dual primal domain decomposition method for uncertainty quantification abstract the spectral stochastic finite element method ssfem may offer an efficient alternative to the traditional monte carlo simulations mcs for uncertainty quantification of large scale numerical simulations in the framework of the intrusive ssfem the main computational challenge involves solving a coupled set of deterministic linear systems for large scale numerical models the computational efficiency of the intrusive ssfem primarily depends on the solution techniques employed to tackle the resulting coupled linear systems in this paper we report a probabilistic version of the dual primal domain decomposition method for the intrusive ssfem in order to exploit high performance computing platforms for uncertainty quantification in particular we formulate a probabilistic version of the dual primal finite element tearing and interconnect feti dp technique to solve the large scale linear systems in the intrusive ssfem in the probabilistic setting the operator of the dual interface system in the dual primal approach contains a coarse problem the introduction of the coarse problem in the probabilistic setting leads to a scalable performance of the dual primal iterative substructuring method for uncertainty quantification of large scale computational models the convergence properties numerical and parallel scalabilities of the probabilistic feti dp method and the recently developed probabilistic version of the balancing domain decomposition by constraints bddc method are contrasted for numerical illustrations we consider flow through porous media and linear elasticity problems with spatially varying system parameters modelled as non gaussian random processes the algorithms are implemented on a linux cluster using mpi and petsc parallel libraries

robust feti dp methods for heterogeneous three dimensional elasticity problems iterative substructuring methods with lagrange multipliers are considered for heterogeneous linear elasticity problems with large discontinuities in the material stiffnesses in particular results for algorithms belonging to the family of dual primal feti methods are presented the core issue of these algorithms is the construction of an appropriate global problem in order to obtain a robust method which converges independently of the material discontinuities in this article several necessary and sufficient conditions arising from the theory are numerically tested and confirmed furthermore results of numerical experiments are presented for situations which are not covered by the theory such as curved edges and material discontinuities not aligned with the interface and an attempt is made to develop rules for these cases

a class of parallel two level nonlinear schwarz preconditioned inexact newton algorithms we propose and test a new class of two level nonlinear additive schwarz preconditioned inexact newton algorithms aspin the two level aspin combines a local nonlinear additive schwarz preconditioner and a global linear coarse preconditioner this approach is more attractive than the two level method introduced in cai keyes and marcinkowski nonlinear additive schwarz preconditioners and applications in computational fluid dynamics international journal numerical methods in fluids 40 2002 1463 1470 which is nonlinear on both levels since the coarse part of the global function evaluation requires only the solution of a linear coarse system rather than a nonlinear coarse system derived from the discretization of original partial dierential equations the overall computational cost is reduced considerably our parallel numerical results based on an incompressible lid driven flow problem show that the new two level aspin is quite scalable with respect to the number of processors and the fine mesh size when the coarse mesh size is fine enough and in addition the convergence is not sensitive to the reynolds numbers

fixed points and stability in linear neutral differential equations with variable delays in this paper we consider the asymptotic stability of a generalized linear neutral differential equation with variable delays by using the fixed point theory an asymptotic stability theorem with a necessary and sufficient condition is proved which improves and generalizes some results due to burton 2003 3 zhang 2005 14 raffoul 2004 13 and jin and luo 2008 12 two examples are also given to illustrate our results

on bendixson s criterion abstract for autonomous differential equations in r n criteria are developed which preclude the existence of invariant closed curves such as periodic or homoclinic trajectories the technique is based on the study of functionals on 2 surfaces results generalize to higher dimensions a criterion of bendixson for the nonexistence of nonconstant periodic solutions in the case n 2 as an example an application to the lorenz system in r 3 is given

a method of finite element tearing and interconnecting and its parallel solution algorithm a novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented the spatial domain is partitioned into a set of totally disconnected subdomains each assigned to an individual processor lagrange multipliers are introduced to enforce compatibility at the interface nodes in the static case each floating subdomain induces a local singularity that is resolved in two phases first the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution next the contributions of these modes are related to the lagrange multipliers through an orthogonality condition a parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and lagrange multipliers which completes the solution of the problem when implemented on local memory multiprocessors this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring it is also suitable for parallel vector computers with shared memory moreover unlike parallel direct solvers it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations

global hopf bifurcation in a neural netlet sufficient conditions are obtained for the global hopf bifurcation of periodic solutions in the dynamics of a neural netlet of activation and inhibition with continuously distributed delays the orbital asymptotic stability of the bifurcating periodic solutions is investigated the results are illustrated with numerical simulations a few problems are formulated in the form of conjectures

a preconditioner for substructuring based on constrained energy minimization a preconditioner for substructuring based on constrained energy minimization concepts is presented the preconditioner is applicable to both structured and unstructured meshes and offers a straightforward approach for the iterative solution of second and fourth order structural mechanics problems the approach involves constraints associated with disjoint sets of nodes on substructure boundaries these constraints provide the means for preconditioning at both the substructure and global levels numerical examples are presented that demonstrate the good performance of the method in terms of iterations compute time and condition numbers of the preconditioned equations

