a complex variable boundary point interpolation method for the nonlinear signorini problem abstract a complex variable boundary point interpolation method cvbpim is presented for numerical analysis of the nonlinear signorini problem to reduce the computational cost and to improve the stability of the point interpolation method a complex variable point interpolation method cvpim is developed to construct meshless shape functions with interpolation properties by using a projection technique to deal with the nonlinear inequality boundary conditions and combining boundary integral equations with the cvpim to establish discrete linear algebraic systems the cvbpim is an easy to implement and boundary only meshless method for nonlinear signorini problems numerical results are presented to show the accuracy and efficiency of the method

the convergence of spline collocation for strongly elliptic equations on curves most boundary element methods for two dimensional boundary value problems are based on point collocation on the boundary and the use of splines as trial functions here we present a unified asymptotic error analysis for even as well as for odd degree splines subordinate to uniform or smoothly graded meshes and prove asymptotic convergence of optimal order the equations are collocated at the breakpoints for odd degree and the internodal midpoints for even degree splines the crucial assumption for the generalized boundary integral and integro differential operators is strong ellipticity our analysis is based on simple fourier expansions in particular we extend results by j saranen and w l wendland from constant to variable coefficient equations our results include the first convergence proof of midpoint collocation with piecewise constant functions i e the panel method for solving systems of cauchy singular integral equations

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 bie method for a nonlinear bvp abstract a nonlinear boundary value problem bvp governed by laplace s equation with a nonlinear boundary condition is considered this paper is concerned with the numerical solvability of a nonlinear boundary integral equation bie obtained by reformulating the nonlinear bvp we give a simple alternative to the standard collocation method for the nonlinear bie which requires less computational effort the method is a bie analogue of the product approximation method used for solving some nonlinear problems by the finite element method we obtain superconvergence of approximate solutions introduced in this paper

a boundary element method for signorini problems in three dimensions in this paper a signorini problem in three dimensions is reduced to a variational inequality on the boundary and a boundary element method is described for the numerical approximation of its solution an optimal error estimate is also given

numerical verification of solutions for a simplified signorini problem abstract we proposed some numerical methods for the automatic proof of existence of solutions for variational inequalities i e obstacle problems and elastoplastic torsion problems it is based on nakao s method by using finite element approximation and its explicit error estimates for the problem in this paper we present a numerical verification algorithm of exact solutions for a simplified signorini problem in a computer a numerical example for the signorini problem is presented

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

primal dual strategy for constrained optimal control problems an algorithm for efficient solution of control constrained optimal control problems is proposed and analyzed it is based on an active set strategy involving primal as well as dual variables for discretized problems sufficient conditions for convergence in finitely many iterations are given numerical examples are given and the role of the strict complementarity condition is discussed

error estimates for the approximation of semicoercive variational inequalities an abstract error estimate for the approximation of semicoercive variational inequalities is obtained provided a certain condition holds for the exact solution this condition turns out to be necessary as is demonstrated analytically and numerically the results are applied to the finite element approximation of poisson s equation with signorini boundary conditions and to the obstacle problem for the beam with no fixed boundary conditions for second order variational inequalities the condition is always satisfied whereas for the beam problem the condition holds if the center of forces belongs to the interior of the convex hull of the contact set applying the error estimate yields optimal order of convergence in terms of the mesh size h the numerical convergence rates observed are in good agreement with the predicted ones

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

numerical verification of solutions for some unilateral boundary value problems abstract in this paper we consider a numerical technique which enables us to verify the existence of solutions for some unilateral boundary value problems using the finite element approximations and explicit a priori error estimates we construct in a computer a set of solutions which satisfies the hypothesis of schauder s fixed point theorem for a compact map on a certain sobolev space further the conditions of verifiability by this method are considered and some numerical examples are presented

on the boundary element method for the signorini problem of the laplacian for the laplace equation with signorini boundary conditions two equivalent boundary variational inequality formulations are deduced we investigate the discretization by a boundary element galerkin method and obtain quasi optimal asymptotic error estimates in the underlying sobolev spaces an algorithm based on the decomposition coordination method is used to solve the discretized problems numerical examples confirm the predicted rate of convergence

stabilized mixed hp bem for frictional contact problems in linear elasticity we analyze stabilized mixed hp boundary element methods for frictional contact problems for the lame equation the stabilization technique circumvents the discrete inf sup condition for the mixed problem and thus allows us to use the same mesh and polynomial degree for the primal and dual variables we prove a priori convergence rates in the case of tresca friction using gauss legendre lagrange polynomials as test and trial functions for the lagrange multiplier additionally a residual based a posteriori error estimate for a more general class of discretizations is derived it in particular applies to discretizations based on bernstein polynomials for the discrete lagrange multiplier which we also analyze the discretization and the a posteriori error estimate are extended to the case of coulomb friction several numerical experiments underline our theoretical results demonstrate the behavior of the method and its insensitivity to the scaling and perturbations of the stabilization term

information theoretic wavelet noise removal for inverse elastic wave scattering theory a discussion of noise removal in ultrasound elastic wave scattering for nondestructive evaluation is given the methods used in this paper include a useful suboptimal wiener filter information theory and orthonormal wavelets the multiresolution analysis mra due to mallat is the key wavelet feature used here whereas fourier transforms have a translational symmetry wavelets have a dilation or affine symmetry which consists of the semi direct product of a translation with a change of scale of the variable the mra describes the scale change features of orthonormal wavelet families first an empirical method of noise removal from scattered elastic waves using wavelets is shown to markedly improve the l 1 and l 2 error norms this suggests that the wavelet scale can act as dial to tune out noise maximization of the kullback liebler information is also shown to provide a scale dependent noise removal technique that supports but does not prove the intuition that certain small energy coefficients that are retained contain large information content the wavelet mra thereby locates islands of information in the phase space of the signal it is conjectured that this method holds more generally

