Cyclic Reduction Research Articles

Some new smoother implementations for discrete-time gaussian reciprocal processes

Abstract The smoothing problem for discrete-time gaussian reciprocal processes is examined. For this class of process, the smoother takes the form of a second-order two-point boundary-value nearest-neighbour model. This smoother is non-causal, but admits causal implementations which extend the two-filter and Rauch-Tung-Striebel smoothing formulae of Gauss-Markov processes. A multi-rate implementation based on a block cyclic reduction method is also described.

Base- p-cyclic reduction for tridiagonal systems of equations

Cyclic reduction is often heralded as a vectorizable and fast method for the solution of weakly diagonally dominant tridiagonal systems of equations on a vector processor. However, in the standard version, where the number of equations is halved in each sweep (recursive doubling), sooner or later the algorithm runs into memory bank conflicts because the stride in the memory access is doubled in each sweep and eventually becomes a multiple of the number of memory banks. In this note we consider variants of the algorithm, in which the stride is tripled after each sweep and/or the reduced system of equations is moved to contoguous locations beyond the original ones. We shall compare performances on CRAY X-MP and NEC SX-2 machines.

Chemistry of platinum(IV), platinum(II) and palladium(II) cyclometallates of benzylthio- or benzosulphinyl-substituted azobenzenes

Cyclometallated platinum(IV) complexes of the type [Pt(RL)Cl 3] incorporating azobenzene (R = H) or 4-methylazobenzene (R = Me) with an SCH 2Ph (L = L 1) or an S(O)CH 2Ph (L = L 2) substituent in an ortho position are described. These were obtained by oxidative addition of chlorine to [Pt II(RL)Ci]. Analogous palladium(II) complexes, however, gave chloro-substituted azobenzenes upon reaction with chlorine. The platinum(IV) complexes show an irreversible cyclic voltammetric metal reduction peak near −0.3 V vs SCE. Constant potential coulometric reduction at −0.5 V or chemical reduction by hydrazine hydrate regenerated [Pt II(RL)Cl]. The structures of [Pt(HL 1)Cl 3], Pt(MeL 2)Cl] and [Pd(MeL(su1)Cl] have been determined by an X-ray diffraction study. The S(O)CH 2Ph group is coordinated via the sulphur atom.

Divide and conquer: a parallel algorithm for the solution of a tridiagonal linear system of equations

We describe a divide and conquer algorithm which solves linear tridiagonal systems with one right-hand side, especially suited for parallel computers. The algorithm is flexible, permits multiprocessing or a combination of vector and multiprocessor implementations, and is adaptable to a wide range of parallelism granularities. This algorithm can also be combined with recursive doubling, cyclic reduction or Wang's partition method, in order to increase the degree of parallelism and vectorizability. The divide and conquer method will be explained. Some results of time measurements on a CRAY X-MP/28, on an Alliant FX/8, and on a Sequent Symmetry S81b, as well as comparisons with the cyclic reduction algorithm and Gaussian elimination will be presented. Finally, numerical results are given.

Memory access problems in block cyclic reduction on vector computers

In the present paper, modifications of block cyclic reduction are introduced by which the often significant memory access problems of this method on vector computers can be drastically reduced. The modifications are discussed in the context of an interval arithmetic application, as interval methods are particularly sensitive to several aspects of memory access. A comparison of computation times from several numerical tests is included.

The effect of sublethal neogregarine infections in the spruce needleminer, Epinotia tedella (Lepidoptera: Tortricidae)