on the use of inexact subdomain solvers for bddc algorithms the standard bddc balancing domain decomposition by constraints preconditioner is shown to be equivalent to a preconditioner built from a partially subassembled finite element model this results in a system of linear algebraic equations which is much easier to solve in parallel than the fully assembled model the cost is then often dominated by that of the problems on the subdomains an important role is also played both in theory and practice by an averaging operator and in addition exact dirichlet solvers are used on the subdomains in order to eliminate the residual in the interior of the subdomains the use of inexact solvers for these problems and even the replacement of the dirichlet solvers by a trivial extension are considered it is established that one of the resulting algorithms has the same eigenvalues as the standard bddc algorithm and the connection of another with the feti dp algorithm with a lumped preconditioner is also considered multigrid methods are used in the experimental work and under certain assumptions it is established that the iteration count essentially remains the same as when exact solvers are used while considerable gains in the speed of the algorithm can be realized since the cost of the exact solvers grows superlinearly with the size of the subdomain problems while the multigrid methods are linear

bifurcation analysis in a neutral differential equation the dynamics of a neural network model in neutral form is investigated we prove that a sequence of hopf bifurcations occurs at the origin as the delay increases the direction of the hopf bifurcations and the stability of the bifurcating periodic solutions are determined by using normal form method and center manifold theory global existence of periodic solutions is established using a global hopf bifurcation result of krawcewicz et al and a bendixson s criterion for higher dimensional ordinary differential equations due to li and muldowney

differentiability of solutions with respect to parameters in neutral differential equations with state dependent delays in this paper we consider a class of nonlinear neutral differential equations with state dependent delays in both the neutral and the retarded terms we study well posedness and continuous dependence issues and differentiability of the parameter map with respect to the initial function and other possibly infinite dimensional parameters in a pointwise sense and also in the c norm

a parallel implementation of dual primal feti methods for three dimensional linear elasticity using a transformation of basis dual primal feti methods for linear elasticity problems in three dimensions are considered these are nonoverlapping domain decomposition methods where some primal continuity constraints across subdomain boundaries are required to hold throughout the iterations whereas most of the constraints are enforced by lagrange multipliers an algorithmic framework for dual primal feti methods is described together with a transformation of basis to implement the primal constraints numerical results obtained from a parallel implementation of these algorithms applied to a model benchmark problem with structured meshes and to problems with more complicated geometries from industrial and biological applications using unstructured meshes are provided these results show that the presented dual primal feti algorithms are numerical and parallel scalable

optimal convergence properties of the feti domain decomposition method abstract the finite element tearing and interconnecting feti method is a practical and efficient domain decomposition dd algorithm for the solution of self adjoint elliptic partial differential equations for large scale structural problems discretized with shell and beam elements this method was found to outperform popular iterative algorithms and direct solvers on both serial and parallel computers and to compare favorably with leading dd methods in this paper we discuss some numerical properties of the feti method that were not addressed before in particular we show that the mathematical treatment of the floating subdomains and the specific conjugate projected gradient algorithm that characterize the feti method are equivalent to the construction and solution of a coarse problem that propagates the error globally accelerates convergence and ensures a performance that is independent of the number of subdomains we also show that when the interface problem is optimally preconditioned and the mesh is partitioned into well structured subdomains with good aspect ratios the performance of the feti method is also independent of the mesh size however we also argue that the feti and other leading dd methods for unstructured problems lose in practice these scalability properties when the mesh contains junctures with rotational degrees of freedom or the decomposition is irregular and characterized by arbitrary subdomain aspect ratios finally we report that for realistic problems optimal preconditioners are not necessarily computationally efficient and can be outperformed by non optimal ones

a neumann neumann domain decomposition algorithm for solving plate and shell problems we present a new neumann neumann type preconditioner of large scale linear systems arising from plate and shell problems the advantage of the new method is a smaller coarse space than those of earlier methods of the authors this improves parallel scalability a new abstract framework for neumann neumann preconditioners is used to prove almost optimal convergence properties of the method the convergence estimates are independent of the number of subdomains coefficient jumps between subdomains and depend only polylogarithmically on the number of elements per subdomain we formulate and prove an approximate parametric variational principle for reissner mindlin elements as the plate thickness approaches zero which makes the results applicable to a large class of nonlocking elements in everyday engineering use the theoretical results are confirmed by computational experiments on model problems as well as examples from real world engineering practice

simultaneous feti and block feti robust domain decomposition with multiple search directions domain decomposition methods often exhibit very poor performance when applied to engineering problems with large heterogeneities in particular for heterogeneities along domain interfaces the iterative techniques to solve the interface problem are lacking an efficient preconditioner recently a robust approach named feti geneo was proposed where troublesome modes are precomputed and deflated from the interface problem the cost of the feti geneo is however high we propose in this paper techniques that share similar ideas with feti geneo but where no pre processing is needed and that can be easily and efficiently implemented as an alternative to standard domain decomposition methods in the block iterative approaches presented in this paper the search space at every iteration on the interface problem contains as many directions as there are domains in the decomposition those search directions originate either from the domain wise preconditioner in the simultaneous feti method or from the block structure of the right hand side of the interface problem block feti we show on 2d structural examples that both methods are robust and provide good convergence in the presence of high heterogeneities even when the interface is jagged or when the domains have a bad aspect ratio the simultaneous feti was also efficiently implemented in an optimized parallel code and exhibited excellent performance compared to the regular feti method

