memetic algorithm based on improved inver over operator and lin kernighan local search for the euclidean traveling salesman problem in this study an improved inver over operator is proposed to solve the euclidean traveling salesman problem tsp problem the improved inver over operator is tested on 14 different tsp examples selected from tsplib the application of the improved inver over operator gives much more effective results regarding to the best and average error values than the basic inver over operator then an effective memetic algorithm based on improved inver over operator and lin kernighan local search is implemented to speed up the convergence capability of the presented algorithm a restart technique is employed we evaluate the proposed algorithm based on standard tsp test problems and show that the proposed algorithm performs better than other memetic algorithm in terms of solution quality and computational effort

homogenization of fully nonlinear uniformly elliptic and parabolic partial differential equations in stationary ergodic media we study the homogenization of fully nonlinear uniformly elliptic and parabolic second order partial differential equations in stationary ergodic media c 2004 wiley periodicals inc

γ convergence of multiscale periodic functionals depending on the time derivative and the curl operator abstract we study the convergence of a family of multiscale periodic quadratic integral functionals defined in a product space whose densities depend on the time derivative and on the curl of solenoidal fields through the multiscale convergence in time space and the multiscale young measures in time space associated with relevant sequences of pairs an explicit representation of the limit density is given by means of an infinite dimensional minimization problem

matrix reordering using multilevel graph coarsening for ilu preconditioning incomplete lu factorization ilu techniques are a well known class of preconditioners often used in conjunction with krylov accelerators for the iterative solution of linear systems of equations however for certain problems ilu factorizations can yield factors that are unstable and in some cases quite dense reordering techniques based on permuting the matrix prior to performing the factorization have been shown to improve the quality of the factorization and the resulting preconditioner in this paper we examine the effect of reordering techniques based on multilevel graph coarsening ideas on one level ilu factorizations such as the level based ilu k or the dual threshold ilut algorithms we consider an aggregation based coarsening idea that implements two main coarsening frameworks a top down approach and a bottom up approach each utilizing one of two different strategies to select the next level coarse graph numerical results are presented to support our findings

symmetric permutations for i matrices to delay and avoid small pivots during factorization in this article we present several new permutations for i matrices making these more suitable for incomplete lu factorization preconditioners used in solving linear systems by iterative methods a general matrix can be transformed by row permutation as well as row and columns scaling into an i matrix i e a matrix having elements of modulus 1 on the diagonal and elements of modulus of no more than 1 elsewhere reordering rows and columns by the same permutation clearly preserves i matrices in this article we consider such reordering techniques which make the permuted matrix more suitable for an incomplete lu factorization preconditioner than the original i matrix we use a multilevel iluc an incomplete lu factorization preconditioner using crout s implementation of gaussian elimination without pivoting to test these reorderings the combination of i matrix preprocessing with the various algorithms presented here and the multilevel incomplete lu factorizations forms a powerful preconditioning method for unsymmetric highly indefinite problems the c code has been made available in the software package ilu

a new pivoting strategy for gaussian elimination this paper discusses a method for determining a good pivoting sequence for gaussian elimination based on an algorithm for solving assignment problems the worst case complexity is o n3 in practice o n2 25 operations are sufficient

divergence measure fields and hyperbolic conservation laws we analyze a class of l infty vector fields called divergence measure fields we establish the gauss green formula the normal traces over subsets of lipschitz boundaries and the product rule for this class of l infty fields then we apply this theory to analyze l infty entropy solutions of initial boundary value problems for hyperbolic conservation laws and to study the ways in which the solutions assume their initial and boundary data the examples of conservation laws include multidimensional scalar equations the system of nonlinear elasticity and a class of m x m systems with affine characteristic hypersurfaces the analysis in l infty also extends to l p

a new hybrid heuristic approach for solving large traveling salesman problem this paper presents a new metaheuristic approach called acomac algorithm for solving the traveling salesman problem tsp we introduce multiple ant clans concept from parallel genetic algorithm to search solution space utilizing various islands to avoid local minima and thus can yield global minimum for solving the traveling salesman problem moreover we present two approaches named the multiple nearest neighbor nn and the dual nearest neighbor dnn to acomac to enhance large tsps to validate the proposed methods numerous simulations were conducted to compare acomac and dorigo s acs with and without the addition of the multiple nearest neighbor nn method or the dual nearest neighbor dnn approach using a range of tsp benchmark problems according to the results of the simulation adding the nn or dnn approach to acomac or acs as initial solutions also significantly enhances the performance of acomac and acs in solving the traveling salesman problem meanwhile using acomac dnn with tsp can yield better solutions than the other stated approaches additionally acomac or acomac nn utilizing five ant clans with a total of 20 ants is verified to yield better solutions furthermore acomac with a local weighting w set to 0 6 can yield better solutions in terms of length