an energy space finite element approach for elliptic dirichlet boundary control problems in this paper we present a finite element analysis for a dirichlet boundary control problem where the dirichlet control is considered in a convex closed subspace of the energy space h 1 2 gamma h 1 2 as an equivalent norm in h 1 2 gamma h 1 2 we use the energy norm induced by the so called steklov poincare operator which realizes the dirichlet to neumann map and which can be implemented by using standard finite element methods the presented stability and error analysis of the discretization of the resulting variational inequality is based on the mapping properties of the solution operators related to the primal and adjoint boundary value problems and their finite element approximations some numerical results are given which confirm on one hand the theoretical estimates but on the other hand indicate the differences when modelling the control in l 2 gamma l 2

a note on a priori mathbf l p error estimates for the obstacle problem this paper is concerned with a priori error estimates for the piecewise linear finite element approximation of the classical obstacle problem we demonstrate by means of two one dimensional counterexamples that the l 2 error between the exact solution u and the finite element approximation u h is typically not of order two even if the exact solution is in h 2 varomega and an estimate of the form vert u u h vert h 1 le ch holds true this shows that the classical aubin nitsche trick which yields a doubling of the order of convergence when passing over from the h 1 to the l 2 norm cannot be generalized to the obstacle problem

smoothing methods and semismooth methods for nondifferentiable operator equations we consider superlinearly convergent analogues of newton methods for nondifferentiable operator equations in function spaces the superlinear convergence analysis of semismooth methods for nondifferentiable equations described by a locally lipschitzian operator in rn is based on rademacher s theorem which does not hold in function spaces we introduce a concept of slant differentiability and use it to study superlinear convergence of smoothing methods and semismooth methods in a unified framework we show that a function is slantly differentiable at a point if and only if it is lipschitz continuous at that point an application to the dirichlet problems for a simple class of nonsmooth elliptic partial differential equations is discussed

coupling regularization and adaptive hp bem for the solution of a delamination problem in this paper we couple regularization techniques with the adaptive hp version of the boundary element method hp bem for the efficient numerical solution of linear elastic problems with nonmonotone contact boundary conditions as a model example we treat the delamination of composite structures with a contaminated interface layer this problem has a weak formulation in terms of a nonsmooth variational inequality the resulting hemivariational inequality is first regularized and then discretized by an adaptive hp bem we give conditions for the uniqueness of the solution and provide an a priori error estimate furthermore we prove the very first a posteriori error estimate for the nonsmooth variational problem utilizing a novel mixed regularized formulation thus enabling hp adaptivity various numerical experiments illustrate the behavior strengths and limitations of the proposed high order approximation scheme

fem and bem coupling for a nonlinear transmission problem with signorini contact this paper is concerned with the interface problem consisting of a nonlinear uniformly monotone partial differential equation in some bounded lipschitz domain omega in reals n n 2 or n geq 3 and the laplace equation with some radiation condition in the unbounded exterior domain omega c reals n setminus overline omega the two problems are coupled by transmission conditions and signorini contact conditions on the interface gamma partial omega the exterior part of the interface problem is rewritten in terms of boundary integral operators this leads to a variational inequality with a nonlinear monotone operator by this approach existence and uniqueness of a solution in appropriate sobolev spaces are obtained its approximation is performed by coupling the finite element method fem in omega and the boundary element method bem the latter living on the interface gamma yielding a discrete monotone variational inequality we present an abstract cea type error estimate and derive asymptotic error estimates

boundary element methods for variational inequalities in this paper we present a priori error estimates for the galerkin solution of variational inequalities which are formulated in fractional sobolev trace spaces i e in widetilde h 1 2 gamma h 1 2 in addition to error estimates in the energy norm we also provide by applying the aubin nitsche trick for variational inequalities error estimates in lower order sobolev spaces including l 2 gamma l 2 the resulting discrete variational inequality is solved by using a semi smooth newton method which is equivalent to an active set strategy a numerical example is given which confirms the theoretical results

solution of nonlinear elliptic equations with boundary singularities by an integral equation method a boundary integral equation bie formulation is presented for the numerical solution of certain two dimensional nonlinear elliptic equations subject to nonlinear boundary conditions by applying the kirchoff transformation all nonlinear aspects are first transferred to the boundary of the solution domain then the accurate solution of problems in which there are boundary singularities is demonstrated by including the analytic nature of the singular solution in only those regions nearest the singularity because of this only a minor modification of the classical nonlinear bie is required and this results in a substantial improvement in the accuracy of the numerical results throughout the entire solution domain

numerical treatment of acoustic problems with boundary singularities by the singular boundary method abstract the singular boundary method sbm is a novel boundary type meshless method based on the fundamental solution of the given governing equation the sbm employs the origin intensity factors to circumvent the singularities resulting from the fundamental solutions in this paper we investigate the acoustic problems with boundary singularities using the sbm this is achieved by combining the sbm with the singularity subtraction techniques where the solution is decomposed into the singular solution and the regular solution the singular solution is derived analytically which satisfies the governing equation and the corresponding boundary conditions containing the singularities then the regular solution is obtained by the sbm numerical examples show the excellent performance of the proposed technique

