Splitting Method Research Articles

Plug-and-Play Split Gibbs Sampler: embedding deep generative priors in Bayesian inference.

This paper introduces a stochastic plug-and-play (PnP) sampling algorithm that leverages variable splitting to efficiently sample from a posterior distribution. The algorithm based on split Gibbs sampling (SGS) draws inspiration from the half quadratic splitting method (HQS) and the alternating direction method of multipliers (ADMM). It divides the challenging task of posterior sampling into two simpler sampling problems. The first problem depends on the likelihood function, while the second is interpreted as a Bayesian denoising problem that can be readily carried out by a deep generative model. Specifically, for an illustrative purpose, the proposed method is implemented in this paper using state-of-the-art diffusion-based generative models. Akin to its deterministic PnP-based counterparts, the proposed method exhibits the great advantage of not requiring an explicit choice of the prior distribution, which is rather encoded into a pre-trained generative model. However, unlike optimization methods (e.g., PnP-ADMM and PnP-HQS) which generally provide only point estimates, the proposed approach allows conventional Bayesian estimators to be accompanied by confidence intervals at a reasonable additional computational cost. Experiments on commonly studied image processing problems illustrate the efficiency of the proposed sampling strategy. Its performance is compared to recent state-of-the-art optimization and sampling methods.

Assessment of Computational Technicalities for 2D/1D Coupling Method and Its Comparison with 3D MOC

The 2D/1D coupling method is recognized as one of the preferred high‐fidelity calculation methods for reactors featured with well homogeneity in the axial direction. However, the choices of the technicalities, such as transverse leakage, axial solver, and transverse leakage splitting method, will lead to different calculational performance. This paper outlines the theory of the 2D/1D coupling method and describes the detailed implementations of the key technicalities. Based on numerical tests, the choices of the related calculational parameters are analyzed, such as the order of the quadrature set, ray spacing, and the axial mesh size. Then, the computational performance of six typical 2D/1D technicalities is compared and assessed. A comparison between 2D/1D coupling method and direct 3D MOC calculation is also made. For the transverse leakage, the Fourier expansion technique could significantly reduce the memory burden and computational cost, but with the similar accuracy as the anisotropic leakage term. It is recommended to use the axial DGFEM SN solver, which has better consistency and leads to higher computational efficiency. Additionally, the results also show that the 2D/1D coupling method can appropriately increase the axial mesh size, which only has slight effect on the accuracy. For nuclear reactors featured with nonstrong heterogeneity in axial direction, the 2D/1D coupling method has significant advantages than the direct 3D MOC calculation.

Splitsing: As a Method for Resolving Teddy Minahasa’s Narcotics Case in Criminal Justice Investigations

Deliberate narcotics crimes typically involve multiple perpetrators or are coordinated to incorporate an element of participation. This is due to the fact that these crimes are carried out in groups and each individual is assigned specific responsibilities in order to accomplish predetermined objectives. In order to identify the principal perpetrators of group crimes, the public prosecutor is compelled to segregate the case files for the purpose of examination. As a result, this article aims to gain a deeper understanding of the splitting mechanism, which is utilized to divide the files of multiple perpetrators. This research employed a normative juridical approach, which is based on cases and legislation, and utilized secondary data in the form of legal materials. The complete dataset, consisting of primary, secondary, and tertiary legal materials, was subsequently gathered and subjected to deductive analysis. The results of this study indicate that case file splitting (splitting) is the division of a single file containing multiple perpetrators' information so that the public prosecutor may divide the file into multiple files based on the number of defendants. This is done in an effort to clear up the case and expedite the judge's evidentiary process. The provision pertaining to the division of case files is outlined in Criminal Procedure Code Article 142. Inspector General of Police Teddy Minahasa and his associates investigated a narcotics trafficking case using the splitting method, which involves the separation of case files. This approach has demonstrated significant efficacy in upholding the principles of fast, simple, and low-cost justice.

