quotient mi groups a many identities group mi group for short is a special algebraic structure in which identity like elements called pseudoidentities are specified and collected into a monoidal substructure in this way many algebraic structures such as monoids of fuzzy intervals numbers or convex bodies possessing behavior very similar to that of a group structure may be well described and investigated using a new approach which seems to be superfluous for the classical structures the concept of mi groups was recently introduced by holcapek and st pnicka in the paper mi algebras a new framework for arithmetics of extensional fuzzy numbers to demonstrate how a standard structure can be generalized in terms of mi algebras this paper is a continuation of the development of mi group theory and is focused on the construction of quotient mi groups and a specification of the conditions under which the isomorphism theorems for groups are fulfilled for mi groups

optimal approximation of sparse hessians and its equivalence to a graph coloring problem we consider the problem of approximating the hessian matrix of a smooth non linear function using a minimum number of gradient evaluations particularly in the case that the hessian has a known fixed sparsity pattern we study the class of direct methods for this problem and propose two new ways of classifying direct methods examples are given that show the relationships among optimal methods from each class the problem of finding a non overlapping direct cover is shown to be equivalent to a generalized graph coloring problem the distance 2 graph coloring problem a theorem is proved showing that the general distance k graph coloring problem is np complete for all fixedk 2 and hence that the optimal non overlapping direct cover problem is also np complete some worst case bounds on the performance of a simple coloring heuristic are given an appendix proves a well known folklore result which gives lower bounds on the number of gradient evaluations needed in any possible approximation method

a numerical study of the limited memory bfgs method and the truncated newton method for large scale optimization this paper examines the numerical performances of two methods for large scale optimization a limited memory quasi newton method l bfgs and a discrete truncated newton method tn various ways of classifying test problems are discussed in order to better understand the types of problems that each algorithm solves well the l bfgs and tn methods are also compared with the polak ribiere conjugate gradient method

optimal estimation of jacobian and hessian matrices that arise in finite difference calculations in this paper the problem of estimating jacobian and hessian matrices arising in the finite difference approximation of partial differential equations is considered using the notion of computational molecule or stencil schemes are developed that require the minimum number of differences to estimate these matrices a procedure applicable to more complicated structures is also given 1 introduction in the past few years there has been a growing interest in studying efficient ways to obtain good estimates of sparse jacobian or hessian matrices two basic approaches have been followed in the first approach it is assumed that one has an estimate of the jacobian hessian which one wishes to revise given arbi trary changes in the variables and the corresponding changes in the functions gradients this has led to the development of several sparse quasi newton methods see 11 14 12 for example in the second approach it is assumed that one can specify the changes in the variables one then forms an estimate of the jacobian hessian matrix from the observed changes in the functions gradients when the dimension of the matrix is large and the evaluation of each function gradient vector is expensive it is necessary to take the sparsity structure of the matrix into account in order to make this approach efficient several methods for doing this for arbitrary sparsity structures have already appeared in the literature 2 5 7 10 13 in this paper we consider the second approach for cases where the sparsity structure arises from finite difference approximations to partial differential equa tions both symmetric and unsymmetric matrices are considered although the algorithms presented do not generalize to every pattern generated by finite dif ferences they cover the most useful structures and in some cases extend existing methods to more complex situations in the next section we describe the method of curtis powell and reid 5 henceforth referred to as the cpr method for estimating sparse jacobian matrices we then show how the so called computational molecule or stencil 1 of the finite difference operator associated with the jacobian matrix under study can be used to determine the groupings i e function differencing directions used by the

on generalized quotient mi groups abstract many identities group mi group for short is an algebraic structure generalizing the group structure where an involutive anti automorphism satisfying certain properties is used instead of the standard group inversion the concept of mi group in a more general form than in this article has been introduced by holcapek and st pnicka in the paper mi algebras a new frame work for arithmetics of extensional fuzzy numbers to describe properties of different approaches to arithmetics of vaguely specified quantities e g stochastic or fuzzy quantities in a unified way this article is a continuation of the effort to develop the theory of mi groups and is focused on a generalization of the construction of quotient mi groups induced by so called normal full mi subgroups which has been introduced by holcapek et al recently in the paper quotient mi groups besides a more general definition of quotient mi groups we prove three isomorphism theorems for mi groups in this new framework