analysis of the element free galerkin method for signorini problems abstract an efficient element free galerkin efg method is developed to solve signorini boundary value problems the nonlinear inequality signorini boundary conditions are reduced to a sequence of linear boundary conditions by using a projection iterative scheme then the efg method is used to deduce discrete system equations convergence of the proposed method is discussed numerical examples are finally provided to verify the convergence and high computational accuracy of the method

a numerical algorithm for the solution of signorini problems in this paper we propose a new iterative algorithm for the solution of a certain class of signorini problems such problems arise in the modelling of a variety of physical phenomena and usually involve the determination of an unknown free boundary here we describe a way of locating the free boundary directly and provide a proof that the algorithm converges when used with analytic methods the advantage of this algorithm is that it can be used in conjunction with any numerical method with minimal development of extra code we demonstrate its application with the boundary element method to some physical problems in both two and three dimensions

two projection methods for the solution of signorini problems abstract two iterative methods based on projection and boundary element methods are considered for signorini problems the regularized problem with the projection boundary condition is first deduced from the signorini problem by using the equivalence between the signorini boundary condition and the projection fixed point problem our methods formulate the signorini boundary condition into a sequence of robin boundary conditions in the new boundary condition there is a penalty parameter the convergence speed of the first method is greatly influenced by the value of which is difficult to choose for individual problems to improve the performance of this method we present a self adaptive projection method which adjusts the parameter automatically per iteration based on the iterative data the main result of this work is to provide the convergence of the methods under mild assumptions as the iteration process is given by the potential and its derivative on the boundary of the domain the unknowns of the problem are computed explicitly by using the boundary element method both theoretical results and numerical experiments indicate efficiency of the methods proposed

differential geometric inference in surface stereo many traditional two view stereo algorithms explicitly or implicitly use the frontal parallel plane assumption when exploiting contextual information since e g the smoothness prior biases toward constant disparity depth over a neighborhood this introduces systematic errors to the matching process for slanted or curved surfaces these errors are nonnegligible for detailed geometric modeling of natural objects such as a human face we show how to use contextual information geometrically to avoid such errors a differential geometric study of smooth surfaces allows contextual information to be encoded in cartan s moving frame model over local quadratic approximations providing a framework of geometric consistency for both depth and surface normals the accuracy of our reconstructions argues for the sufficiency of the approximation in effect cartan s model provides the additional constraint necessary to move beyond the frontal parallel plane assumption in stereo reconstruction it also suggests how geometry can extend surfaces to account for unmatched points due to partial occlusion

on the boundary element method for some nonlinear boundary value problems here we analyse the boundary element galerkin method for two dimensional nonlinear boundary value problems governed by the laplacian in an interior or exterior domain and by highly nonlinear boundary conditions the underlying boundary integral operator here can be decomposed into the sum of a monotoneous hammerstein operator and a compact mapping we show stability and convergence by using leray schauder fixed point arguments due to petryshyn and necas

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

on existence analysis of steady flows of generalized newtonian fluids with concentration dependent power law index abstract we study a system of partial differential equations describing a steady flow of an incompressible generalized newtonian fluid wherein the cauchy stress is concentration dependent namely we consider a coupled system of the generalized navier stokes equations and convection diffusion equation with non linear diffusivity we prove the existence of a weak solution for certain class of models by using a generalization of the monotone operator theory which fits into the framework of generalized sobolev spaces with variable exponent such a framework is involved since the function spaces where we look for the weak solution are dependent of the solution itself and thus we a priori do not know them this leads us to the principal a priori assumptions on the model parameters that ensure the holder continuity of the variable exponent we present here a constructive proof based on the galerkin method that allows us to obtain the result for very general class of models

exact optimization for markov random fields with convex priors we introduce a method to solve exactly a first order markov random field optimization problem in more generality than was previously possible the mrf has a prior term that is convex in terms of a linearly ordered label set the method maps the problem into a minimum cut problem for a directed graph for which a globally optimal solution can be found in polynomial time the convexity of the prior function in the energy is shown to be necessary and sufficient for the applicability of the method

an extrapolation method for a class of boundary integral equations boundary value problems of the third kind are converted into boundary integral equations of the second kind with periodic logarithmic kernels by using green s formulas for solving the induced boundary integral equations a nystrom scheme and its extrapolation method are derived for periodic fredholm integral equations of the second kind with logarithmic singularity asymptotic expansions for the approximate solutions obtained by the nystrom scheme are developed to analyze the extrapolation method some computational aspects of the methods are considered and two numerical examples are given to illustrate the acceleration of convergence

proper orthogonal decomposition variational multiscale element free galerkin pod vmefg meshless method for solving incompressible navier stokes equation abstract recently because of simplicity of meshless methods they have been employed for solving many partial differential equations one of the meshless methods is element free galerkin technique the element free galerkin is very similar to the finite element method with this difference that the test and trial spaces of efg procedure are shape functions of moving least squares approximation using the mentioned shape functions solving a problem on a complex domain is very simple some researchers proposed several modifications and enriched approaches for improving the element free galerkin method that one of them is variational multiscale element free galerkin procedure up to the best of authors knowledge the element free galerkin method based on the shape functions of moving least squares approximation needs more cpu time than the element free galerkin method based on the shape functions of moving kriging interpolation thus in the current paper we employ the variational multiscale element free galerkin based on the shape functions of moving kriging interpolation also for reducing the cpu time of the presented numerical scheme we use the proper orthogonal decomposition pod approach therefore in the current paper we propose the proper orthogonal decomposition variational multiscale element free galerkin pod vmefg method for solving time dependent incompressible navier stokes equation moreover several test problems are given that show the acceptable accuracy and efficiency of the proposed scheme