Forward-backward splitting algorithm with self-adaptive method for finite family of split minimization and fixed point problems in Hilbert spaces

In this paper, we introduce an inertial forward-backward splitting method together with a Halpern iterative algorithm for approximating a common solution of a finite family of split minimization problem involving two proper, lower semicontinuous and convex functions and fixed point problem of a nonexpansive mapping in real Hilbert spaces. Under suitable conditions, we proved that the sequence generated by our algorithm converges strongly to a solution of the aforementioned problems. The stepsizes studied in this paper are designed in such a way that they do not require the Lipschitz continuity condition on the gradient and prior knowledge of operator norm. Finally, we illustrate a numerical experiment to show the performance of the proposed method. The result discussed in this paper extends and complements many related results in literature.

A second-order in time and space model to solve the coupled Reynolds–Rayleigh–Plesset equations for the dynamics of cavitated hydrodynamic journal bearings

Cavitation in hydrodynamic bearings is a fundamental factor in the behavior of the fluid film and in the equilibrium position the journal will have after the application of external loads. The calculation of these parameters by robust tools is an active research topic even though accuracy, stability and conservativeness are not so often discussed. This work shows the results obtained through the implementation of a novel model developed from first principles that allows characterizing the pressure field of a liquid–gas lubricant mixture avoiding the reported instabilities with the time derivative of the mixture density. The formulation also relies on a coupled transport equation describing the evolution of the liquid fraction, ensuring mass conservation. This model is applied to study cases with gaseous, vaporous or pseudo-cavitation through the use of the Rayleigh–Plesset equation to characterize the physical behavior of the bubbles formed taking into account the characteristic time of each phenomenon. High resolution schemes are used for the spatial treatment of the advection and a temporal discretization based on the Strang splitting method which allows for a second order convergence in space and time. The numerical results obtained are contrasted and discussed with experimental data found for static and dynamic bearings for the three types of cavitation mentioned, obtaining a very good agreement.

Energy dissipation and maximum bound principle preserving scheme for solving a nonlocal ternary Allen-Cahn model

We present a new energy dissipation and maximum bound principle preserving scheme for solving a nonlocal ternary Allen-Cahn (NtAC) model, where the standard Laplace operator is deliberately replaced with a spatial convolution term that aims at describing long-range interactions among particles and the structure-preserving scheme is obtained based on the operator splitting method. The discrete maximum bound principle and global convergence of the new scheme are analyzed rigorously. Moreover, the scheme is strictly dissipative for a modified energy, which coincides with the original energy up to O(τ), where τ is the time step. To the best of our knowledge, this is the first work to show that the operator splitting method can guarantee the maximum bound principle for a ternary Allen-Cahn model. Various 2D/3D numerical experiments including the NtAC model with mass conservation are tested to verify the bound-preserving and energy-stable properties of the resulted scheme.

Richardson Iterative Method for Solving Multi-Linear System with M-Tensor

In this paper, Richardson iterative method is employed to solve M-Equation. In order to guarantee the solution can be found, convergence theorems are established and confirmed numerically. The optimal α, which is a parameter of Richardson iterative method that can provide the best convergence rate, is also determined theoretically and numerically. Furthermore, a theorem establishing the range of initial vector for general splitting methods is extended from the range in past study. To further accelerate the convergence rate, Anderson accelerator and three preconditioners are incorporated into Richardson iterative method. Numerical results reveal that by including these accelerators, the convergence rates are enhanced. Finally, we show that Richardson iterative methods with optimal α perform better than the SOR type methods in past studies in terms of number of iterative steps and CPU time.