segregated runge kutta time integration of convection stabilized mixed finite element schemes for wall unresolved les of incompressible flows in this work we develop a high performance numerical framework for the large eddy simulation les of incompressible flows the spatial discretization of the nonlinear system is carried out using mixed finite element fe schemes supplemented with symmetric projection stabilization of the convective term and a penalty term for the divergence constraint these additional terms introduced at the discrete level have been proved to act as implicit les models in order to perform meaningful wall unresolved simulations we consider a weak imposition of the boundary conditions using a nitsche s type scheme where the tangential component penalty term is designed to act as a wall law next segregated runge kutta srk schemes recently proposed by the authors for laminar flow problems are applied to the les simulation of turbulent flows by the introduction of a penalty term on the trace of the acceleration these methods exhibit excellent stability properties for both implicit and explicit treatment of the convective terms srk schemes are excellent for large scale simulations since they reduce the computational cost of the linear system solves by splitting velocity and pressure computations at the time integration level leading to two uncoupled systems the pressure system is a darcy type problem that can easily be preconditioned using a traditional block preconditioning scheme that only requires a poisson solver at the end only coercive systems have to be solved which can be effectively preconditioned by multilevel domain decomposition schemes which are both optimal and scalable the framework is applied to the taylor green and turbulent channel flow benchmarks in order to prove the accuracy of the convection stabilized mixed fes as les models and srk time integrators the scalability of the preconditioning techniques in space only has also been proven for one step of the srk scheme for the taylor green flow using uniform meshes moreover a turbulent flow around a naca profile is solved to show the applicability of the proposed algorithms for a realistic problem

nonlinear feti dp and bddc methods new nonlinear feti dp dual primal finite element tearing and interconnecting and bddc balancing domain decomposition by constraints domain decomposition methods are introduced in all these methods in each iteration local nonlinear problems are solved on the subdomains the new approaches can significantly reduce communication and show a significantly improved performance especially for problems with localized nonlinearities compared to a standard newton krylov feti dp or bddc approach moreover the coarse space of the nonlinear feti dp methods can be used to accelerate the newton convergence it is also found that the new nonlinear feti dp and nonlinear bddc methods are not as closely related as in the linear context numerical results for the p laplace operator are presented

block triangular preconditioners for saddle point problems with a penalty term block triangular preconditioners for a class of saddle point problems with a penalty term are considered an important example is the mixed formulation of the pure displacement problem in linear elasticity it is shown that the spectrum of the preconditioned system is contained in a real positive interval and that the interval bounds can be made independent of the discretization and penalty parameters this fact is used to construct bounds of the convergence rate of the gmres method with respect to an energy norm numerical results are given for gmres and bi cgstab

multilevel bddc for incompressible navier stokes equations we present an approach to the numerical solution of steady navier stokes equations approximation by the finite element method fem leads to a nonlinear saddle point system the system is lineari

eventual norm continuity for neutral semigroups on banach spaces in this paper we prove under suitable hypotheses eventually norm continuity and compactness of the solution semigroups of certain neutral differential equations in banach spaces our approach is based on a general perturbation theorem obtained from closed loop systems of infinite dimensional control systems with unbounded control and observation operators in fact the solution semigroup can be viewed as the semigroup of an appropriate closed loop system in product state spaces

deflated preconditioned conjugate gradient solvers for linear elasticity extensions of deflation techniques previously developed for the poisson equation to static elasticity are presented compared to the scalar poisson equation j comput phys 2008 227 24 10196 10208 int j numer meth engng 2010 doi 10 1002 nme 2932 int j numer meth biomed engng 2010 26 1 73 85 the elasticity equations represent a system of equations giving rise to more complex low frequency modes multigrid elsevier amsterdam 2000 in particular the straightforward extension from the scalar case does not provide generally satisfactory convergence however a simple modification allows to recover the remarkable acceleration in convergence and cpu time reached in the scalar case numerous examples and timings are provided in a serial and a parallel context and show the dramatic improvements of up to two orders of magnitude in cpu time for grids with moderate graph depths compared to the non deflated version furthermore a monotonic decrease of iterations with increasing subdomains as well as a remarkable acceleration for very few subdomains are also observed if all the rigid body modes are included copyright 2011 john wiley sons ltd

symmetric functional differential equations and neural networks with memory we establish an analytic local hopf bifurcation theorem and a topological global hopf bifurcation theorem to detect the existence and to describe the spatial temporal pattern the asymptotic form and the global continuation of bifurcations of periodic wave solutions for functional differential equations in the presence of symmetry we apply these general results to obtain the coexistence of multiple large amplitude wave solutions for the delayed hopfield cohen grossberg model of neural networks with a symmetric circulant connection matrix

hopf bifurcation for neutral type neural network model with two delays in this paper the dynamics of a neutral neural network model with two delays is investigated the condition to ensure the stability of the zero solution of the system is decided by choosing 1 and 2 as parameters respectively then the hopf bifurcation is discussed by using the center manifold theory and normal form method introduced by hassard and kazarinoff global existence of periodic solution is studied by using the global hopf bifurcation theory finally some numerical simulations are carried out to illustrate the analytical results