minimizing nonsubmodular functions with graph cuts a review optimization techniques based on graph cuts have become a standard tool for many vision applications these techniques allow to minimize efficiently certain energy functions corresponding to pairwise markov random fields mrfs currently there is an accepted view within the computer vision community that graph cuts can only be used for optimizing a limited class of mrf energies e g submodular functions in this survey we review some results that show that graph cuts can be applied to a much larger class of energy functions in particular nonsubmodular functions while these results are well known in the optimization community to our knowledge they were not used in the context of computer vision and mrf optimization we demonstrate the relevance of these results to vision on the problem of binary texture restoration

energy corrected fem for optimal dirichlet boundary control problems in the presence of re entrant corners the solution of pde constrained optimization problems in general has singular components even when the given data are smooth consequently the use of standard finite element discretization gives rise to the so called pollution effect this means that only a reduced convergence rate compared to the best approximation error is obtained we discuss how optimal convergence rates in weighted norms can be achieved using the idea of energy corrected finite element methods applied to optimal dirichlet boundary control problem in the energy space we present optimal error estimates in weighted norms for the state variable and for the control several numerical examples illustrate the obtained theoretical results

a priori error estimates for hp penalty bem for contact problems in elasticity the purpose of the paper is to obtain a priori error estimates for the hp version of penalty galerkin bem applied to frictionless contact problems in elasticity the error analysis is divided into two parts at first we consider the error caused by the approximation of the variational inequality or lagrange multiplier formulation with the penalty problem under additional regularity assumptions we derive a linear convergence rate with respect to the penalty parameter then the discretization error between the solution of the penalty problem and its galerkin approximation is considered we show two types of the best approximation property which is similar to the cea s lemma but the estimate depends on the penalty parameter finally we derive an a priori estimate for the error between the exact solution u of the variational inequality and the boundary element galerkin solution of the penalty problem for u h 3 2 c n we obtain the convergence rate o h p 1 when choosing the penalty parameter e c h p 1 for arbitrary fixed 0 1 and c 0

projection methods for a class of hammerstein equations here some new convergence results for a class of hammerstein equations are given a particular example appearing in the solution of a steady state heat conduction problem with a nonlinear boundary condition was recently studied by ruotsalainen and wendland numer math 53 1988 pp 299 314 using the galerkin approximation and by ruotsalainen and saranen jcam 28 1989 pp 339 348 using collocation this paper improves the convergence results in both of these works the general result also applies to some time dependent nonlinear model problems

on the asymptotic convergence of collocation methods we prove quasioptimal and optimal order estimates in various sobolev norms for the approximation of linear strongly elliptic pseudodifferential equations in one independent variable by the method of nodal collocation by odd degree polynomial splines the analysis pertains in particular to many of the boundary element methods used for numerical computation in engineering applications equations to which the analysis is applied include fredholm integral equations of the second kind certain first kind fredholm equations singular integral equations involving cauchy kernels a variety of integro differential equations and two point boundary value problems for ordinary differential equations the error analysis is based on an equivalence which we establish between the collocation methods and certain nonstandard galerkin methods we compare the collocation method with a standard galerkin method using splines of the same degree showing that the galerkin method is quasioptimal in a sobolev space of lower index and furnishes optimal order approximation for a range of sobolev indices containing and extending below that for the collocation method and so the standard galerkin method achieves higher rates of convergence

boundary element free method befm and its application to two dimensional elasticity problems in this study we first discuss the moving least square approximation mls method in some cases the mls may form an ill conditioned system of equations so that the solution cannot be correctly obtained hence in this paper we propose an improved moving least square approximation imls method in the imls method the orthogonal function system with a weight function is used as the basis function the imls has higher computational efficiency and precision than the mls and will not lead to an ill conditioned system of equations combining the boundary integral equation bie method and the imls approximation method a direct meshless bie method the boundary element free method befm for two dimensional elasticity is presented compared to other meshless bie methods befm is a direct numerical method in which the basic unknown quantity is the real solution of the nodal variables and the boundary conditions can be applied easily hence it has higher computational precision for demonstration purpose selected numerical examples are given

a meshless galerkin scheme for the approximate solution of nonlinear logarithmic boundary integral equations utilizing radial basis functions abstract this paper presents a computational scheme to solve nonlinear logarithmic singular boundary integral equations these types of integral equations arise from boundary value problems of laplace s equations with nonlinear robin boundary conditions the discrete galerkin method together with the inverse multiquadric radial basis functions established on scattered points is utilized to approximate the solution the discrete galerkin method for solving boundary integral equations results from the numerical integration of all integrals in the method the proposed scheme uses a special accurate quadrature formula via the nonuniform gauss legendre integration rule to compute logarithm like singular integrals appeared in the scheme since the numerical method developed in the current paper does not require any mesh generations on the boundary of the domain it is meshless and does not depend to the domain form we also investigate the error analysis of the proposed method illustrative examples show the reliability and efficiency of the new scheme and confirm the theoretical error estimates