Simultaneous temporal and spectral evolution of ultrafast optical pulse propagation using a single bidirectional LSTM network

Numerical simulations of optical pulse evolution are often a cumbersome task, mostly due to the presence of nonlinearities, and standard numerical approaches like splitting methods can be too complex and resource-hungry to compute them. However, under certain conditions, deep learning (DL) techniques can provide accurate alternative approaches to predict such propagation behavior. Here, we show a bidirectional long short-term memory (BiLSTM) network that predicts both temporal and spectral evolution of a short optical pulse going through a 13 m highly nonlinear fiber (HNLF). The BiLSTM performed an RMSE metric of 0.004 and 0.012 for temporal and spectral domains, respectively. A supercontinuum generation was also performed in a 20-cm photonic-crystal fiber, and the BiLSTM returns an RMSE metric of 0.018 and 0.013 for temporal and spectral domains, respectively.

Improved error estimates of the time‐splitting methods for the long‐time dynamics of the Klein–Gordon–Dirac system with the small coupling constant

AbstractWe provide improved uniform error estimates for the time‐splitting Fourier pseudo‐spectral (TSFP) methods applied to the Klein–Gordon–Dirac system (KGDS) with the small parameter . We first reformulate the KGDS into a coupled Schrödinger–Dirac system (CSDS) and then apply the second‐order Strang splitting method to CSDS with the spatial discretization provided by Fourier pseudo‐spectral method. Based on rigorous analysis, we establish improved uniform error bounds for the second‐order Strang splitting method at up to the long time at . In addition to the conventional analysis methods, we mainly apply the regularity compensation oscillation technique for the analysis of long time dynamic simulation. The numerical results show that our method and conclusion are not only suitable for one‐dimensional problem, but also can be directly extended to higher dimensional problem and highly oscillatory problem. As far as we know there has not been any relevant long time analysis and any improved uniform error bounds for the TSFP method solving the KGDS. Our methods are novel and provides a reference for analyzing the improved error bounds of other coupled systems similar to the KGDS.

Evaluation of physical education teaching effect using Random Forest model under artificial intelligence

This work aims to optimize the physical education (PE) teaching effect based on deep learning (DL) to cultivate high-level college students better. Firstly, the present situation of college teachers' teaching ability is surveyed to realize the deficiencies in teaching. Secondly, an optimization algorithm is proposed to improve the node splitting mode. This algorithm can solve the problem of single and similar node splitting modes in the Random Forest (RF) algorithm. The independent node splitting method Iterative Dichotomiser 3 and Classification and Regression Tree in the algorithm are recombined, and new splitting rules are obtained through adaptive parameter selection. Finally, the scheme designed is tested. The results suggest: The results suggest: (1) During the training of the proposed algorithm, although the loss curve at 4550 and 6800 points has a small crest, the error of the network loss function shows a downward trend and tends to be flat; (2) Compared with unoptimized Genetic Algorithm (GA) and Genetic Algorithm-Back Propagation (GA-BP), the proposed algorithm shows better performance both in terms of time consumption and accuracy (time consumption is less than 5.4 ms, and accuracy is more than 95 %). In a word, using the GA-BP-RF algorithm proposed to improve the PE teaching effect is feasible. The proposed model provides ideas for applying DL technology to improve teachers' teaching abilities.

A novel flux splitting based on wave-particle splitting for ideal magnetohydrodynamics

A novel flux splitting method based on wave-particle splitting is developed for one-dimensional ideal magnetohydrodynamics. While ideal magnetohydrodynamics (MHD) equations are non-convex with non-homogeneous flux as opposed to their hydrodynamic counterparts, the present flux splitting methods cannot develop Riemann solver. The proposed approach based on wave-particle method referred as Advection Magnetic-direction Wave Particle Splitting (AMWPS) scheme splits the flux into four parts: the wave-like total pressure part, the wave-like normal magnetic part, the particle-like advection part and the particle-like tangential magnetic part. The particle-like parts can be solved by the scheme developed by Toro and Vazquez-Cendon (TV scheme) and we propose a novel approach to the solution of Riemann problem formed by the wave-like parts including total pressure sub-flux and normal magnetic sub-flux. As for the ordered wave foliation in MHD and other limitations in wave structure, we also analyze the relation between plasma β and specific heat ratio γ. The advantage of AMWPS is that it can sharply capture isolated, stationary Alfven wave discontinuity. Several one-dimensional and two-dimensional MHD problems have been tested to highlight the accuracy, positivity preservation and robustness of AMWPS scheme and comparative studies show that AMWPS significantly outperforms the Riemann solver named Harten, Lax and Leer for contact wave (HLLC) in most cases.

