Splitting Method Research Articles

Adaptive fuzzy neighborhood decision tree

Decision tree algorithms have gained widespread acceptance in machine learning, with the central challenge lying in devising an optimal splitting strategy for node sample subspaces. In the context of continuous data, conventional approaches typically involve fuzzifying data or adopting a dichotomous scheme akin to the CART tree. Nevertheless, fuzzifying continuous features often entails information loss, whereas the dichotomous approach can generate an excessive number of classification rules, potentially leading to overfitting. To address these limitations, this study introduces an adaptive growth decision tree framework, termed the fuzzy neighborhood decision tree (FNDT). Initially, we establish a fuzzy neighborhood decision model by leveraging the concept of fuzzy inclusion degree. Furthermore, we delve into the topological structure of misclassified samples under the proposed decision model, providing a theoretical foundation for the construction of FNDT. Subsequently, we utilize conditional information entropy to sift through original features, prioritizing those that offer the maximum information gain for decision tree nodes. By leveraging the conditional decision partitions derived from the fuzzy neighborhood decision model, we achieve an adaptive splitting method for optimal features, culminating in an adaptive growth decision tree algorithm that relies solely on the inherent structure of real-valued data. Experimental evaluations reveal that, compared with advanced decision tree algorithms, FNDT exhibits a simple tree structure, stronger generalization capabilities, and superior performance in classifying continuous data.

Variance reduction in Monte Carlo for Integrals

Abstract. Nowadays, the extensive use of Monte Carlo methods in various fields has promoted the scientific decision-making, improved the efficiency of social operation, and enabled better management of many uncertain factors in daily life. However, in some cases, traditional Monte Carlo methods doesn't work so well because of the excessive variance. This paper aims to explore a new numerical integration method called splitting method to minimize intrinsic variance in Monte Carlo simulations. Through this innovative method, we found that the variance was reduced significantly. Despite some oscillating functions which are still difficult to estimate because they have too many turning points, this study provides new insights into variance reduction in Monte Carlo for integrals to optimize calculations in various fields when the modelling functions are not monotonic.

Interface engineering in Ni(OH)2/NiOOH heterojunction to enhance energy-efficient hydrogen production via urea electrolysis

Electrochemical urea electrolysis has merged as a promising alternative to conventional water splitting methods for hydrogen fuel production due to its cost-effectiveness and superior energy efficiency. The utilization of heterostructures has been proposed as a viable strategy to improve the efficiency of the urea oxidation reaction (UOR) by augmenting the quantity of active sites and optimizing the electronic structure. In this study, a Ni(OH)2/NiOOH heterojunction, referred to as H-Ni, was synthesized via a straightforward hydrothermal synthesis method. The notable performance of H-Ni in UOR is ascribed to the synergistic interaction between Ni(OH)2 and NiOOH, which constitute the principal components of the catalyst. Density functional theory (DFT) calculations reveal that the H-Ni composite is capable of modulating the d-band center, thereby enhancing the adsorption and desorption of reaction intermediates and decreasing the Gibbs free energy (ΔG) associated with the rate-determining step (RDS) of the UOR. Experimental results from catalytic performance tests indicate that the H-Ni-140 catalyst attains a current density of 10 mA·cm−2 in 1 a M KOH electrolyte containing 0.33 M urea at a relatively low potential of 1.341 V versus reversible hydrogen electrode (RHE), thereby highlighting its superior electrocatalytic performance. Furthermore, the catalyst requires only a cell voltage of 1.78 V to achieve a current density of 100 mA·cm−2, which is approximately 120 mV lower than that required for water electrolysis. This work presents a straightforward methodology for the cost-effective development of heterojunction catalysts.

Development and Verification of a Multi-Physics Transport Code of Molten Salt Reactor Fission Products

The transport of fission products in molten salt reactors has attracted much attention. However, few codes can completely describe the transport characteristic, though the migration of fission products in the molten salt reactor is essential to estimate the source term, decay heat, and radiation shielding. This study built a program named ThorFPMC (Thorium Fission Products Migration Code) that can handle the multi-physics transport characteristic based on the flow burnup code ThorMODEc (Thorium MOlten Salt Reactor Specific DEpletion Code). A problem-related depletion chain compression method was applied to decrease the order of the solve matrix. The matrix exponential and splitting methods were applied to solve the steady state and transient calculation, respectively. Error analysis showed that for a specific problem, the simplified depletion chain matrix index method could solve the fission products migration equation with an arbitrary time-step with high speed (s) and high precision (10−4); the splitting method could reach a precision of 10−2 level for the full fuel depletion chain, multi-nodes, and transient problems. Compared to the Strang splitting method, the perturbation splitting method has higher precision and less time consumption. In summary, the developed programmer could describe the migration effect of fission products in molten salt reactors, which provides a significant tool for the design of molten salt reactors.