normal forms for retarded functional differential equations with parameters and applications to hopf bifurcation abstract the paper addresses for retarded functional differential equations fdes with parameters the computation of normal forms associated with the flow on a finite dimensional invariant manifold tangent to an invariant space for the infinitesimal generator of the linearized equation at a singularity the analysis is based on the theory previously developed for fdes without parameters by considering an extension of a phase space appropriate to the computation of these normal forms as an application the generic hopf bifurcation for scalar retarded fdes is considered

periodic solutions of abstract neutral functional differential equations abstract we characterize the existence of periodic solutions of some abstract neutral functional differential equations with finite and infinite delay when the underlying space is a umd space

robust bddc preconditioners for reissner mindlin plate bending problems and mitc elements a balancing domain decomposition method by constraints bddc is constructed and analyzed for the reissner mindlin plate bending problem discretized with mixed interpolation of tensorial components mitc finite elements this bddc algorithm is based on selecting the plate rotations and deflection degrees of freedom at the subdomain vertices as primal continuity constraints after the implicit elimination of the interior degrees of freedom in each subdomain the resulting plate schur complement is solved by the preconditioned conjugate gradient method the preconditioner is based on the solution of local reissner mindlin plate problems on each subdomain with clamping conditions at the primal degrees of freedom and on the solution of a coarse reissner mindlin plate problem for the primal degrees of freedom the main results of the paper are the proof and numerical verification that the proposed bddc plate algorithm is scalable quasi optimal and most important robust with respect to the plate thickness while this result is due to an underlying mixed formulation of the problem both the interface plate problem and the preconditioner are positive definite the numerical results also show that the proposed algorithm is robust with respect to discontinuities of the material properties

modelling practical lot sizing problems as mixed integer programs in spite of the remarkable improvements in the quality of general purpose mixed integer programming software the effective solution of a variety of lot sizing problems depends crucially on the development of tight formulations for the special problem features occurring in practice after reviewing some of the basic preprocessing techniques for handling safety stocks and multilevel problems we discuss a variety of aspects arising particularly in small and large bucket time period models such as start ups changeovers minimum batch sizes choice of one or two set ups per period etc a set of applications is described that contains one or more of these special features and some indicative computational results are presented finally to show another technique that is useful a slightly different supply chain application is presented for which the a priori addition of some simple mixed integer inequalities based on aggregation leads to important improvements in the results

a deluxe feti dp algorithm for a hybrid staggered discontinuous galerkin method for h curl elliptic problems summary convergence theories and a deluxe dual and primal finite element tearing and interconnecting algorithm are developed for a hybrid staggered dg finite element approximation of h curl elliptic problems in two dimensions in addition to the advantages of staggered dg methods the basis functions of the new hybrid staggered dg method are all locally supported in the triangular elements and a lagrange multiplier approach is applied to enforce the global connections of these basis functions the interface problem on the lagrange multipliers is further reduced to the resulting problem on the subdomain interfaces and a dual and primal finite element tearing and interconnecting algorithm with an enriched weight factor is then applied to the resulting problem our algorithm is shown to give a condition number bound ofc 1c log h h 2 independent of the two parameters whereh his the number of triangles across each subdomain numerical results are included to confirm our theoretical bounds copyright 2013 john wiley sons ltd received 12 september 2013 accepted 11 november 2013 in this paper a dual and primal finite element tearing and interconnecting feti dp algorithm is developed for a fast and stable solution of a new hybrid staggered dg hsdg method applied to h curl problems staggered dg sdg methods were first introduced in 1 3 for wave propagation problems the idea was subsequently applied to other problems such as convection diffusion equations 4 electromagnetic problems 5 9 stokes equations 10 and multiscale wave simulations 11 12 similar to 5 the advantages of using sdg for h curl problems are the preservation of the structures of the differential operators the local conservation property and the optimal convergence in particular the discretizations of the two curl operators by the sdg method are adjoint to each other and the null space of the discrete curl operator is exactly the gradients moreover the divergence condition is automatically satisfied in an appropriate weak sense despite the aforementioned advantages the implementation of sdg methods requires careful numbering of the supports of the basis functions to allow an easier implementation of the sdg method the hsdg method is introduced in this paper one distinctive advantage of the hsdg method is that the basis functions are all locally supported in the triangular elements hence the numbering can be easily performed the fact that the basis functions are totally discontinuous requires some global couplings in particular we will use the lagrange multiplier approach to enforce the global connection of the basis functions this idea is also related to the hybridized dg methods 13 16 in fact it is shown that the sdg method is the limit of a hybridized dg method

analysis of feti methods for multiscale pdes in this paper we study a variant of the finite element tearing and interconnecting feti method which is suitable for elliptic pdes with highly heterogeneous multiscale coefficients x in particular coefficients with strong variation within subdomains and or jumps that are not aligned with the subdomain interfaces using energy minimisation and cut off arguments we can show rigorously that for an arbitrary positive coefficient function alpha in l infty omega the condition number of the preconditioned feti system can be bounded by c 1 log h h 2 where h is the subdomain diameter and h is the mesh size and where the function c depends only on the coefficient variation in the vicinity of subdomain interfaces in particular if alpha omega i varies only mildly in a layer i of width near the boundary of each of the subdomains i then c alpha mathcal o h eta 2 independent of the variation of in the remainder i i of each subdomain and independent of any jumps of across subdomain interfaces the quadratic dependence of c on h can be relaxed to a linear dependence under stronger assumptions on the behaviour of in the interior of the subdomains our theoretical findings are confirmed in numerical tests

