Multistep Splitting Research Articles

A modified multi-step splitting iteration and its variants for computing PageRank

A modified multi-step splitting iteration and its variants for computing PageRank

A multi-power and multi-splitting inner-outer iteration for PageRank computation

Abstract As an effective and possible method for computing PageRank problem, the inner-outer (IO) iteration has attracted wide interest in the past few years since it was first proposed by Gleich et al. (2010). In this paper, we present a variant of the IO iteration, which is based on multi-step power and multi-step splitting and is denoted by MPMIO. The description and convergence are discussed in detail. Numerical examples are given to illustrate the effectiveness of the proposed method.

A preprocessed multi-step splitting iteration for computing PageRank

The PageRank algorithm plays an important role in determining the importance of Web pages. The multi-step splitting iteration (MSPI) method for calculating the Pagerank problem is an iterative framework of combining the multi-step classical power method with the inner-outer method. In this paper, we present a preprocessed MSPI method called the Arnoldi-MSPI iteration, which is the MSPI method modified with the thick restarted Arnoldi algorithm. The implementation and convergence of the new method are discussed in detail. Numerical experiments are given to show that our method has a good computational effect when the damping factor is close to 1.

A single-cell encapsulation method based on a microfluidic multi-step droplet splitting system

Abstract Single cell analysis is of great significance to understand the physiological activity of organisms. Microfluidic droplet is an ideal analytical platform for single-cell analysis. We developed a microfluidic droplet splitting system integrated with a flow-focusing structure and multi-step splitting structures to form 8-line droplets and encapsulate single cells in the droplets. Droplet generation frequency reached 1021 Hz with the aqueous phase flow rate of 1 μL/min and the oil phase flow rate of 15 μL/min. Relative standard deviation of the droplet size was less than 5% in a single channel, while less than 6% in all the 8 channels. The system was used for encapsulating human whole blood cells. A single-cell encapsulation efficiency of 31% was obtained with the blood cell concentration of 2.5 × 104 cells/μL, and the multicellular droplet percentage was only 1.3%. The multi-step droplet splitting system for single cell encapsulation featured simple structure and high throughput.

Ultra‐Fast Microwave Synthesis of 3D Flower‐Like Co9S8 Hierarchical Architectures for High‐Performance Supercapacitor Applications

AbstractUltra‐fast microwave irradiation is used to synthesize 3D flower‐like Co9S8 hierarchical architectures. Based on time‐dependent SEM observations, a multistep splitting–growth mechanism is proposed to understand the formation of the observed structures. Moreover, the materials were used as electrodes to fabricate supercapacitors with high specific capacitances of 522, 499, 441, 436, and 397 F g–1 at current densities of 0.5, 0.8,1.0, 1.5, and 2.0 A g–1, respectively. The devices also show high charge–discharge reversibility, with an efficiency of 97.7 % after cycling 1000 times at a current density of 1.0 A g–1.

Unusual Formation of ZnCo2O4 3D Hierarchical Twin Microspheres as a High‐Rate and Ultralong‐Life Lithium‐Ion Battery Anode Material

A facile two‐step strategy involving a polyol method and subsequent thermal annealing treatment is successfully developed for the large‐scale preparation of ZnCo2O4 various hierarchical micro/nanostructures (twin mcrospheres and microcubes) without surfactant assistance. To the best of our knowledge, this is the first report on the synthesis of ZnCo2O4 mesoporous twin microspheres and microcubes. More significantly, based on the effect of the reaction time on the morphology evolution of the precursor, a brand‐new crystal growth mechanism, multistep splitting then in situ dissolution recrystallization accompanied by morphology and phase change, is first proposed to understand the formation of the 3D twin microshperes, providing new research opportunity for investigating the formation of novel micro/nanostructures. When evaluated as anode materials for lithium‐ion batteries (LIBs), ZnCo2O4 hierarchical microstructures exhibit superior capacity retention, excellent cycling stability at the 5 A g−1 rate for 2000 cycles. Surprisingly, the ZnCo2O4 twin microspheres show an exceptionally high rate capability up to the 10 A g−1 rate. It should be noted that such super‐high rate performance and cycling stability at such high charge/discharge rates are significantly higher than most work previously reported on ZnCo2O4 micro/nanostructures and ZnCo2O4‐based heterostructures. The ZnCo2O4 3D hierarchical micro/nanostructures demonstrate the great potential as negative electrode materials for high‐performance LIBs.

Accuracy improvement of a multistep splitting method for nonlinear viscous wave equations

This paper focuses on a multistep splitting method for a class of nonlinear viscous equations in two spaces, which uses second-order backward differentiation formula (BDF2) combined with approximation factorization for time integration, and second-order centred difference approximation to second derivatives for spatial discretization. By the discrete energy method, it is shown that this splitting method can attain second-order accuracy in both time and space with respect to the discrete L2- and H1-norms. Moreover, for improving computational efficiency, we introduce a Richardson extrapolation method and obtain extrapolation solution of order four in both time and space. Numerical experiments illustrate the accuracy and performance of our algorithms.

Combining Computational Fluid Dynamics with a Biokinetic Model for Predicting Ammonia and Phosphate Behavior in Aeration Tanks

The aim of this study was to use computational fluid dynamics for predicting the behavior of reactive pollutants (ammonia and phosphate) in the aerobic zone of the bioreactor located at the Wschod wastewater treatment plant in Gdansk, Poland. The one-dimensional advection-dispersion equation was combined with simple biokinetic models incorporating the Monod-type expressions as source terms for the two pollutants. The problem was solved numerically by a multi-step splitting technique algorithm. The dispersion coefficient, E(L), was estimated using a statistical method and numerical optimization based on experimental data from three tracer studies. With the first method, the values of EL varied within the range 1082 to 1860 m2/h and 695 to 1355 m2/h, respectively, in sections 1 and 2 of the aerobic zone. Except for one case, deviations of the corresponding numerically optimized values of E(L) did not exceed 14%. The maximum specific rates of nitrification [r(n,max,20) = 4.6 g N/(kg VSS h)] and phosphate uptake [r(Pupt,max,20) = 13.5 g P/(kg VSS h)] at T = 20 degrees C were determined based on laboratory batch experiments. With minor adjustments of the kinetic parameters, the model was capable of accurately predicting the longitudinal profiles of ammonia and phosphate in the aerobic zone, and the simulation results were presented using the actual horizontal geometry of the bioreactor as a background.

Generation of monodisperse gel emulsions in a microfluidic device

We demonstrate that high dispersed phase volume fraction emulsions (i.e., gel emulsions) can be prepared in situ for microfluidic applications. Previously, the production of gel-like emulsions in microfluidic devices, where the droplet size is less than the length-scale of the channel, required multistep splitting of larger droplets in a branched microchannel network. Instead, we employ an abrupt change in the aspect ratio of a single microchannel to rapidly destabilize a confined coflowing stream, forming highly monodisperse droplets (coefficient of variance <1.5%). Using this emulsification mechanism, gel emulsions can be prepared in a single production step.

Multistep Splitting Methods of High Order for Initial Value Problems

An attempt is made to construct splitting methods for nonlinear initial-boundary value problems, which are of higher order with respect to the time discretization than the ADI and LOD methods. For semidiscretized parabolic equations third- and fourth-order multistep splitting methods are derived which are unconditionally stable when the Jacobian matrix of the right-hand side admits a splitting into a sum of matrices with a common set of eigenvectors and with negative eigenvalue spectra. For hyperbolic equations we only succeeded in constructing unconditionally stable methods of second order. A few numerical experiments are reported in which the four-step method is compared with a one-step splitting method.