from fuzzy set theory to fuzzy set theory some critical remarks on existing concepts abstract our aim in this paper is to give some foundational remarks on the nature of fuzziness and their modeling by fuzzy sets we will reveal some fundamental misunderstandings which have pervaded fuzzy set theory just from its very beginnings at the end of the paper we give three examples demonstrating the differences and advantages of our new conception our approach in heavily based on an operational and set theoretical background contrasting hisdal s tee model

a chordal preconditioner for large scale optimization we propose an automatic preconditioning scheme for large sparse numerical optimization the strategy is based on an examination of the sparsity pattern of the hessian matrix using a graph theoretic heuristic a block diagonal approximation to the hessian matrix is induced the blocks are submatrices of the hessian matrix furthermore each block is chordal that is under a positive definiteness assumption each block can be cholesky factored without creating new nonzeroes fill therefore the preconditioner is space efficient we conduct a number of numerical experiments to determine the effectiveness of the preconditioner in the context of a linear conjugate gradient algorithm for optimization

algebraic properties and topological properties of the quotient space of fuzzy numbers based on mares equivalence relation abstract in this paper we obtain some algebraic properties and topological properties of the quotient space of fuzzy numbers with respect to the equivalence relation defined by mares every fuzzy number has only one mares core and equivalent fuzzy numbers have the same mares core in addition equivalence classes of fuzzy numbers can be only expressed as the sum of its mares core and the set of symmetric fuzzy numbers which shows the notable difference between the equivalence classes of fuzzy numbers and the cosets of the normal subgroup in a group based on these results we introduce a new concept of convergence under which the quotient space is complete as an application of the main results it is shown that if we identify every fuzzy number with the corresponding equivalence class there would be more differentiable fuzzy functions than what is found in the literature

estimation of sparse jacobian matrices and graph coloring blems given a mapping with a sparse jacobian matrix the problem of minimizing the number of function evaluations needed to estimate the jacobian matrix by differences is investigated this problem can be attacked as a graph coloring problem and this approach leads to very efficient algorithms the behavior of these algorithms is studied and in particular it is proved that two of the algorithms are optimal for band graphs numerical evidence is presented which indicates that these two algorithms are nearly optimal on practical problems

a representation of fuzzy numbers abstract each fuzzy number can be written in a unique way as the sum of a symmetric and a skew fuzzy number these two kinds of fuzzy numbers correspond bijectively to certain functions on the unit interval that are of bounded variation these bijections give rise to a representation of fuzzy numbers by pairs of functions of bounded variation as application mathematical structures on the set of functions of bounded variation are transported to that of fuzzy numbers making the set of fuzzy numbers into a complete metric space a commutative semiring and a commutative residuated lattice

quotients with respect to similarity relations abstract the problem of the quotient w r t a similarity relation is closely related to poincare s paradox and central problems in cluster analysis amazingly this problem remains open for more than 30 years as a generalization of strong m categories we introduce the concept of m valued sets which leads to the definition of the category m set in m set we give a complete solution of the quotient problem

preconditioners for generalized saddle point problems noindent we propose and examine block diagonal preconditioners and variants of indefinite preconditioners for block two by two generalized saddle point problems that is we consider the nonsymmetric nonsingular case where the 2 2 block is small in norm and we are particularly concerned with the case where the 1 2 block is different from the transposed 2 1 block we provide theoretical and experimental analyses of the convergence and eigenvalue distributions of the preconditioned matrices we also extend the results of de sturler and liesen siam j sci comput 26 2005 pp 1598 1619 to matrices with nonzero 2 2 block and to the use of approximate schur complements to demonstrate the effectiveness of these preconditioners we show convergence results spectra and eigenvalue bounds for two model navier stokes problems

fuzzy quantities in networks one of possible applications of the fuzzy sets theory in network analysis is described and discussed it is shown that the uncertainty connected with the length of activities forming the network can be represented by fuzzy quantities summing these fuzzy quantities it is possible to derive the uncertainty connected with more complex paths and reserves an illustrative numerical example is presented at the end of the paper