Request Deadline Split and Interference‐Aware Request Migration in Edge Cloud

ABSTRACTEdge computing extends computing resources from the data center to the edge of the network to better handle latency‐sensitive tasks. However, with the rise of the Internet of Things, edge devices with limited processing capabilities face difficulties in executing requests with fluctuating request peaks. In order to meet the deadline constraints of latency‐sensitive tasks, a feasible solution is to offload some latency‐sensitive tasks to other nearby edge devices. This article studies the problem of request migration in edge computing systems and minimizes the request deadline violation rate based on actual online arrival patterns, performance interference phenomena, and deadline constraints. Since a request contains multiple services and request migration will lead to changes in server resource competition pressure, we split the problem into three sub‐problems, dividing the request deadline to determine the maximum response time of the service, determining the performance of the service under different resource pressures and the request migration strategies. To this end, we propose two deadline splitting methods, a performance interference model under multi‐resource pressure, and two heuristic request migration strategies. Since this article considers online edge scenarios, the number and type of requests are black boxes. We conduct simulation experiments and find that our method has only one‐third the number of request violations of other methods.

Nonlinear Regularized HSS Iteration Method for a Class of Nonlinear Saddle-Point Systems with Applications to Image Deblurring Problems

In this paper, we construct a nonlinear regularized Hermitian and skew-Hermitian splitting (NRHSS) iteration method for a class of nonlinear saddle-point systems. The NRHSS iteration method can be seen as a generalization of the Peaceman-Rachford splitting method with monotone plus skew-symmetric splitting (PRSM/MSS). Besides, if the nonlinear operator reduces to the linear case, the NRHSS iteration method reduces to the regularized Hermitian and skew-Hermitian splitting (RHSS) iteration method. Hence, the NRHSS iteration method can also be seen as a generalization of the RHSS iteration method. To overcome the heavy workload when direct solution methods are used, we also propose an inexact version of the NRHSS iteration method. Detailed theoretical analysis regarding the convergence properties is given. Numerical experiments on image deblurring problems are carried out to illustrate the feasibility and efficiency of the NRHSS iteration method.

Smooth Sigmoid Surrogate (SSS): An Alternative to Greedy Search in Decision Trees

Greedy search (GS) or exhaustive search plays a crucial role in decision trees and their various extensions. We introduce an alternative splitting method called smooth sigmoid surrogate (SSS) in which the indicator threshold function used in GS is approximated by a smooth sigmoid function. This approach allows for parametric smoothing or regularization of the erratic and discrete GS process, making it more effective in identifying the true cutoff point, particularly in the presence of weak signals, as well as less prone to the inherent end-cut preference problem. Additionally, SSS provides a convenient means of evaluating the best split by referencing a parametric nonlinear model. Moreover, in many variants of recursive partitioning, SSS can be reformulated as a one-dimensional smooth optimization problem, rendering it computationally more efficient than GS. Extensive simulation studies and real data examples are provided to evaluate and demonstrate its effectiveness.

The success rate of processed predicted models in molecular replacement: implications for experimental phasing in the AlphaFold era.

The availability of highly accurate protein structure predictions from AlphaFold2 (AF2) and similar tools has hugely expanded the applicability of molecular replacement (MR) for crystal structure solution. Many structures can be solved routinely using raw models, structures processed to remove unreliable parts or models split into distinct structural units. There is therefore an open question around how many and which cases still require experimental phasing methods such as single-wavelength anomalous diffraction (SAD). Here, this question is addressed using a large set of PDB depositions that were solved by SAD. A large majority (87%) could be solved using unedited or minimally edited AF2 predictions. A further 18 (4%) yield straightforwardly to MR after splitting of the AF2 prediction using Slice'N'Dice, although different splitting methods succeeded on slightly different sets of cases. It is also found that further unique targets can be solved by alternative modelling approaches such as ESMFold (four cases), alternative MR approaches such as ARCIMBOLDO and AMPLE (two cases each), and multimeric model building with AlphaFold-Multimer or UniFold (three cases). Ultimately, only 12 cases, or 3% of the SAD-phased set, did not yield to any form of MR tested here, offering valuable hints as to the number and the characteristics of cases where experimental phasing remains essential for macromolecular structure solution.

