orderings for conjugate gradient preconditionings many preconditioners e g ssor ilu for the conjugate gradient method require the solution of sparse triangular systems of equations for elliptic boundary value problems one approach to obtaining additional parallelism in the solution of these systems is the use of red black or multicolor orderings there has been increasing evidence however that these orderings degrade the rate of convergence compared with the natural ordering an alternative is the diagonal ordering which maintains the rate of convergence of the natural ordering but has less parallelism than multicolor orderings this paper reviews these as well as other orderings and then gives some results that help to explain why the red black ordering gives an inferior rate of convergence

convergence analysis without regularity assumptions for multigrid algorithms based on sor smoothing convergence estimates for the standard multigrid algorithms based on sor smoothing are established without regularity assumptions for symmetric and positive definite variational problems the results are applied to second order elliptic equations discretized by using finite element methods it is shown that the convergence is deteriorated no worse than 1 c sigma j 1 for the multigrid algorithms with sor nonsymmetric smoothing where j is the number of levels and sigma the relaxation parameter the results can be regarded as a generalization of the estimates in 10 where only the jacobi smoothing was considered this extension is not trivial since sor smoothing yields a nonsymmetric operator while the jacobi gives a symmetric smoothing operator the estimates are well suited to difficult problems with rough solutions as no regularity is assumed in the analysis

new parallel sor method by domain partitioning in this paper we propose and analyze a new parallel sor method the psor method formulated by using domain partitioning and interprocessor data communication techniques we prove that the psor method has the same asymptotic rate of convergence as the red black r b sor method for the five point stencil on both strip and block partitions and as the four color r b g o sor method for the nine point stencil on strip partitions we also demonstrate the parallel performance of the psor method on four different mimd multiprocessors a ksr1 an intel delta a paragon and an ibm sp2 finally we compare the parallel performance of psor r b sor and r b g o sor numerical results on the paragon indicate that psor is more efficient than r b sor and r b g o sor in both computation and interprocessor data communication

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

on red black sor smoothing in multigrid optimal relaxation parameters are obtained for red black gauss seidel relaxation in multigrid solvers of a family of elliptic equations the resulting relaxation schemes are found to retain very high efficiency over an appreciable range of coefficients of the elliptic differential operator yielding simple inexpensive and fully parallelizable smoothers in many situations where less cost effective block and alternating direction schemes are commonly used

pointwise characterizations of besov and triebel lizorkin spaces and quasiconformal mappings abstract in this paper the authors characterize in terms of pointwise inequalities the classical besov spaces b p q s and triebel lizorkin spaces f p q s for all s 0 1 and p q n n s both in r n and in the metric measure spaces enjoying the doubling and reverse doubling properties applying this characterization the authors prove that quasiconformal mappings preserve f n s q s on r n for all s 0 1 and q n n s a metric measure space version of the above morphism property is also established

besov and triebel lizorkin spaces on metric spaces embeddings and pointwise multipliers abstract in this paper we obtain the franke jawerth embedding property of hajlasz besov and hajlasz triebel lizorkin spaces on a measure metric space x d which is ahlfors regular with dimension q as applications we show that when x d is doubling and satisfies an ahlfors lower bound condition with q then the hajlasz besov space n p q s x with p q s q p 1 and q 0 and the hajlasz triebel lizorkin space m p q s x with p q s q p 1 and q q q s are algebras under pointwise multiplication and moreover when x is ahlfors q regular we characterize the class of all pointwise multipliers on the hajlasz triebel lizorkin space m p q s x for p q s q p 1 and q q q s by its related uniform space

a new parallel gauss seidel method based on alternating group explicit method and domain decomposition method a new parallel gauss seidel method is presented for solution of system of linear equations related to finite difference discretization of partial differential equations this method is based on domain decomposition method and local coupling between interfaces of neighbor sub domains same as alternating group explicit method this method is convergent and number of iterations for achieving convergence criteria is near the original gauss seidel method sometimes better and sometimes worse but difference is very small the convergence theory is discussed in details numerical results are given to justify the convergence and performance of the proposed iterative method

analysis of the sor iteration for the 9 point laplacian the sor iteration for solving linear systems of equations depends upon an overrelaxation factor omega a theory for determining omega was given by young iterative methods for solving partial differential equations of elliptic types trans amer math soc 76 1954 pp 92 111 for consistently ordered matrices here we determine the optimal omega for the 9 point stencil for the model problem of laplace s equation on a square we consider several orderings of the equations including the natural rowwise and multicolor orderings all of which lead to nonconsistently ordered matrices and find two equivalence classes of orderings with different convergence behavior and optimal omega s we compare our results for the natural rowwise ordering to those of garabedian estimation of the relaxation factor for small mesh size math comp 10 1956 pp 183 185 and explain why both results are in a sense correct even though they differ we also analyze a pseudo sor method for the model p

determination of stripe structures for finite element matrics stripe structures were introduced by the author parallel computing to appear as a means for the inclusion of the nonzero elements of a sparse matrix into a regular pattern which allows for efficient parallel manipulation in this paper the stripe structures of stiffness matrices resulting from irregular domains covered by regular grids are analyzed it is proved that the nonzero elements in these matrices may be covered by very few stripes and that these stripes may be nonoverlapping if the nodes of the grids are numbered appropriately the exact number of stripes which is independent of the size of the problem is derived for different types of grids and different numbering schemes the stripe structures of some irregular grids are also examined