perron s method for hamilton jacobi equations on considere l existence des solutions d equations aux derivees partielles non lineaires scalaires d ordre 1 f x u du 0 dans ou est un sous ensemble ouvert de r n f r r n r est continue u r est l inconnue

multiscale convergence and reiterated homogenisation this paper generalises the notion of two scale convergence to the case of multiple separated scales of periodic oscillations it allows us to introduce a multi scale convergence method for the reiterated homogenisation of partial differential equations with oscillating coefficients this new method is applied to a model problem with a finite or infinite number of microscopic scales namely the homogenisation of the heat equation in a composite material finally it is generalised to handle the homogenisation of the neumann problem in a perforated domain

qmr a quasi minimal residual method for non hermitian linear systems the biconjugate gradient bcg method is the natural generalization of the classical conjugate gradient algorithm for hermitian positive definite matrices to general non hermitian linear systems unfortunately the original bcg algorithm is susceptible to possible breakdowns and numerical instabilities in this paper we present a novel bcg like approach the quasi minimal residual qmr method which overcomes the problems of bcg an implementation of qmr based on a look ahead version of the nonsymmetric lanczos algorithm is proposed it is shown how bcg iterates can be recovered stably from the qmr process some further properties of the qmr approach are given and an error bound is presented finally numerical experiments are reported

multi scale young measures we introduce multi scale young measures to deal with problems where multi scale phenomena are relevant we prove some interesting representation results that allow the use of these families of measures in practice and illustrate its applicability by treating from this perspective multi scale convergence and homogenization of multiple integrals

on uniqueness and existence of viscosity solutions of fully nonlinear second order elliptic pde s we prove several comparison and existence theorems for viscosity solutions of fully nonlinear degenerate elliptic equations one of them extends some recent uniqueness results by jensen some establish the uniqueness of solutions for second order isaacs equations and hence include the uniqueness results for bellman equations by p l lions our comparison results apply even for discontinuous solutions and so perron s method readily yields the existence of continuous solutions

guided local search and its application to the traveling salesman problem abstract the traveling salesman problem tsp is one of the most famous problems in combinatorial optimization in this paper we are going to examine how the techniques of guided local search gls and fast local search fls can be applied to the problem gls sits on top of local search heuristics and has as a main aim to guide these procedures in exploring efficiently and effectively the vast search spaces of combinatorial optimization problems gls can be combined with the neighborhood reduction scheme of fls which significantly speeds up the operations of the algorithm the combination of gls and fls with tsp local search heuristics of different eiciency and effectiveness is studied in an effort to determine the dependence of gls on the underlying local search heuristic used comparisons are made with some of the best tsp heuristic algorithms and general optimization techniques which demonstrate the advantages of gls over alternative heuristic approaches suggested for the problem

convergence of the viscosity method for isentropic gas dynamics a convergence theorem for the method of artificial viscosity applied to the isentropic equations of gas dynamics is established convergence of a subsequence in the strong topology is proved without uniform estimates on the derivatives using the theory of compensated compactness and an analysis of progressing entropy waves

a max plus approach to incomplete cholesky factorization preconditioners we present a new method for constructing incomplete cholesky factorization preconditioners for use in solving large sparse symmetric positive definite linear systems this method uses max plus algebra to predict the positions of the largest entries in the cholesky factor and then uses these positions as the sparsity pattern for the preconditioner our method builds on the max plus incomplete lu factorization preconditioner recently proposed in j hook and f tisseur incomplete lu preconditioner based on max plus approximation of lu factorization mims eprint 2016 47 manchester 2016 but applied to symmetric positive definite matrices which comprise an important special case for the method and its application a attractive feature of our approach is that the sparsity pattern of each column of the preconditioner can be computed in parallel numerical comparisons are made with other incomplete cholesky factorization preconditioners using problems from a range of practical applications we demonstrate that the new preconditioner can outperform traditional level based preconditioners and offer a parallel alternative to a serial limited memory based approach

