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 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

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

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

an inexact implicit method for general mixed variational inequalities in the present paper we present an inexact implicit method with a variable parameter for general mixed variational inequalities we use a self adaptive technique to adjust parameter r at each iteration the main advantage of this technique is that the method can adjust the parameter automatically and the numbers of iteration are not very sensitive to different initial parameter r 0

permutation matrices related to sudoku we consider a class of permutation matrices associated with the popular game of sudoku and prove some connectivity and existence results for this matrix class

several path following methods for a class of gradient constrained variational inequalities path following splitting and semismooth newton methods for solving a class of problems related to elasto plastic material deformations are proposed analyzed and tested numerically while the splitting techniques result in alternating minimization schemes which are typically linearly convergent the proposed moreau yosida regularization based semismooth newton technique and an associated lifting step yield local superlinear convergence in function space the lifting step accounts for the fact that the operator associated with the linear system in the newton iteration need not be boundedly invertible uniformly along the iterates for devising an efficient update strategy for the path following parameter regularity properties of the path are studied and utilized within an inexact path following scheme for all approaches the paper ends by a report on numerical tests of the different approaches

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

error estimates of the weakly over penalized symmetric interior penalty method for two variational inequalities in this paper we apply the weakly over penalized symmetric interior penalty method to solve some variational inequalities which include the signorini problem and the obstacle problem optimal a priori error estimates in energy norm are derived some numerical tests are provided to confirm our theoretical analysis

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

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

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

on the symmetric vector quasi equilibrium problems abstract in this paper we using a particular technique consider the symmetric vector quasi equilibrium problems in the hausdorff topological vector space as applications of our existence theorem a coincidence point theorem and the existence of vector optimization problem for a pair of vector valued mappings are obtained moreover we answer an open question raised by fu in j y fu symmetric vector quasi equilibrium problems j math anal appl 285 2003 708 713

path following and augmented lagrangian methods for contact problems in linear elasticity a certain regularization technique for contact problems leads to a family of problems that can be solved efficiently using infinite dimensional semismooth newton methods or in this case equivalently primal dual active set strategies we present two procedures that use a sequence of regularized problems to obtain the solution of the original contact problem first order augmented lagrangian and path following methods the first strategy is based on a multiplier update while path following with respect to the regularization parameter uses theoretical results about the path value function to increase the regularization parameter appropriately comprehensive numerical tests investigate the performance of the proposed strategies for both a 2d as well as a 3d contact problem

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

stability and sensitivity analysis in convex vector optimization in this paper stability and sensitivity of the efficient set in convex vector optimization are considered the perturbation map is defined as a set valued map it associates with each parameter vector the set of all minimal points of the parametrized feasible set with respect to an ordering cone in the objective space sufficient conditions for the upper and lower semicontinuity of the perturbation map are obtained because of the convexity assumptions the conditions obtained are fairly simple if compared to those in the general case moreover a complete characterization of the contingent derivative of the perturbation map is obtained under some assumptions it provides quantitative information on the behavior of the perturbation map

a regularized smoothing newton type algorithm for quasi variational inequalities abstract this paper is concerned with developing an efficient regularized smoothing newton type algorithm for quasi variational inequalities the proposed algorithm takes the advantage of newly introduced smoothing functions and a non monotone line search strategy it is proved to be globally and locally superlinearly quadratically convergent under suitable assumptions numerical results demonstrate that the algorithm generally outperforms the existing interior point method and semismooth method facchinei et al 2014

symmetric vector quasi equilibrium problems the symmetric vector quasi equilibrium problem is introduced under suitable assumptions the symmetric vector quasi equilibrium problem is solvable as its applications a coincidence point theorem and the existence of vector optimization problem for a pair of vector valued mappings are reduced

modified resolvent splitting algorithms for general mixed variational inequalities in this paper we suggest and analyze a number of four step resolvent splitting algorithms for solving general mixed variational inequalities by using the updating technique of the solution the convergence of these new methods requires either monotonicity or pseudomonotonicity of the operator proof of convergence is very simple our new methods differ from the existing splitting methods for solving variational inequalities and complementarity problems the new results are versatile and are easy to implement

calculation of the number of all pairs of disjoint s permutation matrices the concept of s permutation matrix is considered a general formula for counting all disjoint pairs of n2 n2 s permutation matrices as a function of the positive integer n is formulated and proven in this paper to do that the graph theory techniques have been used it has been shown that to count the number of disjoint pairs of n2 n2 s permutation matrices it is sufficient to obtain some numerical characteristics of all n n bipartite graphs

well posedness for convex symmetric vector quasi equilibrium problems abstract in this paper the notion of a generalized levitin polyak well posedness is defined for symmetric vector quasi equilibrium problems sufficient conditions are given for the generalized levitin polyak well posedness moreover it is shown that the results can be refined in the convex case

some developments in general variational inequalities general variational inequalities provide us with a unified natural novel and simple framework to study a wide class of equilibrium problems arising in pure and applied sciences in this paper we present a number of new and known numerical techniques for solving general variational inequalities using various techniques including projection wiener hopf equations updating the solution auxiliary principle inertial proximal penalty function dynamical system and well posedness we also consider the local and global uniqueness of the solution and sensitivity analysis of the general variational inequalities as well as the finite convergence of the projection type algorithms our proofs of convergence are very simple as compared with other methods our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems since the general variational inequalities include quasi variational inequalities and quasi implicit complementarity problems as special cases results presented here continue to hold for these problems several open problems have been suggested for further research in these areas

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

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

numerical verification of solutions for signorini problems using newton like method we proposed a numerical method to verify the existence of solutions for a simplified signorini problem comput math appl 2000 40 1003 1013 using sequential iteration method we numerically constructed a set containing solutions that satisfies the hypothesis of schauder s fixed point theorem in a certain sobolev space it is difficult to apply this method to the problem of which associated operator is not retractive in a neighborhood of the solution in this paper in order to overcome such a difficulty we describe an alternative approach to this problem numerical examples are presented copyright 2007 john wiley sons ltd

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

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

a projection method based on self adaptive rules for stokes equations with nonlinear slip boundary conditions abstract in this work we propose and study a self adaptive projection method for the numerical solution of stokes equations under slip boundary conditions we introduce the projection operator to formulate the slip constraint as a fixed point equation to solve this problem we suggest a projection method by reformulating the nonlinear slip boundary conditions into an iterative form to make the method more efficient we find a self adaptive rule that uses iterative functions to adjust the penalty parameter automatically we show the convergence of the method in function space and give its application in detail finally the numerical results are given to support our theoretical results

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

on the cardinality of a factor set of binary matrices abstract the work considers an equivalence relation in the set of all n m binary matrices in each element of the factor set generated by this relation we define the concept of canonical binary matrix namely the minimal element with respect to the lexicographic order for this purpose the binary matrices are uniquely represented by ordered n tuples of integers we have found a necessary and sufficient condition for an arbitrary matrix to be canonical this condition could be the base for realizing recursive algorithm for finding all n m canonical binary matrices and consequently for finding all bipartite graphs up to isomorphism with cardinality of each part equal to n and m