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

a system of matrix equations and a linear matrix equation over arbitrary regular rings with identity abstract in this paper we consider the classical system of matrix equations a 1 xb 1 c 1 a 2 xb 2 c 2 over r an arbitrary regular ring with identity necessary and sufficient conditions for the existence and the expression of the general solution to the system are derived as an application the linear matrix equation axb cyd e over r is considered

modules and matrices abstract noncommutative ring theory was described in terms of matrices in its earliest days we give some examples showing how matrix theory can now in turn profit from the theory of rings and modules

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

hybrid zero dynamics of planar biped walkers planar underactuated biped walkers form an important domain of applications for hybrid dynamical systems this paper presents the design of exponentially stable walking controllers for general planar bipedal systems that have one degree of freedom greater than the number of available actuators the within step control action creates an attracting invariant set a two dimensional zero dynamics submanifold of the full hybrid model whose restriction dynamics admits a scalar linear time invariant return map exponentially stable periodic orbits of the zero dynamics correspond to exponentially stabilizable orbits of the full model a convenient parameterization of the hybrid zero dynamics is imposed through the choice of a class of output functions parameter optimization is used to tune the hybrid zero dynamics in order to achieve closed loop exponentially stable walking with low energy consumption while meeting natural kinematic and dynamic constraints the general theory developed in the paper is illustrated on a five link walker consisting of a torso and two legs with knees

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

consistency for bi skew symmetric solutions to systems of generalized sylvester equations over a finite central algebra let be a finite dimensional central algebra with an involutorial antiautomorphism and char 2 n n be the set of all n n matrices over a aij n n is called bisymmetric if aij an i 1 n j 1 aji and biskewsymmetric if aij an i 1 n j 1 aji the following systems of generalized sylvester equations over a1x yb1 c1 i asx ybs cs a1xb1 c1xd1 e1 ii asxbs csxds es are considered necessary and sufficient conditions are given for the existence of constant solutions with bi skew symmetric constrains to i and ii as a particular case auxiliary results dealing with the system of sylvester equations are also presented

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

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

roth s theorems for sets of matrices abstract it is shown that roth s theorems on the equivalence and similarity of block diagonal matrices hold for finite sets of matrices over a commutative ring

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

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

the extension of roth s theorem for matrix equations over a ring abstract this paper extends roth s similarity theorem as follows let r be a ring with identity b ki 0 bi i rm q if either r is a division ring and a rn n is algebraic or r is finitely generated as a module over its center then the matrix equation ki 0 aixbi c over r has a solution if and only if

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

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

matrix equivalence and isomorphism of modules abstract it is well known that if a and b are n m matrices over a ring r then coker a coker b does not imply a and b are equivalent an elementary proof is given that the implication does hold if 1 is in the stable range of r furthermore for certain r including commutative rings if a is block diagonal and b is block upper triangular with the same diagonal blocks as a then coker a coker b implies a and b are equivalent under a special equivalence this extends results of roth and gustafson as a corollary a theorem on decomposition of modules is obtained

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

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

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 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

roth s theorems and decomposition of modules it is shown that there is a connection between roth s theorems on similarity and equivalence of block triangular matrices and decomposition of modules the module property is that if m n mn then n is a summand of m this holds for any commutative ring if m is finitely presented new proofs of roth s theorems are given for commutative rings some results are established in the noncommutative case

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

switching and pi control of walking motions of planar biped walkers a companion paper has addressed the problem of designing controllers that induce exponentially stable periodic walking motions at a fixed walking rate for a planar biped robot with one degree of underactuation this note provides two additional control features 1 the ability to compose such controllers to obtain walking at several discrete walking rates with guaranteed stability during the transitions and 2 the ability to regulate the average walking rate to a continuum of values

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

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

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