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

weighted matchings for preconditioning symmetric indefinite linear systems maximum weight matchings have become an important tool for solving highly indefinite unsymmetric linear systems especially in direct solvers in this study we investigate the benefit of reorderings and scalings based on symmetrized maximum weight matchings as a preprocessing step for incomplete mathrm ldl t factorizations the reorderings are constructed such that the matched entries form 1 times 1 or 2 times 2 diagonal blocks in order to increase the diagonal dominance of the system during the incomplete factorization only tridiagonal pivoting is used we report results for this approach and comparisons with other solution methods for a diverse set of symmetric indefinite matrices ranging from nonlinear elasticity to interior point optimization

methods for scaling to doubly stochastic form abstract new methods for scaling square nonnegative matrices to doubly stochastic form are described a generalized version of the convergence theorem of sinkhorn and knopp 1967 is proved and applied to show convergence for these new methods tests indicate that one of the new methods has significantly better average and worst case behavior than the sinkhorn knopp method for one of the 3 3 examples of marshall and olkin 1968 sk requires 130 times as many operations as the new algorithm to achieve row and column sums 1 10 5

linear time approximation for maximum weight matching the maximum cardinality and maximum weight matching problems can be solved in o m n time a bound that has resisted improvement despite decades of research here m and n are the number of edges and vertices in this article we demonstrate that this m n barrier can be bypassed by approximation for any e 0 we give an algorithm that computes a 1 e approximate maximum weight matching in o me 1 log e 1 time that is optimal linear time for any fixed e our algorithm is dramatically simpler than the best exact maximum weight matching algorithms on general graphs and should be appealing in all applications that can tolerate a negligible relative error

preconditioning techniques for large linear systems a survey this article surveys preconditioning techniques for the iterative solution of large linear systems with a focus on algebraic methods suitable for general sparse matrices covered topics include progress in incomplete factorization methods sparse approximate inverses reorderings parallelization issues and block and multilevel extensions some of the challenges ahead are also discussed an extensive bibliography completes the paper

asymptotic decay of besicovitch almost periodic entropy solutions to anisotropic degenerate parabolic hyperbolic equations abstract we prove the well posedness and the asymptotic decay to the mean value of besicovitch almost periodic solutions to nonlinear anisotropic degenerate parabolic hyperbolic equations

a branch and cut algorithm for the resolution of large scale symmetric traveling salesman problems an algorithm is described for solving large scale instances of the symmetric traveling salesman problem stsp to optimality the core of the algorithm is a polyhedral cutting plane procedure that exploits a subset of the system of linear inequalities defining the convex hull of the incidence vectors of the hamiltonian cycles of a complete graph the cuts are generated by several identification procedures that have been described in a companion paper whenever the cutting plane procedure does not terminate with an optimal solution the algorithm uses a tree search strategy that as opposed to branch and bound keeps on producing cuts after branching the algorithm has been implemented in fortran two different linear programming lp packages have been used as the lp solver the implementation of the algorithm and the interface with one of the lp solvers is described in sufficient detail to permit the replication of our experiments computational results are reported with up to 42 stsps with sizes rangin

homogenization of degenerate porous medium type equations in ergodic algebras abstract we consider the homogenization problem for general porous medium type equations of the form u t f x x e u the pressure function f x y may be of two different types in the type 1 case f x y is a general strictly increasing function this is a mildly degenerate case in the type 2 case f x y has the form h x y f u s x y where f u is just a nondecreasing function this is a strongly degenerate case we address the initial boundary value problem for a general bounded or unbounded domain with null or more generally steady pressure condition on the boundary the homogenization is carried out in the general context of ergodic algebras as far as the authors know homogenization of such degenerate quasilinear parabolic equations is addressed here for the first time we also review the existence and stability theory for such equations and establish new results needed for the homogenization analysis further we include some new results on algebras with mean value specially a new criterion establishing the null measure of level sets of elements of the algebra which is useful in connection with the homogenization of porous medium type equations in the type 2 case

the maximum principle for viscosity solutions of fully nonlinear second order partial differential equations we prove that viscosity solutions in w1 of the second order fully nonlinear equation f d2u du u 0 are unique when i f is degenerate elliptic and decreasing in u or ii f is uniformly elliptic and nonincreasing in u we do not assume that f is convex the method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions respectively this method is completely different from that used in lions 8 9 for second order problems with f convex in d 2 u and from that used by crandall lions 3 and crandall evans lions 2 for fully nonlinear first order problems

