eeg and magnetoencephalography meg signals is a typical inverse problem whose solution process requires an accurate and fast forward solver in this paper the method of fundamental solutions is for the first time proposed as a meshfree boundary type and easy to implement alternative to the boundary element method bem for solving the m eeg forward problem the solution of the forward problem is obtained by numerically solving a set of coupled boundary value problems for the three dimensional laplace equation numerical accuracy convergence and computational load are investigated the proposed method is shown to be a competitive alternative to the state of the art bem for m eeg forward solving

a posteriori error estimates for optimal control problems with state and control constraints we discuss the full discretization of an elliptic optimal control problem with pointwise control and state constraints we provide the first reliable a posteriori error estimator that contains only computable quantities for this class of problems moreover we show that the error estimator converges to zero if one has convergence of the discrete solutions to the solution of the original problem the theory is illustrated by numerical tests

an adaptive finite element method for linear elliptic problems on propose une methode adaptative d elements finis basee sur une estimation d erreur optimale de norme maxima pour les problemes elliptiques lineaires

on the stochastic complexity of the asymmetric traveling salesman problem abstract based on the assignment problem relaxation an approach is proposed for the stochastic complexity of the asymmetric traveling salesman problem for an essential subproblem it results that the stochastic complexity is polynomial bounded

an adaptive stabilized finite element scheme for a water quality model abstract residual type a posteriori error estimators are introduced in this paper for an advection diffusion reaction problem with a dirac delta source term the error is measured in an adequately weighted w 1 p norm these estimators are proved to yield global upper and local lower bounds for the corresponding norms of the error they are used to guide adaptive procedures which are experimentally shown to lead to optimal orders of convergence