An adaptive positive preserving numerical scheme based on splitting method for the solution of the CIR model

This paper aims to investigate an adaptive numerical method based on a splitting scheme for the Cox-Ingersoll-Ross (CIR) model. The main challenge associated with numerically simulating the CIR process lies in the fact that most existing numerical methods fail to uphold the positive nature of the solution. Within this article, we present an innovative adaptive splitting scheme. Due to the existence of a square root in the CIR model, the step size is adaptively selected to ensure that, at each step, the value under the square-root does not fall under a given positive level and it is bounded. Moreover, an alternate numerical method is employed if the chosen step size becomes excessively small or the solution derived from the splitting scheme turns negative. This alternative approach, characterized by convergence and positivity preservation, is called the “backstop method”.Furthermore, we prove the proposed adaptive splitting method ensures the positivity of solutions in the sense that it would be possible to find an interval such that for all stepsizes belong, the probability of using the backstop method can be small. Therefore, the proposed adaptive splitting scheme avoids using the backstop method with arbitrarily high probability. We prove the convergence of the scheme and analyze the convergence rate. Finally, we demonstrate the applicability of the scheme through some numerical simulations, thereby corroborating our theoretical findings.

Application of the Cubic Exponential B-spline Collocation Operator Splitting Method for Numerical solutions of Burgers Equation

In this study, the cubic exponential B-spline collocation method has been proposed for the numerical solutions of the Burgers equation with the operator splitting. To apply the operator splitting method, the Burgers' equation has decomposed into two sub-equations based on the time term: the linear part (diffusion) and the nonlinear part (convection). Subsequently, for each sub-equation, Crank-Nicolson finite difference schemes in the temporal direction and cubic exponential B-spline functions and their derivatives, have applied at the x_m nodal points in the spatial direction. The algebraic equation systems obtained have been solved numerically using the Lie-Trotter and Strang splitting schemes to get the solutions of the main equation. Some advantages of the splitting methods include preserving the physical characteristics of the solution, yielding more convergent results over long time intervals, enabling simpler algorithms, and facilitating the storage of solution vectors on computer. To assess the accuracy of the computed numerical results the L_2 and L_∞ error norms have been used. Additionally, the obtained results have been compared with some studies in the literature. The stability analysis of the applied method has been investigated using the von Neumann Fourier series method.

An efficient Newton-type matrix splitting algorithm for solving generalized absolute value equations with application to ridge regression problems

A generalized Newton-based matrix splitting (GNMS) method is proposed for solving the generalized absolute value equations (GAVEs). Under mild conditions, the GNMS method converges to the unique solution of GAVEs. Moreover, we can obtain a few weaker convergence conditions for some existing methods. Numerical results verify the effectiveness of the proposed method.

An RBF Method for Time Fractional Jump-Diffusion Option Pricing Model under Temporal Graded Meshes

This paper explores a numerical method for European and American option pricing under time fractional jump-diffusion model in Caputo scene. The pricing problem for European options is formulated using a time fractional partial integro-differential equation, whereas the pricing of American options is described by a linear complementarity problem. For European option, we present nonuniform discretization along time and the radial basis function (RBF) method for spatial discretization. The stability and convergence analysis of the discrete scheme are carried out in the case of European options. For American option, the operator splitting method is adopted which split linear complementary problem into two simple equations. The numerical results confirm the accuracy of the proposed method.

A novel explicit fully-discrete momentum-preserving scheme of damped nonlinear stochastic wave equation influenced by multiplicative space-time noise

In this paper, the evolution laws of global momentum and averaged global momentum for the damped nonlinear stochastic wave equation (DNSWE) influenced by multiplicative space-time noise are derived. We innovatively combine the second-order central finite difference method and discrete gradient method in space, and integrate the splitting method and Störmer-Verlet type method in time. Both the novel spatial semi-discrete scheme and fully-discrete scheme constructed in this way can successfully preserve the corresponding evolution laws for discrete global momentum and discrete averaged global momentum. Numerical experiments on DNSWE with cubic nonlinearity validate the theoretical results.

SURFACE ENERGY OF GRAPHENE

The «sessile drop» method for determining the surface energy of solids – γ is considered. For graphite and graphene, this method yielded average values of γ2 = 53.6 ± 2.1 and γ1 = 44.8 ± 14.7 mJ/m2. Using the crystal cleavage method for graphite, the average value of γ2 = 3250 mJ/m2 was obtained, which is 60 times higher than the value of γ2 obtained by the "sessile drop" method. Analysis of both methods showed that the "sessile drop" method for determining the surface energy of solids cannot be used without modification. We propose new methods for determining the surface energy of solids: 1) through the melting point of a solid; 2) through the size dependence of a physical property on the thickness of the deposited coating; 3) through the measured value of the contact potential difference. The following values were obtained for graphene: γ1 = 947.1; γ1 = 974; γ1 = 960 mJ/m2, which is an order of magnitude higher than in the sessile drop method. The values of γ1 must be multiplied by 3 and we obtain the values for graphite – γ2 = 2841; γ2 = 2922, γ2 = 2880 mJ/m2, which differs slightly from the crystal splitting method.