reiterated homogenization of the vector potential formulation of a magnetostatic problem in anisotropic composite media abstract we present an explicit characterization of the effective coefficients of a family of boundary value problems with multiscale periodic oscillatory coefficients which correspond to the vector potential formulation of a magnetostatic problem in anisotropic composite media with periodic microstructures moreover we study the convergence of sequences of multiscale periodic integral functionals depending on the curl of divergence free fields applying the properties of multiscale young measures associated with sequences of divergence free fields

homogenization of linear and nonlinear transport equations we study the homogenization of the linear and nonlinear transport equations with oscillatory velocity fields two types of homogenized equations are derived for general n dimensional linear and nonlinear problems we derive homogenized equations by introducing additional independent variables to represent the small scales for the two dimensional linear transport equations we derive effective equations for the averaged quantities such equations take the form of either a degenerate non local diffusion equation with memory or a higher order hyperbolic equation to study the nonlinear transport equations we introduce the concept of two scale young measure and extend diperna s method to prove that it reduces to a family of dirac measures

multiscale young measures in homogenization of continuous stationary processes in compact spaces and applications abstract we introduce a framework for the study of nonlinear homogenization problems in the setting of stationary continuous processes in compact spaces the latter are functions f t r n q q with f t x f t x where q is a compact hausdorff topological space f c q and t x q q x r n is an n dimensional continuous dynamical system endowed with an invariant radon probability measure it can be easily shown that for almost all q the realization f t x belongs to an algebra with mean value that is an algebra of functions in buc r n containing all translates of its elements and such that each of its elements possesses a mean value this notion was introduced by zhikov and krivenko v v zhikov e v krivenko homogenization of singularly perturbed elliptic operators mat zametki 33 1983 571 582 english transl in math notes 33 1983 294 300 we then establish the existence of multiscale young measures in the setting of algebras with mean value where the compactifications of r n provided by such algebras plays an important role these parametrized measures are useful in connection with the existence of correctors in homogenization problems we apply this framework to the homogenization of a porous medium type equation in r n with a stationary continuous process as a stiff oscillatory external source this application seems to be new even in the classical context of periodic homogenization

effective hamiltonians and averaging for hamiltonian dynamics i this paper building upon ideas of mather moser fathi e and others applies pde partial differential equation methods to understand the structure of certain hamiltonian flows the main point is that the cell or corrector pde introduced and solved in a weak sense by lions papanicolaou and varadhan in their study of periodic homogenization for hamilton jacobi equations formally induces a canonical change of variables in terms of which the dynamics are trivial we investigate to what extent this observation can be made rigorous in the case that the hamiltonian is strictly convex in the momenta given that the relevant pde does not usually in fact admit a smooth solution

multilevel ilu with reorderings for diagonal dominance this paper presents a preconditioning method based on combining two sided permutations with a multilevel approach the nonsymmetric permutation exploits a greedy strategy to put large entries of the matrix in the diagonal of the upper leading submatrix the method can be regarded as a complete pivoting version of the incomplete lu factorization this leads to an effective incomplete factorization preconditioner for general nonsymmetric irregularly structured sparse linear systems

algebraic multilevel preconditioner for the helmholtz equation in heterogeneous media an algebraic multilevel ml preconditioner is presented for the helmholtz equation in heterogeneous media it is based on a multilevel incomplete ldl t factorization and preserves the inherent complex symmetry of the helmholtz equation the ml preconditioner incorporates two key components for efficiency and numerical stability symmetric maximum weight matchings and an inverse based pivoting strategy the former increases the block diagonal dominance of the system whereas the latter controls l 1 for numerical stability when applied recursively their combined effect yields an algebraic coarsening strategy similar to algebraic multigrid methods even for highly indefinite matrices the ml preconditioner is combined with a krylov subspace method and applied as a black box solver to a series of challenging two and three dimensional test problems mainly from geophysical seismic imaging the numerical results demonstrate the robustness and efficiency of the ml preconditioner even at higher frequency regimes

pivoting strategies leading to diagonal dominance by rows pivoting strategies for gaussian elimination leading to upper triangular matrices which are diagonally dominant by rows are studied forward error analysis of triangular systems whose coefficient matrices are diagonally dominant by rows is performed we also obtain small bounds of the backward errors for the pivoting strategies mentioned above our examples of matrices include h matrices and some generalizations of diagonally dominant matrices and scaled partial pivoting for the 1 norm is an example of these pivoting strategies in the case of an n times n m matrix a pivoting strategy of computational complexity o n 2 is proposed which satisfies all the results of the paper

