a nonlinear parallel algorithm with application to the stefan problem the algebraic problem b gamma e a beta e eta which may evolve from elliptic or parabolic partial differential equations is studied conditions are given so that a parallel algorithm of a nonlinear gauss seidel type converges to the unique solution an application is given to the solidification of a channel in which the continuum problem is discretized by the finite element method these computations show a significant decrease in computing times as compared to serial algorithms

convergence of parallel multisplitting iterative methods for m matrices abstract despite the major recent activity in parallel processing few effective new algorithms designed exclusively for multiprocessors have been put forth one such is the multisplitting iterative algorithm suggested by o leary and white although they have given some sufficient conditions for convergence a general convergence theory has not been developed even for the classical situation where the coefficient matrix a is an m matrix or is symmetric positive definite in this paper we study the m matrix case in detail the multisplitting process for a rn n is recast as an ordinary iterative process for a certain block matrix a r kn kn where k is the number of processors and standard convergence results are used to develop a convergence theory for multisplitting iterative methods where a is an m matrix comparison results between multisplitting methods are established in terms of monotonic norms and for the case where a is irreducible in terms of the asymptotic convergence rate a key observation here is that in certain cases the rate of global convergence of these parallel iterative methods is inherent in the splitting of a and is independent of the manner in which the work is distributed among the processors thus in general one can distribute the work for load balancing purposes without affecting the convergence rate

abelian quotients of triangulated categories abstract we study abelian quotient categories a t j where t is a triangulated category and j is an ideal of t under the assumption that the quotient functor is cohomological we show that it is representable and give an explicit description of the functor we give technical criteria for when a representable functor is a quotient functor and a criterion for when j gives rise to a cluster tilting subcategory of t we show that the quotient functor preserves the ar structure as an application we show that if t is a finite 2 calabi yau category then with very few exceptions j is a cluster tilting subcategory of t

generalized multisplitting asynchronous iteration abstract we give a generalized asynchronous iteration based on the multisplitting of linear or nonlinear operators to solve systems of equations the condition for convergence can be imposed only on the components which actually enter the computation the results contain synchronous multisplitting iterations as a special case

asynchronous parallel nonlinear multisplitting relaxation methods for large sparse nonlinear complementarity problems in accordance with the principle of sufficiently using delayed information and making use of the nonlinear multisplitting and the nonlinear relaxation techniques we present in this paper a class of asynchronous parallel nonlinear multisplitting successive overrelaxation sor methods for solving large sparse nonlinear complementarity problems on high speed mimd multiprocessor systems these new methods particularly include the so called asynchronous parallel nonlinear multisplitting sor newton method asynchronous parallel nonlinear multisplitting sor chord method and asynchronous parallel nonlinear multisplitting sor steffensen method under suitable conditions we establish the local convergence theory of this class of new methods numerical imitations show that our new methods are feasible and efficient for solving the nonlinear complementarity problems on the mimd multiprocessor systems

improved comparison theorem for the nonlinear multisplitting relaxation method abstract a new comparison theorem is given for the nonlinear multisplitting relaxation method 1 and an important modification is proposed for it too

a class of parallel nonlinear multisplitting relaxation methods for the large sparse nonlinear complementarity problems abstract by making use of the nonlinear multisplitting and the nonlinear relaxation techniques we present in this paper a class of parallel nonlinear multisplitting successive overrelaxation methods for solving the large sparse nonlinear complementarity problems on the modern high speed multiprocessors these new methods particularly include the so called nonlinear multisplitting sor newton method under suitable conditions we establish the local convergence theories of the new methods and investigate their asymptotic convergence rates a lot of numerical results show that our new methods are feasible and efficient for parallel solving the nonlinear complementarity problems

parallel nonlinear multisplitting methods linear multisplitting methods are known as parallel iterative methods for solving a linear systemax b we extend the idea of multisplittings to the problem of solving a nonlinear system of equationsf x 0 our nonlinear multisplittings are based on several nonlinear splittings of the functionf in a parallel computing environment each processor would have to calculate the exact solution of an individual nonlinear system belonging to his nonlinear multisplitting and these solutions are combined to yield the next iterate although the individual systems are usually much less involved than the original system the exact solutions will in general not be available therefore we consider important variants where the exact solutions of the individual systems are approximated by some standard method such as newton s method several methods proposed in literature may be regarded as special nonlinear multisplitting methods as an application of our systematic approach we present a local convergence analysis of the nonlinear multisplitting methods and their variants one result is that the local convergence of these methods is determined by an induced linear multisplitting of the jacobian off

a class of asynchronous parallel nonlinear accelerated overrelaxation methods for the nonlinear complementarity problems in accordance with the principle of using sufficiently the delayed information and by making use of the nonlinear multisplitting and the nonlinear relaxation techniques we present in this paper a class of asynchronous parallel nonlinear multisplitting accelerated overrelaxation aor methods for solving the large sparse nonlinear complementarity problems on the high speed mimd multiprocessor systems these new methods in particular include the so called asynchronous parallel nonlinear multisplitting aor newton method the asynchronous parallel nonlinear multisplitting aor chord method and the asynchronous parallel nonlinear multisplitting aor steffensen method under suitable constraints on the nonlinear multisplitting and the relaxation parameters we establish the local convergence theory of this class of new methods when the jacobi matrix of the involved nonlinear mapping at the solution point of the nonlinear complementarity problem is an h matrix

on the convergence of parallel chaotic nonlinear multisplitting newton type methods abstract a class of parallel chaotic nonlinear multisplitting newton type methods for solving the nonlinear system of equations f x 0 f d r n r n is established and its local convergence theory is presented