nonlinear parallel in time schur complement solvers for ordinary differential equations abstract in this work we propose a parallel in time solver for linear and nonlinear ordinary differential equations the approach is based on an efficient multilevel solver of the schur complement related to a multilevel time partition for linear problems the scheme leads to a fast direct method next two different strategies for solving nonlinear odes are proposed first we consider a newton method over the global nonlinear ode using the multilevel schur complement solver at every nonlinear iteration second we state the global nonlinear problem in terms of the nonlinear schur complement at an arbitrary level and perform nonlinear iterations over it numerical experiments show that the proposed schemes are weakly scalable i e we can efficiently exploit increasing computational resources to solve for more time steps the same problem

three level bddc in two dimensions balancing domain decomposition by constraints bddc methods are non overlapping iterative substructuring domain decomposition methods for the solution of large sparse linear algebraic systems arising from discretization of elliptic boundary value problems they are similar to the balancing neumann neumann algorithm however in bddc methods a small number of continuity constraints are enforced across the interface and these constraints form a new coarse global component an important advantage of using such constraints is that the schur complements that arise in the computation will all be strictly positive definite the matrix of the coarse problem is generated and factored by direct solvers at the beginning of the computation however this problem can ultimately become a bottleneck if the number of subdomains is very large in this paper two three level bddc methods are introduced for solving the coarse problem approximately in two dimensional space while still maintaining a good convergence rate estimates of the condition numbers are provided for the two three level bddc methods and numerical experiments are also discussed copyright 2006 john wiley sons ltd

newton krylov bddc solvers for nonlinear cardiac mechanics abstract the aim of this work is to design and study a balancing domain decomposition by constraints bddc solver for the nonlinear elasticity system modeling the mechanical deformation of cardiac tissue the contraction relaxation process in the myocardium is induced by the generation and spread of the bioelectrical excitation throughout the tissue and it is mathematically described by the coupling of cardiac electro mechanical models consisting of systems of partial and ordinary differential equations in this study the discretization of the electro mechanical models is performed by q1 finite elements in space and semi implicit finite difference schemes in time leading to the solution of a large scale linear system for the bioelectrical potentials and a nonlinear system for the mechanical deformation at each time step of the simulation the parallel mechanical solver proposed in this paper consists in solving the nonlinear system with a newton krylov bddc method based on the parallel solution of local mechanical problems and a coarse problem for the so called primal unknowns three dimensional parallel numerical tests on different machines show that the proposed parallel solver is scalable in the number of subdomains quasi optimal in the ratio of subdomain to mesh sizes and robust with respect to tissue anisotropy

inexact newton methods with restricted additive schwarz based nonlinear elimination for problems with high local nonlinearity the classical inexact newton algorithm is an efficient and popular technique for solving large sparse nonlinear systems of equations when the nonlinearities in the system are well balanced a near quadratic convergence is often observed however if some of the equations are much more nonlinear than the others in the system the convergence is much slower the slow convergence or sometimes divergence is often determined by the small number of equations in the system with the highest nonlinearities the idea of nonlinear preconditioning has been proven to be very useful through subspace nonlinear solves the local high nonlinearities are removed and the fast convergence can then be restored when the inexact newton algorithm is called after the preconditioning recently a left preconditioned inexact newton method was proposed in which the nonlinear function is replaced by a preconditioned function with more balanced nonlinearities in this paper we combine an inexact newton method with a restricted additive schwarz based nonlinear elimination the new approach is easier to implement than the left preconditioned method since the nonlinear function does not have to be replaced and furthermore the nonlinear elimination step does not have to be called at every outer newton iteration we show numerically that it performs well for as an example solving the incompressible navier stokes equations with high reynolds numbers and on machines with large numbers of processors

a new dantzig wolfe reformulation and branch and price algorithm for the capacitated lot sizing problem with setup times although the textbook dantzig wolfe decomposition reformulation for the capacitated lot sizing problem as already proposed by manne manne a s 1958 programming of economic lot sizes management sci 4 2 115 135 provides a strong lower bound it also has an important structural deficiency imposing integrality constraints on the columns in the master program will not necessarily give the optimal integer programming solution manne s model contains only production plans that satisfy the wagner whitin property and it is well known that the optimal solution to a capacitated lot sizing problem will not necessarily satisfy this property the first contribution of this paper answers the following question unsolved for almost 50 years if manne s formulation is not equivalent to the original problem what is then a correct reformulation we develop an equivalent mixed integer programming mip formulation to the original problem and show how this results from applying the dantzig wolfe decomposition to the original mip formulation the set of extreme points of the lot size polytope that are needed for this mip dantzig wolfe reformulation is much larger than the set of dominant plans used by manne we further show how the integrality restrictions on the original setup variables translate into integrality restrictions on the new master variables by separating the setup and production decisions our new formulation gives the same lower bound as manne s reformulation second we develop a branch and price algorithm for the problem computational experiments are presented on data sets available from the literature column generation is accelerated by a combination of simplex and subgradient optimization for finding the dual prices the results show that branch and price is computationally tractable and competitive with other state of the art approaches found in the literature