new insertion and postoptimization procedures for the traveling salesman problem this paper describes a new insertion procedure and a new postoptimization routine for the traveling salesman problem the combination of the two methods results in an efficient algorithm genius which outperforms known alternative heuristics in terms of solution quality and computing time

cell centered finite volume method for heterogeneous anisotropic poromechanics problem abstract the article introduces a finite volume method for the poromechanics problem for heterogeneous anisotropic media the media are characterized by the full tensors of elasticity permeability and biot coefficient the pressure and displacement unknowns are collocated at centers of cells poromechanics represents a multi physics problem of darcy flow and linear elasticity coupled via biot terms based on the flux and the displacement and pressure continuity conditions we derive the coupled block expression for the flux the flux expression is split into a block harmonic part a transversal part and an additional contribution due to gravity the block harmonic part is equivalent to the two point flux approximation method for the darcy problem for the approximation of the transversal part we derive an interpolation method for the coupled pressure and displacement fields valid over discontinuities of the tensors finally we derive the flux expression for generalized boundary conditions the flux expression does not require introduction of auxiliary unknowns at the boundary we evaluate the performance of the proposed method on a number of numerical tests and apply the method to a problem on a grid corresponding to a real oil field

multiscale young measures in almost periodic homogenization and applications we prove the existence of multiscale young measures associated with almost periodic homogenization we give applications of this tool in the homogenization of nonlinear partial differential equations with an almost periodic structure such as scalar conservation laws nonlinear transport equations hamilton jacobi equations and fully nonlinear elliptic equations motivated by the application in nonlinear transport equations we also prove basic results on flows generated by lipschitz almost periodic vector fields which are of interest in their own in our analysis an important role is played by the so called bohr compactification mathbb g n of mathbb r n this is a natural parameter space for the young measures our homogenization results provide also the asymptotic behavior for the whole set of mathbb g n translates of the solutions which is in the spirit of recent studies on the homogenization of stationary ergodic processes

homogenization of a stochastic nonlinear reaction diffusion equation with a large reaction term the almost periodic framework abstract homogenization of a stochastic nonlinear reaction diffusion equation with a large nonlinear term is considered under a general besicovitch almost periodicity assumption on the coefficients of the equation we prove that the sequence of solutions of the said problem converges in probability towards the solution of a rather different type of equation namely the stochastic nonlinear convection diffusion equation which we explicitly derive in terms of appropriate functionals we study some particular cases such as the periodic framework and many others this is achieved under a suitable generalized concept of convergence for stochastic processes

interior a priori estimates for solutions of fully non linear equations on etend une theorie de perturbations aux solutions d equations uniformement elliptiques d ordre 2 totalement non lineaires

an effective implementation of the lin kernighan traveling salesman heuristic abstract this paper describes an implementation of the lin kernighan heuristic one of the most successful methods for generating optimal or near optimal solutions for the symmetric traveling salesman problem tsp computational tests show that the implementation is highly effective it has found optimal solutions for all solved problem instances we have been able to obtain including a 13 509 city problem the largest non trivial problem instance solved to optimality today

homogenization of a generalized stefan problem in the context of ergodic algebras abstract we address the deterministic homogenization in the general context of ergodic algebras of a doubly nonlinear problem which generalizes the well known stefan model and includes the classical porous medium equation it may be represented by the differential inclusion for a real valued function u x t t u x e x u x x e x t u u f x e x t u on a bounded domain r n t 0 t together with initial boundary conditions where z x is strictly convex and z x t u is a c 1 convex function both with quadratic growth satisfying some additional technical hypotheses as functions of the oscillatory variable x u x t u and f x t u belong to the generalized besicovitch space b 2 associated with an arbitrary ergodic algebra a the periodic case was addressed by visintin 2007 based on the two scale convergence technique visintin s analysis for the periodic case relies heavily on the possibility of reducing two scale convergence to the usual l 2 convergence in the cartesian product r n where is the periodic cell this reduction is no longer possible in the case of a general ergodic algebra to overcome this difficulty we make essential use of the concept of two scale young measures for algebras with mean value associated with bounded sequences in l 2