comparisons of weak regular splittings and multisplitting methods comparison results for weak regular splittings of monotone matrices are derived as an application we get upper and lower bounds for the convergence rate of iterative procedures based on multisplittings this yields a very simple proof of results of neumann plemmons on upper bounds and establishes lower bounds which has in special cases been conjectured by these authors

the monotone convergence of a class of parallel nonlinear relaxation methods for nonlinear complementarity problems abstract we set up a class of parallel nonlinear multisplitting aor methods by directly multisplitting the nonlinear mapping involved in the nonlinear complementarity problems the different choices of the relaxation parameters can yield all the known and a lot of new relaxation methods as well as a lot of new relaxed parallel nonlinear multisplitting methods for solving the nonlinear complementarity problems the two sided approximation properties and the influences on the convergence rates from the relaxation parameters about our new methods are shown and sufficient conditions guaranteeing the methods to converge globally are discussed finally a lot of numerical results show that our new methods are feasible and efficient

a general asynchronous block iterative model with related convergence conditions abstract this paper formulates a general time varying asynchronous block iterative model a convergence condition for asynchronous block iterations based on this model is given compared to existing conditions for similar and shown to be strictly weaker

new comparison theorem for the nonlinear multisplitting relaxation method for the nonlinear complementarity problems abstract a new comparison theorem on the monotone convergence rates of the parallel nonlinear multisplitting accelerated overrelaxation aor method for solving the large scale nonlinear complementarity problem is established thus the monotone convergence theory of this class of method is completed

d abelian quotients of d 2 angulated categories abstract let t be a triangulated category if t is a cluster tilting object and i add t is the ideal of morphisms factoring through an object of add t then the quotient category t i is abelian this is an important result of cluster theory due to keller reiten and konig zhu more general conditions which imply that t i is abelian were determined by grimeland and the first author now let t be a suitable d 2 angulated category for an integer d 1 if t is a cluster tilting object in the sense of oppermann thomas and i add t is the ideal of morphisms factoring through an object of add t then we show that t i is d abelian the notions of d 2 angulated and d abelian categories are due to geiss keller oppermann and jasso they are higher homological generalisations of triangulated and abelian categories which are recovered in the special case d 1 we actually show that if end t t is the endomorphism algebra of t then t i is equivalent to a d cluster tilting subcategory of mod in the sense of iyama this implies that t i is d abelian moreover we show that is a d gorenstein algebra more general conditions which imply that t i is d abelian will also be determined generalising the triangulated results of grimeland and the first author

on the parallel gsaor method for block diagonally dominant matrices in this paper we consider the parallel generalized saor iterative method based on the generalized aor iterative method presented by james for solving large nonsingular system we obtain some convergence theorems for the case when coefficient matrix is a block diagonally dominant matrix or a generalized block diagonal dominant matrix a numerical example is given to illustrate to our results

n angulated categories we dene n angulated categories by modifying the axioms of triangulated categories in a natural way we show that heller s parametrization of pre triangulations extends to pre n angulations we obtain a large class of examples of n angulated cate gories by considering n 2 cluster tilting subcategories of triangulated categories which are stable under the n 2 nd power of the suspension functor as an application we show how n angulated calabi yau categories yield triangulated calabi yau categories of higher calabi yau dimension finally we sketch a link to algebraic geometry and string theory

on multisplitting methods for band matrices we present new theoretical results on two classes of multisplitting methods for solving linear systems iteratively these classes are based on overlapping blocks of the underlying coefficient matrix a which is assumed to be a band matrix we show that under suitable conditions the spectral radius rho h of the iteration matrix h does not depend on the weights of the method even if these weights are allowed to be negative for a certain class of splittings we prove an optimality result for rho h with respect to the weights provided that a is an m matrix this result is based on the fact that the multisplitting method can be represented by a single splitting a m n which in our situation surprisingly turns out to be a regular splitting furthermore we show by numerical examples that weighting factors alpha not in 0 1 may considerably improve the convergence

n representation finite algebras and n apr tilting we introduce the notion of n representation finiteness generalizing representationfinite hereditary algebras we establish the procedure of n apr tilting and show that it preserves n representation finiteness we give some combinatorial description of this procedure and use this to completely describe a class of n representation finite algebras called type a

analysis of the projected coupled cluster method in electronic structure calculation the electronic schrodinger equation plays a fundamental role in molcular physics it describes the stationary nonrelativistic behaviour of an quantum mechanical n electron system in the electric field generated by the nuclei the projected coupled cluster method has been developed for the numerical computation of the ground state energy and wave function it provides a powerful tool for high accuracy electronic structure calculations the present paper aims to provide a rigorous analytical treatment and convergence analysis of this method if the discrete hartree fock solution is sufficiently good the quasi optimal convergence of the projected coupled cluster solution to the full ci solution is shown under reasonable assumptions also the convergence to the exact wave function can be shown in the sobolev h 1 norm the error of the ground state energy computation is estimated by an aubin nitsche type approach although the projected coupled cluster method is nonvariational it shares advantages with the galerkin or ci method in addition it provides size consistency which is considered as a fundamental property in many particle quantum mechanics

parallel interval multisplittings we introduce interval multisplittings to enclose the sets a 1b a a b b where a denotes an interval matrix and b an interval vector the resulting iterative multisplitting methods have a natural parallelism we investigate these methods with respect to convergence speed of convergence and quality of the resulting enclosure fors

convergence of relaxed parallel multisplitting methods abstract multisplitting methods are parallel methods for the solution of a linear system ax b it has been observed that the convergence of multisplitting methods is often improved if some kind of relaxation is used we investigate two different variants of relaxed multisplitting methods if a is an h matrix these methods converge if the relaxation parameter is from an interval 0 0 with 0 1