. 1 A hitherto unknown factor causing cyclic reduction in population fertility of Epirzotia tedella (Cl.) was identified as sublethal infections of the neogregarine Mattesia sp. 2 Infection probably takes place when larvae enter hibernation, but does not develop into spore production until the formation of pupae and adult moths. 3 At the individual level, the sublethal neogregarine infection causes a slight delay in adult emergence, a decreased adult lifespan, a suppression of egg development, and thus a reduction in fertility. 4 At the population level, fertility reduction has a delayed density dependent component and, through correlation, Mattesia is assumed to be causative. Since fertility reduction is a key-factor in E.tedella, Mattesia may even be a dominant factor in the dynamics of this species. 5 Mattesia infections were found in no other dominant insect species at the locations, and consequently the interaction between E.tedella and Mattesia sp. seems to be specific. Such delayed density dependent interaction can create host oscillations, as seen in many forest insects. However, host-parasitoid interaction alone leads to oscillations with the observed period of 6–8 years length, and the neogregarine infections apparently act in parallel with the parasitoids and enhance the amplitude of the oscillations.

Real-time optical observation of morphological change of small supported copper particles during redox treatments

In situ optical observation was used as a technique to study the effects of cyclic oxidation and reduction (redox) treatment made at temperatures below 673 K on small copper particles supported on porous glass. The in situ real-time observation revealed decreases both in size of copper particles and volume fraction of metallic copper during the cyclic redox treatment. The decreases correspond to an increase in fraction of nonmetallic copper species on the support surface. This is confirmed by comparing the measured with calculated optical spectra and tracing the change in the rate of oxidation of copper particles. This phenomenon is enhanced by using 500 ppm O 2 rather than 1% O 2, and at higher temperatures. The role of a lower oxygen concentration in the mechanism of the phenomenon is discussed.

Algorithms for Special Tridiagonal Systems

Algorithms for the solution of symmetric diagonally dominant tridiagonal systems of linear equations with constant diagonals are considered. Such systems occur, for example, when solving certain constant-coefficient elliptic partial differential equations by the Fourier method. In particular, the specialized $LU$ factorization method of Malcolm and Palmer (SpLU), the cyclic reduction method of Hockney (CR), and the reversed triangular factorization method of Evans (RTF) are considered. An interesting property of the first two algorithms is that they may be terminated early for highly diagonally dominant systems. A new implementation of RTF that also has this property is presented, significantly reducing its operation count in many cases. The slightly perturbed systems that arise from problems with Neumann or periodic boundary conditions are also considered, with extensions given for SpLU to the periodic case and RTF to the Neumann case. Floating-point operation counts are given for each method, and the results of experiments on a single scalar processor are reported.

Using cyclic reduction on a parallel computer to improve the performance of an underwater sound implicit finite difference model

Using cyclic reduction on a parallel computer to improve the performance of an underwater sound implicit finite difference model

Induction of Oxidant Stress by Iron Available in Advanced Forms ofPlasmodium Falciparum

Oxidative stress has been incriminated as a deleterious factor in the development of malaria parasites. Various chemical reductones which can undergo cyclic oxidation and reduction, such as ascorbate have been shown to cause oxidative stress to red blood cells. This, naturally-occurring and redox-active compound, can induce the formation of active oxygen derived species, such as superoxide radicals (.O2-), hydrogen peroxide (H2O2) and hydroxyl radical (OH.). The formation of the hydroxyl radical, the ultimate deleterious species, is mediated by the redox-active and available transition metals iron and copper in the Haber-Weiss reaction. During the development of the parasite, hemoglobin is progressively digested and a concurrent release of high levels of iron-containing breakdown products takes place within the red blood cell. Indications for the progressive increase in redox-active iron during the growth of P. falciparum have been recently found in our lab: a) adventitious ascorbate proved highly detrimental to the parasite when added to the mature forms. In contrast, if the parasitized erythrocytes were in the early phase following invasion, and only low levels of iron-containing structures had been liberated, then the observed effect was a small promotion of parasite development. b) erythrocytes containing mature parasites were more potent than erythrocytes containing ring forms as a source for redox-active iron in the ascorbate-driven metal-mediated degradation of DNA. The addition of extracts from parasitized erythrocytes and ascorbate to DNA caused a dose and time dependent DNA degradation. Non-infected erythrocytes had no effect.(ABSTRACT TRUNCATED AT 250 WORDS)

The cyclic fixation and reduction of molecular nitrogen with [WH4(Ph2PCH2CH2PPh2)2] in γ-irradiated solutions

Yields of NH3 and N2H4 in radiation-catalytic reduction of N2 by means of a hydride complex of WIV with dppe in thf solution and in mixtures of thf with other solvents have been estimated. A mechanism proposed earlier of reduction of coordinated N2 has been confirmed and the role of solvent in reduction of N2 to NH3 and amines has been explained.