measure density and embeddings of hajlasz besov and hajlasz triebel lizorkin spaces abstract in this paper we investigate the relation between sobolev type embeddings of hajlasz besov spaces and also hajlasz triebel lizorkin spaces defined on a metric measure space x d and lower bound for the measure we prove that if the measure satisfies b x r c r q for some q 0 and for any ball b x r x then the sobolev type embeddings hold on balls for both these spaces on the other hand if the sobolev type embeddings hold in a domain x then we prove that the domain satisfies the so called measure density condition i e b x r c r q holds for any ball b x r x where x x d is an ahlfors q regular and geodesic metric measure space

image recovery using partitioned separable paraboloidal surrogate coordinate ascent algorithms iterative coordinate ascent algorithms have been shown to be useful for image recovery but are poorly suited to parallel computing due to their sequential nature this paper presents a new fast converging parallelizable algorithm for image recovery that can be applied to a very broad class of objective functions this method is based on paraboloidal surrogate functions and a concavity technique the paraboloidal surrogates simplify the optimization problem the idea of the concavity technique is to partition pixels into subsets that can be updated in parallel to reduce the computation time for fast convergence pixels within each subset are updated sequentially using a coordinate ascent algorithm the proposed algorithm is guaranteed to monotonically increase the objective function and intrinsically accommodates nonnegativity constraints a global convergence proof is summarized simulation results show that the proposed algorithm requires less elapsed time for convergence than iterative coordinate ascent algorithms with four parallel processors the proposed algorithm yields a speedup factor of 3 77 relative to single processor coordinate ascent algorithms for a three dimensional 3 d confocal image restoration problem

sobolev embeddings extensions and measure density condition there are two main results in the paper in the first one theorem 1 we prove that if the sobolev embedding theorem holds in in any of all the possible cases then satisfies the measure density condition the second main result theorem 5 provides several characterizations of the wm p extension domains for 1 p as a corollary we prove that the property of being a w1 p extension domain 1 p is invariant under bi lipschitz mappings theorem 8

relativized polynomial time hierarchies having exactly k levels it is proved that for every integer k geqq 0 there is an oracle a k relative to which the polynomial time hierarchy collapses so that it has exactly k levels furthermore sets b k and c k may be constructed so that relative to b k the polynomial time hierarchy has exactly k levels and the class pspace coincides with the polynomial time hierarchy and relative to c k the polynomial time hierarchy has exactly k levels and the class pspace is different from the polynomial time hierarchy

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

a general method to construct oracles realizing given relationships between complexity classes we present a method to prove oracle results of the following type let k 1 ldots k 2n and l 1 ldots l 2m be complexity classes our method provides a general framework for constructing an oracle a such that k a 2i 1 neq k a 2i for i 1 ldots n and l a 2j 1 l a 2j for j 1 ldots m using that method we prove several results of this kind the hardest of them is the existence of an oracle a such that p a neq np a p a bpp a and both conp a sets and np a sets are p a separable we exhibit also two theorems that cannot be proved by that method

the sor method on parallel computers the jacobi overrelaxation jor method is usually cited as a perfect parallel algorithm whereas the successive overrelaxation sor method is considered as quite the opposite for linear systems with dense matrices an algorithm for the sor method is presented which is suited for parallelization nearly in the same way as jor for systems with band matrices an algorithm is described which using a pipeline principle yields a speedups p 1 p 1 p m 1 wherep denotes the number of processors andm the number of sor iterations thus form p there is no speedup whereas the speedup tends to its maximal valuep ifm tends to infinity

a new block parallel sor method and its analysis as the development of the psor method a point parallel sor method by mesh domain partitioning proposed in siam j sci comput 20 1999 pp 2261 2281 this paper introduces a new mesh domain partition and ordering the multitype partition and ordering and proposes a new block parallel sor bpsor method for numerically solving 2 dimensional 2d or three dimensional 3d elliptic boundary problems a general mathematical analysis shows that the bpsor method can have the same asymptotic convergence rate as the corresponding sequential block sor method if the coefficient matrix of the block linear system is consistently ordered it also shows that the original sequential ordering can be maintained in the parallel implementation of the bpsor method so that the bpsor method can be effectively applied to solve complex elliptic boundary problems furthermore three particular multitype orderings are proposed based on strip and block mesh partitions which lead to three effective bpsor methods for solving the five point like linear systems in two dimensions and the seven point like linear systems in three dimensions in addition it is shown that the psor method can be generated from bpsor if each block equation is solved approximately by only one point sor iteration thus the psor method is well defined without involving any interprocessor data communication operations and extended to solving three dimensional 3d problems finally numerical results are presented which confirm the theoretical results and show that bpsor has a good parallel performance on a parallel mimd computer

sobolev embedding for m1 p spaces is equivalent to a lower bound of the measure abstract it has been known since 1996 that a lower bound for the measure b x r b r s implies sobolev embedding theorems for sobolev spaces m 1 p defined on metric measure spaces we prove that in fact sobolev embeddings for m 1 p spaces are equivalent to the lower bound of the measure

parallel domain decomposition method with non blocking communication for flow through porous media this paper introduces a domain decomposition method for numerically solving the stokes equation for very large complex geometries examples arise from realistic porous media the computational method is based on the simple semi implicit method for pressure linked equations algorithm which uses a finite differences approach for discretizing the underlying equations it achieves comparable speed and efficiency as lattice boltzmann methods the domain decomposition method splits a large three dimensional region into slices that can be processed in parallel on multi processor computation environments with only minimal communication between the computation nodes with this method the flow through a porous medium with grid sizes up to 20483 voxel has been calculated