a class of codimension two free boundary problems this review collates a wide variety of free boundary problems which are characterized by the uniform proximity of the free boundary to a prescribed surface such situations can often be approximated by mixed boundary value problems in which the boundary data switches across a codimension two free boundary namely the edge of the region obtained by projecting the free boundary normally onto the prescribed surface as in the parent problem the codimension two free boundary needs to be determined as well as the solution of the relevant field equations but no systematic methodology has yet been proposed for nonlinear problems of this type after presenting some examples to illustrate the surprising behavior that can sometimes occur we discuss the relevance of traditional ideas from the theories of moving boundary problems singular integral equations variational inequalities and stability finally we point out the ways in which further refinement of these techniques is needed if a coherent theory is to emerge

the boundary element free method for 2d interior and exterior helmholtz problems abstract the boundary element free method befm is developed in this paper for numerical solutions of 2d interior and exterior helmholtz problems with mixed boundary conditions of dirichlet and neumann types a unified boundary integral equation is established for both interior and exterior problems by using the improved interpolating moving least squares method to form meshless shape functions mixed boundary conditions in the befm can be satisfied directly and easily detailed computational formulas are derived to compute weakly and strongly singular integrals over linear and higher order integration cells three numerical integration procedures are developed for the computation of strongly singular integrals numerical examples involving acoustic scattering and radiation problems are presented to show the accuracy and efficiency of the meshless method

finite element error estimates in non energy norms for the two dimensional scalar signorini problem this paper is concerned with error estimates for the piecewise linear finite element approximation of the two dimensional scalar signorini problem on a convex polygonal domain varomega using a cea type lemma a supercloseness result and a non standard duality argument we prove w 1 p varomega l infty varomega w 1 infty varomega and h 1 2 partial varomega error estimates under reasonable assumptions on the regularity of the exact solution and l p varomega error estimates under comparatively mild assumptions on the involved contact sets the obtained orders of convergence turn out to be optimal for problems with essentially bounded right hand sides our results are accompanied by numerical experiments which confirm the theoretical findings

fast approximate energy minimization via graph cuts many tasks in computer vision involve assigning a label such as disparity to every pixel a common constraint is that the labels should vary smoothly almost everywhere while preserving sharp discontinuities that may exist e g at object boundaries these tasks are naturally stated in terms of energy minimization the authors consider a wide class of energies with various smoothness constraints global minimization of these energy functions is np hard even in the simplest discontinuity preserving case therefore our focus is on efficient approximation algorithms we present two algorithms based on graph cuts that efficiently find a local minimum with respect to two types of large moves namely expansion moves and swap moves these moves can simultaneously change the labels of arbitrarily large sets of pixels in contrast many standard algorithms including simulated annealing use small moves where only one pixel changes its label at a time our expansion algorithm finds a labeling within a known factor of the global minimum while our swap algorithm handles more general energy functions both of these algorithms allow important cases of discontinuity preserving energies we experimentally demonstrate the effectiveness of our approach for image restoration stereo and motion on real data with ground truth we achieve 98 percent accuracy

using the wpg method for solving integral equations of the second kind abstract in this paper we use the wavelet petrov galerkin wpg method based on discontinuous orthogonal multiwavelets for solving fredholm integral equations of the second kind that yields linear system that the condition number of its coefficient matrix is bounded and for showing efficiency of method we use numerical examples

on the asymptotic convergence of collocation methods with spline functions of even degree we investigate the collocation of linear one dimensional strongly elliptic integro dif ferential or more generally pseudo differential equations on closed curves by even degree polynomial splines the equations are collocated at the respective midpoints subject to uniform nodal grids of the even degree s splines we prove quasioptimal and optimal order asymptotic error estimates in a scale of sobolev spaces the results apply in particular to boundary element methods used for numerical computations in engineering applications the equations considered include fredholm integral equations of the second and the first kind singular integral equations involving cauchy kernels and integro differential equations having convolutional or constant coefficient principal parts respectively the error analysis is based on an equivalence between the collocation and certain varia tional methods with different degree splines as trial and as test functions we further need to restrict our operators essentially to pseudo differential operators having convolutional prin cipal part this allows an explicit fourier analysis of our operators as well as of the spline spaces in terms of trigonometric polynomials providing babuska s stability condition based on strong ellipticity our asymptotic error estimates extend partly those obtained by d n arnold and w l wendland from the case of odd degree splines to the case of even degree splines 1 introduction in this paper we investigate the asymptotic convergence of the collocation method using even degree polynomial splines applied to strongly elliptic systems of pseudo differential equations on closed curves with convolutional prin cipal part the collocation here employs the gauss points of one point integration as collocation points which corresponds to the usual boundary element collocation in applications 5 12 17 19 26 31 34 this is in contrast to the method investigated by g schmidt in 29 where one collocates at the break points of even order splines the asymptotic convergence properties for the standard galerkin method with splines of arbitrary orders are well known 22 for the collocation method however asymptotic convergence up to now has been shown for strongly elliptic systems only in the case of odd order splines by d n arnold and w l wendland 7

a hypersingular boundary integral method for two dimensional screen and crack problems we analyze hypersingular integral equations on a curved open smooth arc in r 2 that model either curved cracks in an elastic medium or the scattering of acoustic and elastic waves at a hard screen by using the mellin transformation we obtain sharp regularity results for the solution of these equations in sobolev spaces in the form of singular expansions in particular we show that the expansions do not contain logarithmic singularities