self sustained oscillations in a ring array of coupled lossless transmission lines abstract in this paper we derive a neutral difference differential system with diffusion which arises from a ring array of coupled lossless transmission lines we investigate the problem of self sustained oscillations of the considered transmission lines and apply a global hopf bifurcation theorem to establish the existence of multiple large amplitude phase locked periodic solutions in the corresponding neutral system

the two level feti method for static and dynamic plate problems part i an optimal iterative solver for biharmonic systems abstract we present a lagrange multiplier based substructuring method for solving iteratively large scale systems of equations arising from the finite element discretization of static and dynamic plate bending problems the proposed method is essentially an extension of the feti domain decomposition algorithm to fourth order problems the main idea is to enforce exactly the continuity of the transverse displacement field at the substructure corners throughout the preconditioned conjugate projected gradient iterations this results in a two level feti substructuring method where the condition number of the preconditioned interface problem does not grow with the number of substructures and grows at most polylogarithmically with the number of elements per substructure these theoretically proven optimal convergence properties of the new feti method are numerically demonstrated for several finite element plate bending static and transient problems the two level iterative solver presented in this paper is applicable to a large family of biharmonic time independent as well as time dependent systems it is also extendible to shell problems

spectral element feti dp and bddc preconditioners with multi element subdomains abstract the discrete systems generated by spectral or hp version finite elements are much more ill conditioned than the ones generated by standard low order finite elements or finite differences this paper focuses on spectral elements based on gauss lobatto legendre gll quadrature and the construction of primal and dual non overlapping domain decomposition methods belonging to the family of balancing domain decomposition methods by constraints bddc and dual primal finite element tearing and interconnecting feti dp algorithms new results are presented for the spectral multi element case and also for inexact feti dp methods for spectral elements in the plane theoretical convergence estimates show that these methods have a convergence rate independent of the number of subdomains and coefficient jumps of the elliptic operator while there is only a polylogarithmic dependence on the spectral degree p and the ratio h h of subdomain and element sizes parallel numerical experiments on a linux cluster confirm these results for tests with spectral degree up to p 32 thousands of subdomains and coefficient jumps up to 8 orders of magnitude

set partitioning and column generation heuristics for capacitated dynamic lotsizing abstract this paper discusses set partitioning and column generation heuristics for the multi item single level capacitated dynamic lotsizing problem most previous research efforts reported in literature use the single item uncapacitated problem solutions as candidate plans in this paper a different approach is proposed in which candidate plans are also generated by some well known multi item capacitated dynamic lotsizing heuristics the lp relaxation of a set partitioning model is solved and rounded to obtain an integer solution a number of heuristics are subsequently applied to improve upon this initial solution heuristics are also used to convert the possibly fractional solution from the column generation step to a feasible integer one our methods are compared with other common sense heuristics computational experience shows that our methods perform best for problems with fewer periods than items but are quite good on average

unconditionally stable operator splitting algorithms for the incompressible magnetohydrodynamics system discretized by a stabilized finite element formulation based on projections summary in this article we propose different splitting procedures for the transient incompressible magnetohydrodynamics mhd system that are unconditionally stable we consider two levels of splitting on one side we perform the segregation of the fluid pressure and magnetic pseudo pressure from the vectorial fields computation at the second level the fluid velocity and induction fields are also decoupled this way we transform a fully coupled indefinite multi physics system into a set of smaller definite ones clearly reducing the cpu cost with regard to the finite element approximation we stick to an unconditionally convergent stabilized finite element formulation because it introduces convection stabilization allows to circumvent inf sup conditions clearly simplifying implementation issues and is able to capture non smooth solutions of the magnetic subproblem however residual based finite element formulations are not suitable for segregation because they lose the skew symmetry of the off diagonal blocks therefore in this work we have proposed a novel term by term stabilization of the mhd system based on projections that is still unconditionally convergent copyright 2012 john wiley sons ltd

a domain decomposition method with lagrange multipliers and inexact solvers for linear elasticity a new domain decomposition method with lagrange multipliers for elliptic problems is introduced it is based on a reformulation of the well known finite element tearing and interconnecting feti method as a saddle point problem with both primal and dual variables as unknowns the resulting linear system is solved with block structured preconditioners combined with a suitable krylov subspace method this approach allows the use of inexact subdomain solvers for the positive definite subproblems it is shown that the condition number of the preconditioned saddle point problem is bounded independently of the number of subregions and depends only polylogarithmically on the number of degrees of freedom of individual local subproblems numerical results are presented for a plane stress cantilever membrane problem

nonlinear feti dp and bddc methods a unified framework and parallel results parallel newton krylov feti dp finite element tearing and interconnecting dual primal domain decomposition methods are fast and robust solvers e g for nonlinear implicit problems in structural mechanics in these methods the nonlinear problem is first linearized and then decomposed into loosely coupled linear problems which can be solved in parallel by changing the order of the operations new parallel communication can be constructed where the loosely coupled local problems are nonlinear we discuss different nonlinear feti dp methods which are equivalent when applied to linear problems but which show a different performance for nonlinear problems moreover a new unified framework is introduced which casts all nonlinear feti dp domain decomposition approaches discussed in the literature into a single algorithm furthermore the equivalence of nonlinear feti dp methods to specific nonlinearly right preconditioned newton krylov methods is shown for the methods using nested newton iteration