Numerical model of Earth ionosphere F region based on three-dimensional transport and ambipolar diffusion equations

Abstract The paper provides a detailed description of the numerical implementation of the transport scheme in the Earth’s ionosphere three-dimensional dynamical model of the Institute of Numerical Mathematics (INMIM). The presented version of INM-IM model takes into account the global dynamical processes of ion transport and ambipolar diffusion in the altitude range between 100 and 500 km (corresponding to F region). Based upon the splitting method the model solves the equations of ambipolar diffusion on the first split step and incorporates specifically designed advective transport scheme on the second step. The accuracy of transport scheme implementation in the model has been investigated through analytical solutions. The stability of the numerical algorithm has been demonstrated even for cases when transport velocities approaching extreme values.

Incorporating history and deviations in forward–backward splitting

We propose a variation of the forward–backward splitting method for solving structured monotone inclusions. Our method integrates past iterates and two deviation vectors into the update equations. These deviation vectors bring flexibility to the algorithm and can be chosen arbitrarily as long as they together satisfy a norm condition. We present special cases where the deviation vectors, selected as predetermined linear combinations of previous iterates, always meet the norm condition. Notably, we introduce an algorithm employing a scalar parameter to interpolate between the conventional forward–backward splitting scheme and an accelerated O1n2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}\\left( \\frac{1}{n^2}\\right) $$\\end{document}-convergent forward–backward method that encompasses both the accelerated proximal point method and the Halpern iteration as special cases. The existing methods correspond to the two extremes of the allowed scalar parameter range. By choosing the interpolation scalar near the midpoint of the permissible range, our algorithm significantly outperforms these previously known methods when addressing a basic monotone inclusion problem stemming from minimax optimization.

Efficient numerical simulation of the conserved Allen–Cahn type flow-coupled binary fluid-surfactant model by a dimension splitting method

Efficient numerical simulation of the conserved Allen–Cahn type flow-coupled binary fluid-surfactant model by a dimension splitting method

Why projection over convex sets works and how to make it better

Projection over convex sets (POCS) is one of the most widely used algorithms to interpolate seismic data sets. A formal understanding of the underlying objective function and the associated optimization process is, however, lacking to date in the literature. Here, POCS is shown to be equivalent to the application of the half-quadratic splitting (HQS) method to the [Formula: see text] norm of an orthonormal projection of the sought after data, constrained on the available traces. Similarly, the apparently heuristic strategy of using a decaying threshold in POCS is revealed to be the result of the continuation strategy that HQS must use to converge to a solution of the minimizer. In light of this theoretical understanding, another methods able to solve this convex optimization problem, namely the Chambolle-Pock primal-dual algorithm, is shown to lead to a new POCS-like method with superior interpolation capabilities at nearly the same computational cost of the industry-standard POCS method.

Convergence analysis of operator splitting methods for Maxwell’s equations in dispersive media of Debye type

Convergence analysis of operator splitting methods for Maxwell’s equations in dispersive media of Debye type

Geometric Particle-In-Cell discretizations of a plasma hybrid model with kinetic ions and mass-less fluid electrons

We explore the possibilities of applying structure-preserving numerical methods to a plasma hybrid model with kinetic ions and mass-less fluid electrons satisfying the quasi-neutrality relation. The numerical schemes are derived by finite element methods in the framework of finite element exterior calculus (FEEC) for field variables, Particle-In-Cell (PIC) methods for the Vlasov equation, and splitting methods in time based on an anti-symmetric bracket proposed. Conservation properties of energy, quasi-neutrality relation, positivity of density, and divergence-free property of the magnetic field are given irrespective of the used resolution and metric. Local quasi-interpolation is used for dealing with the current terms in order to make the proposed methods more efficient. The implementation has been done in the framework of the Python package STRUPHY [1], and has been verified by extensive numerical experiments.