O-221 - Effect of DV-7028, a new 5-HT2 receptor antagonist, on cyclic blood flow reduction in the canine coronary artery.

O-221 - Effect of DV-7028, a new 5-HT2 receptor antagonist, on cyclic blood flow reduction in the canine coronary artery.

A new quadrant interlocking factorization for parallel solution of tridiagonal linear systems

We introduce a new quadrant interlocking factorization (Q.I.F.) for use with the partition method for the solution of tridiagonal linear systems. A given linear system of size N is partitioned into r blocks each of size n (N = rn n even). The new factorization WZ of the coefficient matrix in each block has the properties that the vector is invariant under the transformation W and the solution process with coefficient matrix Z proceeds from the first and the last unknowns towards the middle ones. These properties of the new factorization help us uncouple the partitioned systems for parallel execution once a core system of order 2r is solved. The scalar count for the resulting algorithm is O(17N) which is the count for cyclic reduction and the partition method of Wang [9]. We also note that for the parallel algorithm based on modified Q.I.F. given in Evans et al. [6] the number of processors required for the algorithm is in terms of semibandwidth and the size of the system; on the other hand, the number of processors required for the present algorithm is in terms of the number of blocks into which the system is partitioned.

Iterative methods for cyclically reduced nonselfadjoint linear systems. II

We perform an analytic and experimental study of line iterative methods for solving linear systems arising from finite difference discretizations of non-self-adjoint elliptic partial differential equations on two-dimensional domains. The methods consist of performing one step of cyclic reduction, followed by solution of the resulting reduced system by line relaxation. We augment previous analyses of one-line methods, and we derive a new convergence analysis for two-line methods, showing that both classes of methods are highly effective for solving the convection-diffusion equation. In addition, we compare the experimental performance of several variants of these methods, and we show that the methods can be implemented efficiently on parallel architectures.

A full parallel preconditioner for a class of M-matrices

A full parallel preconditioner for preconditioned conjugate gradient (PCG) methods is derived, by using an incomplete cyclic reduction, for a class of M-matrices.

On recursive decoupling method for solving tridiagonal linear systems

A recursive decoupling method was introduced by Evans [2] to form the basis of a parallel tridiagonal linear system solver. Recently, Evans [3] has described another version of the recursive decoupling strategy making it into an efficient parallel algorithm by using a series of rank one updating stages performed in parallel to combine the solutions of the 2 ×2 subsystems. However, in his version of the decoupling method the diagonal elements of the coefficient matrix of the system are modified. In the present paper we present an alternative implementation of the recursive decoupling method which is in the same vein with the essential difference that our implementation obviates the need to modify the diagonal elements of the coefficient matrix. The total arithmetical operations count for the present version of recursive decoupling method is , while that for the parallel variant of cyclic reduction, termed PARACR, is (Hockney and Jesshope [4]).

A tridiagonal solver for massively parallel computer systems

This paper describes a parallel solver of tridiagonal systems appropriate for distributed memory computers and implemented on an array of chain-connected T800 Transputers. Each processor in the chain uses the same program to solve its own subset of equations. This implementation is suited, for instance, for solving the heat conduction equation in one-dimensional hydrodynamic codes. The procedure performs a parallel cyclic reduction, a recursive Gaussian elimination on a reduced number of equations and a parallel backward unfolding scheme, with a direct substitution in the reduced equations. The code has been written in Occam2 language. A one-way communication of values between adjacent processors is required at each cycle of both the reduction and the unfolding steps. Due to the number of floating point operations and the amount of communications the implementation described here works efficiently on arrays with more than 4 processors and for more than ∼ 50 equations per processor.