nonlinear calderon zygmund theory in the limiting case we prove a maximal differentiability and regularity result for solutions to nonlinear measure data problems specifically we deal with the limiting case of the classical theory of calderon and zygmund in the setting of nonlinear possibly degenerate equations and we show a complete linearization effect with respect to the differentiability of solutions a prototype of the results obtained here tells for instance that if rm div du p 2 du mu quad mbox in omega subset mathbb r n with mu being a borel measure with locally finite mass on the open subset omega subset mathbb r n and p 2 1 n then du p 2 du in w sigma 1 rm loc omega quad mbox for every sigma in 0 1 the case sigma 1 is obviously forbidden already in the classical linear case of the poisson equation triangle u mu

partitioning group correction cholesky techniques for large scale sparse unconstrained optimization this paper studies a successive partitioning group correction cholesky algorithm and its modified algorithm for solving large scale sparse unconstrained optimization problems these methods employ an initial cholesky factorization of their approximation hessians and then correct the partitioning group s of their diagonal factor and lower triangular factor directly and successively at each step iterations are generated using forward and backward substitution employing the update factorizations a self correcting property a q superlinear convergence result and an r convergence rate estimate show that the two methods both have good local convergence properties the numerical results show that the two methods especially the modified algorithm may be competitive with some current used algorithms

the efficient computation of sparse jacobian matrices using automatic differentiation this paper is concerned with the efficient computation of sparse jacobian matrices of nonlinear vector maps using automatic differentiation ad specifically we propose the use of a graph coloring technique bicoloring to exploit the sparsity of the jacobian matrix j and thereby allow for the efficient determination of j using ad software we analyze both a direct scheme and a substitution process we discuss the results of numerical experiments indicating significant practical potential of this approach

additive decomposition of fuzzy quantities abstract the object of this paper is to define a more refined equivalent relation than mares 3 and to improve mares result about additive decomposition of fuzzy quantities concerning an equivalent relation this result proves one of his open questions

recent developments in constrained optimization constrained optimization problems occur in many applications of science engineering social science and medicine and hence it is useful for practitioners in these areas to be informed about numerical methods for solving such problems however improvements in optimization typically do not spread quickly to the non specialist literature this paper contains a brief summary of two recent developments in the solution of nonlinearly constrained problems direct use of the lagrangian function and linearization of nonlinear constraints in order to highlight the new approaches we also discuss techniques used in older methods to illustrate the capabilities of modern optimization methods a brief description is given of the development of methods to solve large nonlinearly constrained problems that arise in the electrical power industry the general nonlinear programming problem may be stated in the following form

on the convergence of partitioning group correction algorithms this paper studies a successive partitioning group correction algorithm and its some modified algorithms for solving large scale sparse unconstrained optimization problems the methods depend on a symmetric consistent partition of the columns of the hessian matrix a q superlinear convergence result and an r convergence rate estimate show that the methods have good local convergence properties the numerical results show that the methods especially the modified algorithms may be competitive with some current used algorithms

new acyclic and star coloring algorithms with application to computing hessians acyclic and star coloring problems are specialized vertex coloring problems that arise in the efficient computation of hessians using automatic differentiation or finite differencing when both sparsity and symmetry are exploited we present an algorithmic paradigm for finding heuristic solutions for these two np hard problems the underlying common technique is the exploitation of the structure of two colored induced subgraphs for a graph g on n vertices and m edges the time complexity of our star coloring algorithm is o n overline d 2 where overline d k a generalization of vertex degree denotes the average number of distinct paths of length at most k edges starting at a vertex in g the time complexity of our acyclic coloring algorithm is larger by a multiplicative factor involving the inverse of ackermann s function the space complexity of both algorithms is o m to the best of our knowledge our work is the first practical algorithm for the acyclic coloring problem for the star coloring problem our algorithm uses fewer colors and is considerably faster than a previously known o n overline d 3 time algorithm computational results from experiments on various large size test graphs demonstrate that the algorithms are fast and produce highly effective solutions the use of these algorithms in hessian computation is expected to reduce overall runtime drastically