continuous 3d label stereo matching using local expansion moves we present an accurate stereo matching method using local expansion moves based on graph cuts this new move making scheme is used to efficiently infer per pixel 3d plane labels on a pairwise markov random field mrf that effectively combines recently proposed slanted patch matching and curvature regularization terms the local expansion moves are presented as many alpha expansions defined for small grid regions the local expansion moves extend traditional expansion moves by two ways localization and spatial propagation by localization we use different candidate alpha labels according to the locations of local alpha expansions by spatial propagation we design our local alpha expansions to propagate currently assigned labels for nearby regions with this localization and spatial propagation our method can efficiently infer mrf models with a continuous label space using randomized search our method has several advantages over previous approaches that are based on fusion moves or belief propagation it produces submodular moves deriving a subproblem optimality it helps find good smooth piecewise linear disparity maps it is suitable for parallelization it can use cost volume filtering techniques for accelerating the matching cost computations even using a simple pairwise mrf our method is shown to have best performance in the middlebury stereo benchmark v2 and v3

on the stability of the moving least squares approximation and the element free galerkin method in this paper the stability of the moving least squares mls approximation and a stabilized mls approximation is analyzed theoretically and verified numerically it is shown that the stability of the mls approximation deteriorates severely as the nodal spacing decreases while the stability of the stabilized mls approximation is not affected by the nodal spacing the stabilized mls approximation is introduced into the element free galerkin efg method to produce a stabilized efg method theoretical error analysis of the stabilized efg method is provided for boundary value problems with mixed boundary conditions of dirichlet and robin type numerical examples confirm the theoretical results and show that the stabilized efg method has higher computational precision and better stability than the original efg method

boundary integral equation methods for solving laplace s equation with nonlinear boundary conditions the smooth boundary case a nonlinear boundary value problem for laplace s equation is solved numerically by using a reformulation as a nonlinear boundary integral equation two numerical methods are proposed and analyzed for discretizing the integral equation both using product integration to approximate the singu lar integrals in the equation the first method uses the product simpson s rule and the second is based on trigonometric interpolation iterative methods in cluding two grid methods for solving the resulting nonlinear systems are also discussed extensively numerical examples are included

fusion moves for markov random field optimization the efficient application of graph cuts to markov random fields mrfs with multiple discrete or continuous labels remains an open question in this paper we demonstrate one possible way of achieving this by using graph cuts to combine pairs of suboptimal labelings or solutions we call this combination process the fusion move by employing recently developed graph cut based algorithms so called qpbo graph cut the fusion move can efficiently combine two proposal labelings in a theoretically sound way which is in practice often globally optimal we demonstrate that fusion moves generalize many previous graph cut approaches which allows them to be used as building blocks within a broader variety of optimization schemes than were considered before in particular we propose new optimization schemes for computer vision mrfs with applications to image restoration stereo and optical flow among others within these schemes the fusion moves are used 1 for the parallelization of mrf optimization into several threads 2 for fast mrf optimization by combining cheap to compute solutions and 3 for the optimization of highly nonconvex continuous labeled mrfs with 2d labels our final example is a nonvision mrf concerned with cartographic label placement where fusion moves can be used to improve the performance of a standard inference method loopy belief propagation

joint entropy of continuously differentiable ultrasonic waveforms this study is based on an extension of the concept of joint entropy of two random variables to continuous functions such as backscattered ultrasound for two continuous random variables x and y the joint probability density p x y is ordinarily a continuous function of x and y that takes on values in a two dimensional region of the real plane however in the case where x f t and y g t are both continuously differentiable functions x and y are concentrated exclusively on a curve t f t g t in the x y plane this concentration can only be represented using a mathematically singular object such as a schwartz distribution its use for imaging requires a coarse graining operation which is described in this study subsequently removal of the coarse graining parameter is accomplished using the ergodic theorem the resulting expression for joint entropy is applied to several data sets showing the utility of the concept for both materials characterization and detection of targeted liquid nanoparticle ultrasonic contrast agents in all cases the sensitivity of these techniques matches or exceeds sometimes by a factor of two that demonstrated in previous studies that employed signal energy or alternate entropic quantities

a comparison of shannon entropy versus signal energy for acoustic detection of artificially induced defects in plexiglas an experimental comparison of scanned images formed from shannon entropy total energy and log energy measurements of acoustic wave propagation in a plexiglas bar with drill hole defects shows that shannon entropy in the instance investigated is approximately two to three times as sensitive to the presence of small defects than are the two energy measures this suggests that the shannon entropy technique may be useful for flaw detection in materials that are too thick or lossy for the conventional use of acoustic energy for flaw detection

a fem bem coupling method for a nonlinear transmission problem modelling coulomb friction contact a nonlinear transmission problem modelling elastoplastic interface problems with contact and coulomb friction is reduced to a boundary domain variational inequality we present a corresponding fem bem coupling procedure and derive for its galerkin solution an a priori error estimate furthermore we reformulate the problem as an equivalent saddle point problem whose discretization can be solved by the uzawa algorithm convergence of the fem bem coupling method is proved and numerical results are given

on the convergence of the collocation method for nonlinear boundary integral equations abstract recently galerkin and collocation methods have been analysed for some nonlinear boundary integral equations for the collocation method it has been assumed that the nonlinearity is asymptotically linear in this paper we remove this restriction we shall prove the convergence of the collocation method for nonlinear boundary integral equations when the nonlinearity has a polynomial growth condition in addition to this the optimal order error estimates follow in l q norm

