performance evaluation of independent superbasic sets on nonlinear replicated networks abstract in this paper we describe a new type of network flow problems that basically consists of the classical transshipment problem with the following extensions 1 the replication of a network by producing subnetworks with identical structure being linked by so called linking arcs 2 the objective function terms related to the linking arcs are non differentiable nonlinear functions by using a networks specialized implementation of a linearly constrained nonlinear programming algorithm described elsewhere we report the computational performance of the new concept of independent superbasic sets that allows to obtain in parallel independent pieces of the solution at each iteration

a variational meshfree method for solving time discrete diffusion equations a meshfree method is developed for solving time discrete diffusion equations that arise in models in brain research important criteria for a suitable method are flexibility with respect to domain geometry and the ability to work with very small moving sources requiring easy refinement possibilities one part of the work concerns a meshfree discretization of the modified helmholtz equation based on the related minimization problem and a local least squares function approximation in a second part a node choosing algorithm is presented that moves around randomly distributed nodes for optimizing the node distribution and varying the node density as needed the method is illustrated by two numerical tests

simultaneous lotsizing and scheduling by combining local search with dual reoptimization abstract the contribution of this paper is twofold on the one hand the particular problem of integrating lotsizing and scheduling of several products on a single capacitated production line is modelled and solved taking into account sequence dependent setup times thereby continuous lotsizes meeting deterministic dynamic demands are to be determined and scheduled with the objective of minimizing inventory holding costs and sequence dependent setup costs on the other hand a new general algorithmic approach is presented a dual reoptimization algorithm is combined with a local search heuristic for solving a mixed integer programming problem this idea is applied to the above lotsizing and scheduling problem by embedding a dual network flow algorithm into threshold accepting and simulated annealing respectively computational tests show the effectiveness of the new solution method

reoptimization procedures for bounded variable primal simplex network algorithms abstract many algorithms and applications call for the use of a network subprogram which must be optimized numerous times with slight changes to the problem data bound and right hand side changes to an existing basis tree may yield an infeasible basic solution this report gives a sequence of steps that modify an existing basic solution to reflect such changes and streamline the reoptimization process

a point collocation method based on reproducing kernel approximations a reproducing kernel particle method with built in multiresolution features in a very attractive meshfree method for numerical solution of partial differential equations the design and implementation of a galerkin based reproducing kernel particle method however faces several challenges such as the issue of nodal volumes and accurate and efficient implementation of boundary conditions in this paper we present a point collocation method based on reproducing kernel approximations we show that in a point collocation approach the assignment of nodal volumes and implementation of boundary conditions are not critical issues and points can be sprinkled randomly making the point collocation method a true meshless approach the point collocation method based on reproducing kernel approximations however requires the calculation of higher order derivatives that would typically not be required in a galerkin method a correction function and reproducing conditions that enable consistency of the point collocation method are derived the point collocation method is shown to be accurate for several one and two dimensional problems and the convergence rate of the point collocation method is addressed copyright 2000 john wiley sons ltd

a stabilized finite point method for analysis of fluid mechanics problems in this paper a meshless procedure termed the finite point method for solving convection diffusion and fluid flow type problems is presented the method is based on the use of a weighted least square interpolation procedure together with point collocation for evaluating the approximation integrals special emphasis is given to the stabilization of the convective terms and the neumann boundary condition which has been found to be essential to obtain accurate results some examples of application to diffusive and convective transport and compressible flow problems using quadratic fp interpolations are presented

dual algorithms for pure network problems this paper reports the development of a new algorithmic implementation of the dual algorithm for the capacitated minimum cost network flow problem furthermore it introduces dual reoptimization procedures and compares primal and dual algorithms for the optimization and reoptimization of network problems the implementation makes use of cut sets for the efficient execution of the entering variable selection and the selection of the leaving variable is tied to the basis structure at an iteration empirical testing of the dual algorithm for optimization shows that it dominates the primal procedure for assignment problems with 400 nodes transportation problems with more than 600 nodes and transshipment problems with more than 1500 nodes computational testing with typical changes found in decomposition and relaxation procedures indicates the superiority of dual reoptimization over primal reoptimization for a sequence of parametric changes as would be typical in large scale programming techniques on the