A multilevel parallel solver for block tridiagonal and banded linear systems

This paper describes an efficient algorithm for the parallel solution of systems of linear equations with a block tridiagonal coefficient matrix. The algorithm comprises a multilevel LU-factorization based on block cyclic reduction and a corresponding solution algorithm. The paper includes a general presentation of the parallel multilevel LU-factorization and solution algorithms, but the main emphasis is on implementation principles for a message passing computer with hypercube topology. Problem partitioning, processor allocation and communication requirement are discussed for the general block tridiagonal algorithm. Band matrices can be cast into block tridiagonal form, and this special but important problem is dealt with in detail. It is demonstrated how the efficiency of the general block tridiagonal multilevel algorithm can be improved by introducing the equivalent of two-way Gaussian elimination for the first and the last partitioning and by carefully balancing the load of the processors. The presentation of the multilevel band solver is accompanied by detailed complexity analyses. The properties of the parallel band solver were evaluated by implementing the algorithm on an Intel iPSC hypercube parallel computer and solving a larger number of banded linear equations using 2 to 32 processors. The results of the evaluation include speed-up over a sequential processor, and the measure values are in good agreement with the theoretical values resulting from complexity analysis. It is found that the maximum asymptotic speed-up of the multilevel LU-factorization using p processors and load balancing is approximated well by the expression ( p +6)/4. Finally, the multilevel parallel solver is compared with solvers based on row and column interleaved organization.

Optimizing Tridiagonal Solvers for Alternating Direction Methods on Boolean Cube Multiprocessors

Sets of tridiagonal systems occur in many applications. Fast Poisson solvers and Alternate Direction Methods make use of tridiagonal system solvers. Network-based multiprocessors provide a cost-effective alternative to traditional supercomputer architectures. The complexity of concurrent algorithms for the solution of multiple tridiagonal systems on Boolean-cube-configured multiprocessors with distributed memory are investigated. Variations of odd-even cyclic reduction, parallel cyclic reduction, and algorithms making use of data transposition with or without substructuring and local elimination, or pipelined elimination, are considered. A simple performance model is used for algorithm comparison, and the validity of the model is verified on an Intel iPSC/ 1. For many combinations of machine and system parameters, pipelined elimination, or equation transposition with or without substructuring is optimum. Hybrid algorithms that at any stage choose the best algorithm among the considered ones for the remainder of the problem are presented. It is shown that the optimum partitioning of a set of independent tridiagonal systems among a set of processors yields the embarrassingly parallel case. If the systems originate from a lattice and solutions are computed in alternating directions, then to first order the aspect ratio of a computational lattice shall be the same as that of the lattice forming the base for the equations. The experiments presented here demonstrate the importance of combining in the communication system for architectures with a relatively high communications start-up time.

Incomplete Block Cyclic Reduction as a Preconditioner for Polynomial Iterative Methods

Preconditionings based on incomplete block odd-even cyclic reduction for the Chebyshev and conjugate gradient polynomial iterative methods have been shown to be effective for accelerating the convergence of the solution of linear systems that arise from discrete approximations of elliptic partial differential equations. The main contribution of this paper is to extend these methods to the important case of red/black partitioning of equations and unknowns and to demonstrate that a significant reduction in computations can be realized by applying preconditioned polynomial methods to the “reduced systems” obtained from the original systems by eliminating approximately half of the unknowns. This paper also compares the performance of several preconditioned polynomial methods for solving a two-dimensional elliptic equation on the CYBER 205 vector computer. The most effective methods tested are shown to be (i) the new reduced system conjugate gradient method with preconditioner based on incomplete block odd-even cyclic reduction, and (ii) the classical reduced system conjugate gradient method with preconditioner $D_B $, the tridiagonal matrix associated with the black unknowns. Of these methods, the former is better in most cases, especially for some slowly converging problems.