Improved uniform error estimates for the two-dimensional nonlinear space fractional Dirac equation with small potentials over long-time dynamics

We develop improved uniform error bounds on a second-order Strang splitting method for the long-time dynamics of the nonlinear space fractional Dirac equation (NSFDE) in two dimension (2D) with small electromagnetic potentials. First, a Strang splitting approach is implemented to discretize NSFDE in time. Afterwards the Fourier pseudospectral method is used to complete the discretization of NSFDE in space. With the aid of a second-order Strang splitting approach employed to the Dirac equation, the major local truncation error of the indicated numerical methods is established. Moreover, for the semi-discrete scheme and full-discretization, we rigorously demonstrate the improved, sharp uniform error estimates are O(ετ2) and O(h1m+h2m+ετ2) in virtue of the regularity compensation oscillation (RCO) technique. In the formulations, τ is the time step, hi(i=1,2) stands for spatial sizes in xi-directions, m is dependent on the regularity of solutions, and ε∈(0,1]. In order to verify our error bounds and to illustrate some fascinating long-time dynamical behaviors of the NSFDE with honeycomb lattice potentials for varied ε, numerical investigations are presented.

Customized Douglas-Rachford splitting methods for structured inverse variational inequality problems

Recently, structured inverse variational inequality (SIVI) problems have attracted much attention. In this paper, we propose new splitting methods to solve SIVI problems by employing the idea of the classical Douglas-Rachford splitting method (DRSM). In particular, the proposed methods can be regarded as a novel application of the DRSM to SIVI problems by decoupling the linear equality constraint, leading to smaller and easier subproblems. The main computational tasks per iteration are the evaluations of certain resolvent operators, which are much cheaper than those methods without taking advantage of the problem structures. To make the methods more implementable in the general cases where the resolvent operator is evaluated in an iterative scheme, we further propose to solve the subproblems in an approximate manner. Under quite mild conditions, global convergence, sublinear rate of convergence, and linear rate of convergence results are established for both the exact and the inexact methods. Finally, we present preliminary numerical results to illustrate the performance of the proposed methods.

The theoretical insights on the mechanism for Fe-N4 improving the HCOOH dissociation on Pd13/Gra

The reaction mechanism of HCOOH dissociation on the Fe-N4-Gra supported Pd13 (Pd13/Fe-N-Gra) is systematically investigated by using the density functional method. For Pd13/Fe-N-Gra, the two possible active sites, i.e., the top site (T) and interface (synergy) site (S site), are all considered. The results show that, besides pathway via HCOO (HCOOH → HCOO → CO2), there is a new favorable channel (HCOOH → COOH → CO2) for producing the CO2, which is different from that of pure Pd (prefer HCOO pathway). More importantly, the new pathway at T site shows higher selectivity and activity for HCOOH dissociation to CO2, compared with the S site. Further analysis based on energy splitting method and electronic structure indicate that the higher electron transfer during the intermediate COOH dissociation brings the deformation of Pd13, which increases the selectivity for CO. In conclusion, the Fe-N4 can not directly act as the active center for the reaction, while it can effectively keep the number of active electrons in Pd13, especially for the T site, to keep high activity and selectivity for the HCOOH dissociation to CO2. Meanwhile, the results also propose that the effective method for reducing the adsorption of OH species at the interface site or fast release the OH species can further improve the catalytic activity and life of Pd13/Fe-N-Gra for HCOOH dissociation.