an adaptive finite element material point method and its application in extreme deformation problems abstract taking advantages of both lagrangian and eulerian methods material point method mpm is suitable for modeling problems with extreme deformation however mpm is less accurate and less efficient than finite element method fem for small deformation problems due to particle quadrature and mappings between particles and background grid applied in mpm to take advantages of both fem and mpm an adaptive finite element material point method is developed for modeling the dynamic behavior of material under extreme loading bodies are initially discretized by finite elements and then the elements with large strain are adaptively converted into mpm particles based on their degree of distortion or plastic strain during the solution process the interaction between the remaining finite elements and mpm particles is implemented based on the background grid in mpm framework several numerical examples are presented to validate the efficiency and accuracy of the proposed method and the numerical results are in good agreement with experiments while the efficiency of the method is higher than that of both mpm and fem

an assessment of nonmonotone linesearch techniques for unconstrained optimization the purpose of this paper is to discuss the potential of nonmonotone techniques for enforcing convergence of unconstrained minimization algorithms from starting points distant from the solution linesearch based algorithms are considered for both small and large problems and extensive numerical experiments show that this potential is sometimes considerable a new variant is introduced in order to limit some of the identified drawbacks of the existing techniques this variant is again numerically tested and appears to be competitive finally the impact of preconditioning on the considered methods is examined

efficient line search methods for convex functions in this paper we propose two new line search methods for convex functions these new methods exploit the convexity property of the function contrary to existing methods the first method is an improved version of the golden section method for the second method it is proven that after two evaluations the objective gap is at least halved the practical efficiency of the methods is shown by applying our methods to a real life bus and buffer size optimization problem and to several classes of convex functions

line search procedures for the logarithmic barrier function barrier methods for constrained optimization widely applied in the 1960s and 1970s have recently enjoyed a revival of popularity a line search with respect to a logarithmic barrier function or related potential function is often part of modern interior methods standard line search procedures tend to be inefficient in this context for two reasons an inappropriate choice of initial trial step if any constraints are nonlinear and poor approximation of the barrier function by loworder polynomial interpolants this paper discusses line search strategies specifically designed for the logarithmic barrier function

algorithms for network piecewise linear programs a comparative study abstract piecewise linear programming plp is an important area of mathematical programming and concerns the minimisation of a convex separable piecewise linear objective function subject to linear constraints in this paper a subarea of plp called network piecewise linear programming nplp is explored the paper presents four specialised algorithms for nplp strongly feasible primal simplex dual method out of kilter and strongly polynomial cost scaling and their relative efficiency is studied a statistically designed experiment is used to perform a computational comparison of the algorithms the response variable observed in the experiment is the cpu time to solve randomly generated network piecewise linear problems classified according to problem class transportation transshipment and circulation problem size extent of capacitation and number of breakpoints per arc results and conclusions on performance of the algorithms are reported

finite cloud method a true meshless technique based on a fixed reproducing kernel approximation we introduce fixed moving and multiple fixed kernel techniques for the construction of interpolation functions over a scattered set of points we show that for a particular choice of nodal volumes the fixed moving and multiple fixed kernel approaches are identical to the fixed moving and multiple fixed least squares approaches a finite cloud method which combines collocation with a fixed kernel technique for the construction of interpolation functions is presented as a true meshless technique for the numerical solution of partial differential equations numerical results are presented for several one and two dimensional problems including examples from elasticity heat conduction thermoelasticity stokes flow and piezoelectricity copyright 2001 john wiley sons ltd

a finite point method in computational mechanics applications to convective transport and fluid flow the paper presents a fully meshless procedure fo solving partial differential equations the approach termed generically the finite point method is based on a weighted least square interpolation of point data and point collocation for evaluating the approximation integrals some examples showing the accuracy of the method for solution of adjoint and non self adjoint equations typical of convective diffusive transport and also to the analysis of compressible fluid mechanics problem are presented

a new implementation of the element free galerkin method abstract element free galerkin methods efg are methods for solving partial differential equations with moving least squares interpolants efg methods require only nodal data no element connectivity is needed in a previous implementation of the efg method lagrange multipliers were used to enforce the essential boundary condition however the use of lagrange multipliers increases the cost of solving the linear algebraic equations a new implementation is developed based on a modified variational principle in which the lagrange multipliers are replaced at the outset by their physical meaning so that the discrete equations are banded in addition weighted orthogonal basis functions are constructed so the need for solving equations at each quadrature point is eliminated numerical examples show that the present implementation effectively computes stress concentrations and stress intensity factors at cracks with very irregular arrangements of nodes the latter makes it very advantageous for modelling progressive cracking