pspike a family of parallel hybrid sparse linear system solvers abstract we present pspike as a family of parallel hybrid linear system solvers that are more robust than other available preconditioned iterative methods and more scalable than parallel sparse direct solvers a critical step in using pspike is the extraction of a preconditioner that is in the form of a generalized central band which consists of a set of overlapping diagonal blocks that encapsulate as many of the heaviest elements of the coefficient matrix to address this issue we propose the g paver reordering scheme that produces an effective preconditioner in the form of overlapping diagonal blocks while simultaneously accommodating load balancing on distributed memory architectures and controlling overlap sizes in each outer krylov subspace iteration for solving the linear system we need to solve linear systems involving the preconditioner this is accomplished using a parallel tearing based method which may be regarded as an algebraic domain decomposition scheme the tearing based method requires the solution of a much smaller linear system we call the balance system whose size is equal to the sum of the overlaps in this paper we form the balance system explicitly and solve it directly our parallel implementation of pspike based on g paver pardiso 5 0 0 and solving the explicit balance system results in impressive robustness and good parallel scalability on computing platforms with a limited number of multi core nodes

a distributed memory algorithm for computing a heavy weight perfect matching on bipartite graphs we design and implement an efficient parallel algorithm for finding a perfect matching in a weighted bipartite graph such that weights on the edges of the matching are large this problem differs f

optimized annealing of traveling salesman problem from the nth nearest neighbor distribution abstract we report a new statistical general property in traveling salesman problem that the n th nearest neighbor distribution of optimal tours verifies with very high accuracy an exponential decay as a function of the order of neighbor n defining the energy function as deviation from this exponential decay which is different to the tour length d in normal annealing processes we propose a distinct highly optimized annealing scheme which is performed in space and d space by turns the simulation results of some standard traveling salesman problems in tsplib95 are presented it is shown that our annealing recipe is superior to the canonical simulated annealing

polynomial time approximation schemes for euclidean traveling salesman and other geometric problems we present a polynomial time approximation scheme for euclidean tsp in fixed dimensions for every fixed c 1 and given any n nodes in r 2 a randomized version of the scheme finds a 1 1 c approximation to the optimum traveling salesman tour in o n log n o c time when the nodes are in r d the running time increases to o n log n o d c d 1 for every fixed c d the running time is n poly log n that is nearly linear in n the algorithmm can be derandomized but this increases the running time by a factor o n d the previous best approximation algorithm for the problem due to christofides achieves a 3 2 aproximation in polynomial time we also give similar approximation schemes for some other np hard euclidean problems minimum steiner tree k tsp and k mst the running times of the algorithm for k tsp and k mst involve an additional multiplicative factor k the previous best approximation algorithms for all these problems achieved a constant factor approximation we also give efficient approximation schemes for euclidean min cost matching a problem that can be solved exactly in polynomial time all our algorithms also work with almost no modification when distance is measured using any geometric norm such as e p for p 1 or other minkowski norms they also have simple parallel i e nc implementations

the two scale convergence method applied to generalized besicovitch spaces the twoscale convergence method has proved to be a very useful tool for dealing with periodic homogenization problems in the present paper we develop this theory to generalized besicovitch spaces

exact and approximate correctors for stochastic hamiltonians the 1 dimensional case we perform a qualitative investigation of critical hamilton jacobi equations with stationary ergodic hamiltonian in dimension 1 we show the existence of approximate correctors give characterizing conditions for the existence of correctors provide lax type representation formulae and establish comparison principles the results are applied to look into the corresponding effective hamiltonian and to study a homogenization problem in the analysis a crucial role is played by tools from stochastic geometry such as for instance closed random stationary sets

efficient preconditioning of sequences of nonsymmetric linear systems we present a new approach for approximate updates of factorized nonsymmetric preconditioners for solving sequences of linear algebraic systems this approach is algebraic and it is theoretically motivated it generalizes diagonal updates introduced by benzi and bertaccini bit 43 2003 pp 231 244 and bertaccini electron trans numer anal 18 2004 pp 49 64 it is shown experimentally that this approach can be very beneficial for example it is successful in significantly decreasing the number of iterations of a preconditioned iterative method for solving subsequent systems of a sequence when compared with freezing the preconditioner from the first system of the sequence in some cases the updated preconditioners offer a rate of convergence similar to or even higher than the rate obtained when preconditioning with recomputed preconditioners since the updates are typically cheap and straightforward their use is of practical interest they can replace recomputing preconditioners which is often expensive especially in parallel and matrix free environments