distributed memory parallel algorithms for distance 2 coloring and related problems in derivative computation the distance 2 graph coloring problem aims at partitioning the vertex set of a graph into the fewest sets consisting of vertices pairwise at distance greater than 2 from each other its applications include derivative computation in numerical optimization and channel assignment in radio networks we present efficient distributed memory parallel heuristic algorithms for this np hard problem as well as for two related problems used in the computation of jacobians and hessians parallel speedup is achieved through graph partitioning speculative iterative coloring and a bulk synchronous parallel like organization of parallel computation results from experiments conducted on a pc cluster employing up to 96 processors and using large size real world as well as synthetically generated test graphs show that the algorithms are scalable in terms of quality of solution the algorithms perform remarkably well the numbers of colors used by the parallel algorithms are observed to be very close to the numbers used by their sequential counterparts which in turn are quite often near optimal moreover the experimental results show that the parallel distance 2 coloring algorithm compares favorably with the alternative approach of solving the distance 2 coloring problem on a graph mathcal g by first constructing the square graph mathcal g 2 and then applying a parallel distance 1 coloring algorithm on mathcal g 2 implementations of the algorithms are made available via the zoltan toolkit

what color is your jacobian graph coloring for computing derivatives graph coloring has been employed since the 1980s to efficiently compute sparse jacobian and hessian matrices using either finite differences or automatic differentiation several coloring problems occur in this context depending on whether the matrix is a jacobian or a hessian and on the specifics of the computational techniques employed we consider eight variant vertex coloring problems here this article begins with a gentle introduction to the problem of computing a sparse jacobian followed by an overview of the historical development of the research area then we present a unifying framework for the graph models of the variant matrix estimation problems the framework is based upon the viewpoint that a partition of a matrix into structurally orthogonal groups of columns corresponds to distance 2 coloring an appropriate graph representation the unified framework helps integrate earlier work and leads to fresh insights enables the design of more efficient algorithms for many problems leads to new algorithms for others and eases the task of building graph models for new problems we report computational results on two of the coloring problems to support our claims most of the methods for these problems treat a column or a row of a matrix as an atomic entity and partition the columns or rows or both a brief review of methods that do not fit these criteria is provided we also discuss results in discrete mathematics and theoretical computer science that intersect with the topics considered here

additive decomposition of fuzzy quantities with finite supports abstract the results on group properties of the addition operation over fuzzy quantities presented in a previous paper by the author 5 imply some interesting consequences concerning the equivalence relation between fuzzy quantities the equivalence is necessary for the validity of the group axioms for the class of fuzzy quantities more detailed analysis of the equivalence properties implies some conclusions regarding the interpretation of different sources of fuzziness entering the fuzzy quantities model the method used to prove the main theorem of this paper demands to limit the investigation to fuzzy quantities with finite support this limitation however unpleasant does not influence the applicability of the presented results in a wide class of practical situations

estimation of sparse hessian matrices and graph coloring problems large scale optimization problems often require an approximation to the hessian matrix if the hessian matrix is sparse then estimation by differences of gradients is attractive because the number of required differences is usually small compared to the dimension of the problem the problem of estimating hessian matrices by diferences can be phrased as follows given the sparsity structure of a symmetric matrix a obtain vectors d 1 d 2 ldots d p such that ad 1 ad 2 ldots ad p determine a uniquely with p as small as possible we approach this problem from a graph theoretic point of view and show that both direct and indirect approaches to this problem have a natural graph coloring interpretation the complexity of the problem is analyzed and efficient practical heuristic procedures are developed numerical results illustrate the differences between the various approaches

acyclic and star coloring of p4 reducible and p4 sparse graphs an acyclic coloring of a graph g is a proper vertex coloring such that g contains no bicolored cycles the more restricted notion of star coloring of g is an acyclic coloring in which each path of length 3 is not bicolored in this paper we mainly study on the acyclic and star coloring of p4 reducible and p4 sparse graphs moreover we list polynomial time algorithms for giving an optimal acyclic or star coloring of a p4 reducible or p4 sparse graph

representation theorems for l fuzzy quantities abstract in this paper representation theorems are given for l fuzzy quantities which permit a better understanding of fuzziness in particular representation theorems due to negoita and ralescu 17 and to sherwood and taylor 23 are extended to the scope of complete and brouwerian lattices

on fuzzy differential equations in the quotient space of fuzzy numbers since addition possesses an inverse in the quotient space of fuzzy numbers based on mares equivalence relation in this paper we introduce a metric on the quotient space of fuzzy numbers and then deal with fuzzy mappings of a real variable whose values are equivalence classes of fuzzy numbers we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation finally some examples are given to illustrate the main theorems