obtaining a good primal solution to the uncapacitated transportation problem abstract sharma and sharma european journal of operational research 122 3 2000 37 have given a new procedure to solve the dual of the well known uncapacitated transportation problem this is expected to enhance the performance of dual based optimizing algorithms for solving the transportation problems in this paper we give a heuristic that obtains a very good starting solution with a duality gap of less than 2 for the primal transportation problem in o n 3 time and this is expected to enhance the performance of network simplex algorithm that obtains the optimal solution and runs in o n 3 log n time empirical investigation revealed that our heuristic gave significantly better solutions than the well known vogel s approximation method that runs in o n 2 time

new concepts in meshless methods meshless methods have been extensively popularized in literature in recent years due to their flexibility in solving boundary value problems two kinds of truly meshless methods the meshless local boundary integral equation mlbie method and the meshless local petrov galerkin mlpg approach are presented and discussed both methods use the moving least squares approximation to interpolate the solution variables while the mlbie method uses a local boundary integral equation formulation and the mlpg employs a local symmetric weak form the two methods are truly meshless ones as both of them do not need a finite element or boundary element mesh either for purposes of interpolation of the solution variables or for the integration of the energy all integrals can be easily evaluated over regularly shaped domains in general spheres in three dimensional problems and their boundaries numerical examples presented in the paper show that high rates of convergence with mesh refinement are achievable in essence the present meshless method based on the lswf is found to be a simple efficient and attractive method with a great potential in engineering applications copyright 2000 john wiley sons ltd

natural neighbour galerkin methods summary natural neighbour co ordinates sibson co ordinates is a well known interpolation scheme for multivariate data tting and smoothing the numerical implementation of natural neighbour co ordinates in a galerkin method is known as the natural element method nem in the natural element method natural neighbour co ordinates are used to construct the trial and test functions recent studies on nem have shown that natural neighbour co ordinates which are based on the voronoi tessellation of a set of nodes are an appealing choice to construct meshless interpolants for the solution of partial dierential equations in belikov et al computational mathematics and mathematical physics 1997 37 1 9 15 a new interpolation scheme non sibsonian interpolation based on natural neighbours was proposed in the present paper the nonsibsonian interpolation scheme is reviewed and its performance in a galerkin method for the solution of elliptic partial dierential equations that arise in linear elasticity is studied a methodology to couple nite elements to nem is also described two signicant advantages of the non sibson interpolant over the sibson interpolant are revealed and numerically veried the computational eciency of the non sibson algorithm in 2 dimensions which is expected to carry over to 3 dimensions and the ability to exactly impose essential boundary conditions on the boundaries of convex and non convex domains copyright 2001 john wiley sons ltd

an optimal adaptive algorithm for the approximation of concave functions motivated by the study of parametric convex programs we consider approximation of concave functions by piecewise affine functions using dynamic programming we derive a procedure for selecting the knots at which an oracle provides the function value and one supergradient the procedure is adaptive in that the choice of a knot is dependent on the choice of the previous knots it is also optimal in that the approximation error in the integral sense is minimized in the worst case

a new dual based procedure for the transportation problem abstract we pose the transportation problem in a slightly different form to obtain a dual problem that has a special structure we exploit the structure of the dual problem to give a new heuristic that runs in o cn 2 time n is number of nodes in the network and c is a constant to improve the dual solution our heuristic obtained the optimal solution to several small sized transportation problems that we attempted and for large sized problems it obtained solutions that were within 82 of the optimal solution on average our approach gives good starting solutions for dual based approaches used for solving transportation problems and thus is expected to enhance their computational performance

moving least square reproducing kernel methods i methodology and convergence this paper formulates the moving least square interpolation scheme in a framework of the so called moving least square reproducing kernel mlsrk representation in this study the procedure of constructing moving least square interpolation function is facilitated by using the notion of reproducing kernel formulation which as a generalization of the early discrete approach establishes a continuous basis for a partition of unity this new formulation possesses the quality of simplicity and it is easy to implement moreover the reproducing kernel formula proposed is not only able to reproduce any mth order polynomial exactly on an irregular particle distribution but also serves as a projection operator that can approximate any smooth function globally with an optimal accuracy in this contribution a generic m consistency relation has been found which is the essential property of the mlsrk approximation an interpolation error estimate is given to assess the convergence rate of the approximation it is shown that for sufficiently smooth function the interpolant expansion in terms of sampled values will converge to the original function in the sobolev norms as a meshless method the convergence rate is measured by a new control variable dilation parameter of the window function instead of the mesh size h as usually done in the finite element analysis to illustrate the procedure convergence has been shown for the numerical solution of the second order elliptic differential equations in a galerkin procedure invoked with this interpolant in the numerical example a two point boundary problem is solved by using the method and an optimal convergence rate is observed with respect to various norms