singular boundary method for heat conduction problems with certain spatially varying conductivity the singular boundary method sbm is a recent boundary type meshless collocation method in which the solution of a given problem is expanded as a linear combination of the fundamental solutions in terms of the source points the method circumvents the fictitious boundary long perplexing the method of fundamental solution mfs by the introduction of the concept of origin intensity factors oifs this paper documents the first attempt to extend the method to heat conduction problems in nonhomogeneous materials we derive the fundamental solutions of heat conduction problems with the thermal conductivity of the quadratic exponential and trigonometric material variations in three directions furthermore we firstly theoretically derive the value of the oif for the natural logarithm function which is later extended to the oifs for a group of fundamental solutions the feasibility accuracy and stability of the presented sbm formulation are confirmed for both two 2d and three dimensional 3d examples

global stereo reconstruction under second order smoothness priors second order priors on the smoothness of 3d surfaces are a better model of typical scenes than first order priors however stereo reconstruction using global inference algorithms such as graph cuts has not been able to incorporate second order priors because the triple cliques needed to express them yield intractable nonsubmodular optimization problems this paper shows that inference with triple cliques can be effectively performed our optimization strategy is a development of recent extensions to alpha expansion based on the ldquo qpbordquo algorithm the strategy is to repeatedly merge proposal depth maps using a novel extension of qpbo proposal depth maps can come from any source for example frontoparallel planes as in alpha expansion or indeed any existing stereo algorithm with arbitrary parameter settings

on the collocation method for a nonlinear boundary integral equation abstract in this paper we study a potential problem with a nonlinear boundary condition using the green representation formula for a harmonic function we reformulate the nonlinear boundary value problem as a nonlinear boundary integral equation we shall give a brief discussion of the solvability of the integral equation the aim of this paper however is to analyse the collocation method for finding an approximate solution to this equation using the theory of a proper and a stable mappings we prove the unique solvability of the collocation equations and the asymptotic error estimates to do this we assume that the nonlinearity is strongly monotone

a direct boundary element method for signorini problems in this paper a signorini problem is reduced to a variational inequality on the boundary and a direct boundary element method is presented for its solution furthermore error estimates for the approximate solutions of signorini problems are given in addition we show that the signorini problem may be formulated as a saddle point problem on the boundary

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 comparative study of energy minimization methods for markov random fields with smoothness based priors among the most exciting advances in early vision has been the development of efficient energy minimization algorithms for pixel labeling tasks such as depth or texture computation it has been known for decades that such problems can be elegantly expressed as markov random fields yet the resulting energy minimization problems have been widely viewed as intractable algorithms such as graph cuts and loopy belief propagation lbp have proven to be very powerful for example such methods form the basis for almost all the top performing stereo methods however the trade offs among different energy minimization algorithms are still not well understood in this paper we describe a set of energy minimization benchmarks and use them to compare the solution quality and runtime of several common energy minimization algorithms we investigate three promising methods graph cuts lbp and tree reweighted message passing in addition to the well known older iterated conditional mode icm algorithm our benchmark problems are drawn from published energy functions used for stereo image stitching interactive segmentation and denoising we also provide a general purpose software interface that allows vision researchers to easily switch between optimization methods the benchmarks code images and results are available at http vision middlebury edu mrf

a mesh independence result for semismooth newton methods for a class of semismooth operator equations a mesh independence result for generalized newton methods is established the main result states that the continuous and the discrete newton process when initialized properly converge q linearly with the same rate the problem class considered in the paper includes mcp function based reformulations of first order conditions of a class of control constrained optimal control problems for partial differential equations for which a numerical validation of the theoretical results is given

guided image filtering in this paper we propose a novel explicit image filter called guided filter derived from a local linear model the guided filter computes the filtering output by considering the content of a guidance image which can be the input image itself or another different image the guided filter can be used as an edge preserving smoothing operator like the popular bilateral filter 1 but it has better behaviors near edges the guided filter is also a more generic concept beyond smoothing it can transfer the structures of the guidance image to the filtering output enabling new filtering applications like dehazing and guided feathering moreover the guided filter naturally has a fast and nonapproximate linear time algorithm regardless of the kernel size and the intensity range currently it is one of the fastest edge preserving filters experiments show that the guided filter is both effective and efficient in a great variety of computer vision and computer graphics applications including edge aware smoothing detail enhancement hdr compression image matting feathering dehazing joint upsampling etc

a meshless integration free and boundary only rbf technique based on the radial basis function rbf nonsingular general solution and dual reciprocity method drm this paper presents an inherently meshless integration free boundary only rbf collocation technique for numerical solution of various partial differential equation systems the basic ideas behind this methodology are very mathematically simple in this study the rbfs are employed to approximate the inhomogeneous terms via the drm while nonsingular general solution leads to a boundary only rbf formulation for homogenous solution the present scheme is named as the boundary knot method bkm to differentiate it from the other numerical techniques in particular due to the use of nonsingular general solutions rather than singular fundamental solutions the bkm is different from the method of fundamental solution in that the former does not require the artificial boundary and results in the symmetric system equations under certain conditions the efficiency and utility of this new technique are validated through a number of typical numerical examples completeness concern of the bkm due to the sole use of the nonsingular part of complete fundamental solution is also discussed