meta heuristics for dynamic lot sizing a review and comparison of solution approaches proofs from complexity theory as well as computational experiments indicate that most lot sizing problems are hard to solve because these problems are so difficult various solution techniques have been proposed to solve them in the past decade meta heuristics such as tabu search genetic algorithms and simulated annealing have become popular and efficient tools for solving hard combinatorial optimization problems we review the various meta heuristics that have been specifically developed to solve lot sizing problems discussing their main components such as representation evaluation neighborhood definition and genetic operators further we briefly review other solution approaches such as dynamic programming cutting planes dantzig wolfe decomposition lagrange relaxation and dedicated heuristics this allows us to compare these techniques understanding their respective advantages and disadvantages gives insight into how we can integrate elements from several solution approaches into more powerful hybrid algorithms finally we discuss general guidelines for computational experiments and illustrate these with several examples

a two level nonoverlapping schwarz algorithm for the stokes problem numerical study abstract a general framework of a two level nonoverlapping schwarz algorithm for the stokes problem is developed by relaxing average zero condition on pressure unknowns this framework allows both discontinuous and continuous pressure finite element spaces the coarse problem is built by algebraic manipulation after selecting appropriate primal unknowns just like in bddc algorithms performance of the suggested algorithm is presented depending on the selection of finite elements and primal unknowns under the same set of primal unknowns the algorithm for the case with discontinuous pressure functions outperforms one with continuous pressure functions for the two dimensional stokes problem the algorithm with a set of primal unknowns consisting of velocity unknowns at corners averages of velocity components over common edges and pressure unknowns at corners presents good scalability when continuous pressure test functions are used in both two and three dimensional stokes problems an improvement can be made for the case with continuous pressure test functions by applying the suggested algorithm to the interface problem which is obtained by eliminating velocity unknowns and pressure unknowns interior to each subdomains

double hopf bifurcation in a container crane model with delayed position feedback in this paper we study dynamics in a container crane model with delayed position feedback with particular attention focused on non resonant double hopf bifurcation by using multiple time scales and center manifold reduction methods we obtain the equivalent normal forms near a double hopf critical point in this neutral delayed differential system moreover bifurcations are classified in a two dimensional parameter space near the critical point numerical simulations are presented to demonstrate the applicability of the theoretical results

a preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems this paper provides a preconditioned iterative technique for the solution of saddle point problems these problems typically arise in the numerical approximation of partial differential equations by lagrange multiplier techniques and or mixed methods the saddle point problem is reformulated as a symmetric positive definite system which is then solved by conjugate gradient iteration applications to the equations of elasticity and stokes are discussed and the results of numerical experiments are given

a study on the lumped preconditioner and memory requirements of feti and related primal domain decomposition methods in recent years domain decomposition methods ddms have emerged as advanced solvers in several areas of computational mechanics in particular during the last decade in the area of solid and structural mechanics they reached a considerable level of advancement and were shown to be more efficient than popular solvers like advanced sparse direct solvers the present contribution follows the lines of a series of recent publications on the relationship between primal and dual formulations of ddms in some of these papers the effort to unify primal and dual methods led to a family of ddms that was shown to be more efficient than the previous methods the present paper extends this work presenting a new family of related ddms thus enriching the theory of the relations between primal and dual methods with the primal methods which correspond to the dual ddm that uses the lumped preconditioner the paper also compares the numerical performance of the new methods with that of the previous ones and focuses particularly on memory requirement issues related to the use of the lumped preconditioner suggesting a particularly memory efficient formulation

bddc preconditioners for spectral element discretizations of almost incompressible elasticity in three dimensions balancing domain decomposition by constraints bddc algorithms are constructed and analyzed for the system of almost incompressible elasticity discretized with gauss lobatto legendre spectral elements in three dimensions initially mixed spectral elements are employed to discretize the almost incompressible elasticity system but a positive definite reformulation is obtained by eliminating all pressure degrees of freedom interior to each subdomain into which the spectral elements have been grouped appropriate sets of primal constraints can be associated with the subdomain vertices edges and faces so that the resulting bddc methods have a fast convergence rate independent of the almost incompressibility of the material in particular the condition number of the bddc preconditioned operator is shown to depend only weakly on the polynomial degree n the ratio h h of subdomain and element diameters and the inverse of the inf sup constants of the subdomains and the underlying mixed formulation while being scalable i e independent of the number of subdomains and robust i e independent of the poisson ratio and young s modulus of the material considered these results also apply to the related dual primal finite element tearing and interconnect feti dp algorithms defined by the same set of primal constraints numerical experiments carried out on parallel computing systems confirm these results

primal and dual interface concentrated iterative substructuring methods this paper is devoted to the fast solution of interface concentrated finite element equations the interface concentrated finite element schemes are constructed on the basis of a nonoverlapping domain decomposition where a conforming boundary concentrated finite element approximation is used in every subdomain similar to boundary element domain decomposition methods the total number of unknowns per subdomain behaves like o h h d 1 where h h and d denote the usual scaling parameter of the subdomains the average discretization parameter of the subdomain boundaries and the spatial dimension respectively we propose and analyze primal and dual substructuring iterative methods which asymptotically exhibit the same or at least almost the same complexity as the number of unknowns in particular the so called all floating finite element tearing and interconnecting solvers are highly parallel and very robust with respect to large coefficient jumps