polyhedrization of an arbitrary 3d point set abstract given a 3d point set the problem of defining the volume associated dividing it into a set of regions elements and defining a boundary surface is tackled several physical problems need to define volume domains boundary surfaces and approximating functions from a given point distribution this is for instance the case of particle methods in which all the information is the particle positions and there are not boundary surfaces definition until recently all the fem mesh generators were limited to the generation of simple elements as tetrahedral or hexahedral elements or triangular and quadrangular in 2d problems the reason of this limitation was the lack of any acceptable shape function to be used in other kind of geometrical elements nowadays there are several acceptable shape functions for a very large class of polyhedra these new shape functions together with a generalization of the delaunay tessellation presented in this paper gives an optimal marriage and a powerful tool to solve a large variety of physical problems by numerical methods the domain partition into polyhedra presented here is not a standard mesh generation the problem here is for a given node distribution to find a suitable boundary surface and a suitable mesh to be used in the solution of a physical problem by a numerical method to include new nodes or change their positions is not allowed

curvilinear linesearch for tensor methods this paper presents a curvilinear linesearch for use when solving nonlinear equations with tensor methods standard tensor methods use a combination of the tensor step and newton step in a linesearch but they are handled separately and in an ad hoc manner our curvilinear linesearch combines the two directions in a single parametric step which guarantees a monotonic decrease on the tensor model and also asymptotically approaches the newton direction as the step length shrinks to zero thus guaranteeing descent on the nonlinear equations numerical experiments on a set of 35 small scale problems drawn primarily from the constrained and unconstrained testing environment show an 18 23 improvement in terms of function evaluations over previous tensor linesearches when using quadratic backtracking and a 41 83 improvement when halving lambda before each trial our results also suggest that the curvilinear linesearch is more robust than linesearch based newton s method or the standard tensor linesearch producing fewer failures than either method experiments on two large scale fluid flow problems complement the small scale results and give preliminary indication of the applicability of the tensor method and the efficiency of the curvilinear linesearch the theoretical properties coupled with the better performance make this a desirable improvement over previous tensor method linesearches

a nonlinear network flow algorithm for maximization of benefits in a hydroelectric power system a nonlinear network flow algorithm for maximization of benefits in a multireservoir hydroelectric power system is presented the problem was posed by tennessee valley authority tva as an optimization model with a nonseparable nonlinear objective function and with linear network flow constraints the proposed algorithm is based on reduced gradient methodology with somewhat nonstandard modifications and on primal linear network flows with simplifications resulting from the special structure of the problem network an unusual feature of the algorithm is an integer programming subproblem whose exact solution determines the superbasic set and the search directions the algorithm is coded by means of an efficient basis tree labeling system which consists of a single node length array and which is specifically designed for the physical context of the problem test problems on a 6 reservoir tva subsystem were solved with computer costs well within the affordable range

the meshless finite element method a meshless method is presented which has the advantages of the good meshless methods concerning the ease of introduction of node connectivity in a bounded time of order n and the condition that the shape functions depend only on the node positions furthermore the method proposed also shares several of the advantages of the finite element method such as a the simplicity of the shape functions in a large part of the domain b c0 continuity between elements which allows the treatment of material discontinuities and c ease of introduction of the boundary conditions copyright 2003 john wiley sons ltd

representing inverses in pure network flow optimization abstract a basis for a pure network flow problem always exhibits special structure whose exploitation has led to the development of extremely efficient network optimization software the backbone of these programs is a data structure which represents the basis as a tree and is used to perform the linear algebraic steps required for optimization we demonstrate that the basis inverse also possesses a special structure which we call the inverse tree we develop a data structure to represent the inverse tree and we show that it obeys a symmetric relationship with the basis tree data structure the predecessor function of the inverse tree is the preorder link function of the basis tree and vice versa we develop efficient computational procedures for exploiting these results and show that using the inverse tree requires computation of the same order as the basis tree

centroidal voronoi tessellations applications and algorithms a centroidal voronoi tessellation is a voronoi tessellation whose generating points are the centroids centers of mass of the corresponding voronoi regions we give some applications of such tessellations to problems in image compression quadrature finite difference methods distribution of resources cellular biology statistics and the territorial behavior of animals we discuss methods for computing these tessellations provide some analyses concerning both the tessellations and the methods for their determination and finally present the results of some numerical experiments