lp estimates of boundary integral equations for some nonlinear boundary value problems recently galerkin and collocation methods have been analyzed for boundary integral equation formulations of some potential problems in the plane with nonlinear boundary conditions and stability results and error estimates in theh 1 2 norm have been proved ruotsalainen and wendland and ruotsalainen and saranen we show that these results extend tol p setting without any extra conditions these extensions are proved by studying the uniform boundedness of the inverses of the linearized integral operators and then considering the nonlinear equations the fact that inh 1 2 setting the nonlinear operator is a homeomorphism with lipschitz continuous inverse plays a crucial role optimal error estimates for the galerkin and collocation method inl p space then follow

the numerical solution of a nonlinear hypersingular boundary integral equation in this paper we consider a direct hypersingular integral approach to solve harmonic problems with nonlinear boundary conditions by using a practical variant of the galerkin boundary element method the proposed approach provides an almost optimal balance between the order of convergence and the numerical effort of work to compute the approximate solution numerical examples confirm the theoretical results

a galerkin boundary node method and its convergence analysis the boundary node method bnm exploits the dimensionality of the boundary integral equation bie and the meshless attribute of the moving least square mls approximations however since mls shape functions lack the property of a delta function it is difficult to exactly satisfy boundary conditions in bnm besides the system matrices of bnm are non symmetric a galerkin boundary node method gbnm is proposed in this paper for solving boundary value problems in this approach an equivalent variational form of a bie is used for representing the governing equation and the trial and test functions of the variational formulation are generated by the mls approximation as a result boundary conditions can be implemented accurately and the system matrices are symmetric total details of numerical implementation and error analysis are given for a general bie taking the dirichlet problem of laplace equation as an example we set up a framework for error estimates of gbnm some numerical examples are also given to demonstrate the efficacity of the method

an optimal a priori error estimate for nonlinear multibody contact problems nonconforming domain decomposition methods provide a powerful tool for the numerical approximation of partial differential equations for the discretization of a nonlinear multibody contact problem we use linear mortar finite elements based on dual lagrange multipliers under some regularity assumptions on the solution an optimal convergence order of h 0 5 nu 0 nu leq 0 5 can be established in two dimensions 2d and three dimensions 3d compared with a standard linear saddle point formulation two additional terms which provide a measure for the nonconformity and the nonlinearity of the approach have to be taken in account numerical examples illustrating the performance of the nonconforming method and confirming our theoretical result are presented

boundary element methods for potential problems with nonlinear boundary conditions galerkin boundary element methods for the solution of novel first kind steklov poincare and hypersingular operator boundary integral equations with nonlinear perturbations are investigated to solve potential type problems in two and three dimensional lipschitz domains with nonlinear boundary conditions for the numerical solution of the resulting newton iterate linear boundary integral equations we propose practical variants of the galerkin scheme and give corresponding error estimates we also discuss the actual implementation process with suitable preconditioners and propose an optimal hybrid solution strategy

on the convergence of the galerkin method for nonsmooth solutions of integral equations it is shown that the stability region of the galerkin method includes solutions not lying in the conventional energy space optimal order error estimates for these nonsmooth solutions are derived the new result is compared with the classical statement by means of the basic potential problem

properties of an entropy based signal receiver with an application to ultrasonic molecular imaging qualitative and quantitative properties of the finite part hf of the shannon entropy of a continuous waveform f t in the continuum limit are derived in order to illuminate its use for waveform characterization simple upper and lower bounds on hf based on features of f t are defined quantitative criteria for a priori estimation of the average case variation of hf and logef where ef is the signal energy of f t are also derived these provide relative sensitivity estimates that could be used to prospectively choose optimal imaging strategies in real time ultrasonic imaging machines where system bandwidth is often pushed to its limits to demonstrate the utility of these sensitivity relations for this application a study designed to assess the feasibility of identification of angiogenic neovasculature targeted with perfluorocarbon nanoparticles that specifically bind to v 3 integrin expression in tumors was performed the outcome of this study agrees with the prospective sensitivity estimates tha

meshless projection iterative analysis of signorini problems using a boundary element free method this paper presents a meshless method based on coupling a boundary element free method befm with a new projection iterative scheme for signorini problems using the new iterative scheme the nonlinear complementary signorini boundary conditions are transformed into neumann boundary conditions then mixed elliptic problems are formed and are solved by the befm in this method only a nodal data structure on the boundary is required and the coefficient matrices of the discrete system equations only have to be computed once in the whole iterative process convergence of the iterative scheme is discussed theoretically numerical examples involving beach percolation and an electropaint process are given to illustrate the convergence rate and high computational efficiency of the method

projection and self adaptive projection methods for the signorini problem with the bem abstract we propose two new projection methods for solving the signorini problem and study the relationship between them since the signorini boundary condition is equivalent to a projection fixed point problem the methods formulate the signorini boundary condition into a sequence of robin boundary conditions and only require solving a variational problem with simple boundary conditions at each iterative step the convergence speed of the first method depends on the parameter heavily and it is difficult to choose a proper for individual problems to improve the efficiency of the method we propose an adaptive projection method which adjusts the parameter automatically per iteration we establish the convergence analysis of the methods because the signorini boundary condition is given in an iterative form by the potential and its derivative on the boundary of the domain we can easily provide the boundary element approximation of the problem finally we present some numerical simulations which illustrate the performance of the methods

from finite differences to finite elements a short history of numerical analysis of partial differential equations abstract this is an account of the history of numerical analysis of partial differential equations starting with the 1928 paper of courant friedrichs and lewy and proceeding with the development of first finite difference and then finite element methods the emphasis is on mathematical aspects such as stability and convergence analysis