inexact data sparse boundary element tearing and interconnecting methods the boundary element tearing and interconnecting beti methods have recently been introduced as boundary element counterparts of the well established finite element tearing and interconnecting feti methods in this paper we present inexact data sparse versions of the beti methods which avoid the elimination of the primal unknowns and dense matrices however instead of symmetric and positive definite systems we finally have to solve twofold saddle point problems the proposed iterative solvers and preconditioners result in almost optimal solvers whose complexity is proportional to the number of unknowns on the skeleton up to some polylogarithmical factor moreover the solvers are robust with respect to large coefficient jumps

a three level bddc algorithm for mortar discretizations in this paper a three level balancing domain decomposition by constraints bddc algorithm is developed for the solutions of large sparse algebraic linear systems arising from the mortar discretization of elliptic boundary value problems the mortar discretization is considered on geometrically nonconforming subdomain partitions in two level bddc algorithms the coarse problem needs to be solved exactly however its size will increase with the increase of the number of the subdomains to overcome this limitation the three level algorithm solves the coarse problem inexactly while a good rate of convergence is maintained this is an extension of previous work the three level bddc algorithms for standard finite element discretization estimates of the condition numbers are provided for the three level bddc method and numerical experiments are also discussed

choosing the forcing terms in an inexact newton method an inexact newton method is a generalization of newton s method for solving f x 0 f mathbb r n to mathbb r n in which at the kth iteration the step s k from the current approximate solution x k is required to satisfy a condition f x k f x k s k leqslant eta k f x k for a forcing term eta k in 0 1 in typical applications the choice of the forcing terms is critical to the efficiency of the method and can affect robustness as well promising choices of the forcing terms are given their local convergence properties are analyzed and their practical performance is shown on a representative set of test problems

bddc and feti dp preconditioners with adaptive coarse spaces for three dimensional elliptic problems with oscillatory and high contrast coefficients abstract bddc and feti dp algorithms are developed for three dimensional elliptic problems with adaptively enriched coarse components it is known that these enriched components are necessary in the development of robust preconditioners to form the adaptive coarse components carefully designed generalized eigenvalue problems are introduced for each faces and edges and the coarse components are formed by using eigenvectors with their corresponding eigenvalues larger than a given tolerance t o l upper bounds for condition numbers of the preconditioned systems are shown to be c t o l with the constant c depending only on the maximum number of edges and faces per subdomain and the maximum number of subdomains sharing an edge numerical results are presented to test the robustness of the proposed approach

a new heuristic approach for the multi item dynamic lot sizing problem abstract in this paper a framework for a new heuristic approach for solving the single level multi item capacitated dynamic lot sizing problem is presented the approach uses an iterative item by item strategy for generating solutions to the problem in each iteration a set of items are scheduled over the planning horizon and the procedure terminates when all items are scheduled an algorithm that implements this approach is developed in which in each iteration a single item is selected and scheduled over the planning horizon each item is scheduled by the solution of a bounded single item lot sizing problem where bounds on inventory and production levels are used to ensure feasibility of the overall problem the performance of this algorithm is compared to some well known heuristics over a set of test problems the computational results demonstrated that on the average our algorithm outperforms other algorithms the suggested algorithm especially appears to outperform other algorithm for problems with many periods and few items in the literature these problems are considered as hard

hopf bifurcation analysis in a delayed nicholson blowflies equation abstract the dynamics of a nicholson s blowflies equation with a finite delay are investigated we prove that a sequence of hopf bifurcations occur at the positive equilibrium as the delay increases explicit algorithm for determining the direction of the hopf bifurcations and the stability of the bifurcating periodic solutions are derived using the theory of normal form and center manifold global existence of periodic solutions are established using a global hopf bifurcation result of wu trans amer math soc 350 1998 4799 and a bendixson criterion for higher dimensional ordinary differential equations due to li and muldowney j differential equations 106 1994 27

solving multi item capacitated lot sizing problems using variable redefinition mixed integer programming models are typically not used to solve realistic sized production scheduling problems because they require exorbitant solution times we impose a useful taxonomy on production scheduling problems and develop alternative formulations for a wide variety of problems within the taxonomy the linear programming relaxation of the new models is very effective in generating bounds we show that these bounds are equal to those that could be generated using lagrangian relaxation or column generation the linear programming bounds increase in effectiveness as the problems become larger perhaps of greatest significance is that practitioners can obtain our results using only standard off the shelf codes such as lindo or mpsx 370 we report computational experience in several computing environments hardware and software on problems with up to 200 products and 10 time periods 2000 0 1 variables

feti dp bddc and block cholesky methods the feti dp and bddc algorithms are reformulated using block cholesky factorizations an approach which can provide a useful framework for the design of domain decomposition algorithms for solving symmetric positive definite linear system of equations instead of introducing lagrange multipliers to enforce the coarse level primal continuity constraints in these algorithms a change of variables is used such that each primal constraint corresponds to an explicit degree of freedom with the new formulation of these algorithms a simplified proof is provided that the spectra of a pair of feti dp and bddc algorithms with the same set of primal constraints are essentially the same numerical experiments for a two dimensional laplace s equation also confirm this result copyright 2005 john wiley sons ltd