Blind Deblurring Method for CASEarth Multispectral Images Based on Inter-Band Gradient Similarity Prior.

Multispectral remote sensing images contain abundant information about the distribution and reflectance of ground objects, playing a crucial role in target detection, environmental monitoring, and resource exploration. However, due to the complexity of the imaging process in multispectral remote sensing, image blur is inevitable, and the blur kernel is typically unknown. In recent years, many researchers have focused on blind image deblurring, but most of these methods are based on single-band images. When applied to CASEarth satellite multispectral images, the spectral correlation is unutilized. To address this limitation, this paper proposes a novel approach that leverages the characteristics of multispectral data more effectively. We introduce an inter-band gradient similarity prior and incorporate it into the patch-wise minimal pixel (PMP)-based deblurring model. This approach aims to utilize the spectral correlation across bands to improve deblurring performance. A solution algorithm is established by combining the half-quadratic splitting method with alternating minimization. Subjectively, the final experiments on CASEarth multispectral images demonstrate that the proposed method offers good visual effects while enhancing edge sharpness. Objectively, our method leads to an average improvement in point sharpness by a factor of 1.6, an increase in edge strength level by a factor of 1.17, and an enhancement in RMS contrast by a factor of 1.11.

Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization

Abstract It is of strong theoretical significance and application prospects to explore three-block nonconvex optimization with nonseparable structure, which are often modeled for many problems in machine learning, statistics, and image and signal processing. In this article, by combining the Bregman distance and Peaceman-Rachford splitting method, we propose a novel three-block Bregman Peaceman-Rachford splitting method (3-BPRSM). Under a general assumption, global convergence is presented via optimality conditions. Furthermore, we prove strong convergence when the augmented Lagrange function satisfies Kurdyka-Łojasiewicz property. In addition, if the association function possessing the Kurdyka-Łojasiewicz property exhibits a distinctive structure, then linear and sublinear convergence rate of 3-BPRSM can be guaranteed. Finally, a preliminary numerical experiment demonstrates the effectiveness.

Development of predictive model for post-stroke depression at discharge based on decision tree algorithm: A multi-center hospital-based cohort study

ObjectivePost-stroke depression (PSD) is one of the most common and severe neuropsychological sequelae after stroke. Using a prediction model composed of multiple predictors may be more beneficial than verifying the predictive performance of any single predictor. The primary objective of this study was to construct practical prediction tools for PSD at discharge utilizing a decision tree (DT) algorithm. MethodsA multi-center prospective cohort study was conducted from May 2018 to October 2019 and stroke patients within seven days of onset were consecutively recruited. The independent predictors of PSD at discharge were identified through multivariate logistic regression with backward elimination. Classification and regression tree (CART) algorithm was employed as the DT model's splitting method. ResultsA total of 876 stroke patients who were discharged from the neurology departments of three large general Class A tertiary hospitals in Wuhan were eligible for analysis. Firstly, we divided these 876 patients into PSD and non-PSD groups, history of coronary heart disease (OR = 1.835; 95 % CI, 1.106–3.046; P = 0.019), length of hospital stay (OR = 1.040; 95 % CI, 1.013–1.069; P = 0.001), NIHSS score (OR = 1.124; 95 % CI, 1.052–1.201; P = 0.001), and Mini mental state examination (MMSE) score (OR = 0.935; 95 % CI, 0.893–0.978; P = 0.004) were significant predictors. The subgroup analysis results have shown that hemorrhagic stroke, history of hypertension and higher modified Rankin Scale score (mRS) score were associated with PSD at discharge in the young adult stroke patients. ConclusionsSeveral predictors of PSD at discharge were identified and convenient DT models were constructed to facilitate clinical decision-making.

Operator splitting for coupled linear port-Hamiltonian systems

Operator splitting methods tailored to coupled linear port-Hamiltonian systems are developed. We present algorithms that are able to exploit scalar coupling, as well as multirate potential of these coupled systems. The obtained algorithms preserve the dissipative structure of the overall system and are convergent of second order. Numerical results for coupled mass–spring–damper chains illustrate the computational efficiency of the splitting methods compared to a straight-forward application of the implicit midpoint rule to the overall system.

SS-DNN: A hybrid strang splitting deep neural network approach for solving the Allen–Cahn equation

The Allen–Cahn equation is a fundamental partial differential equation that describes phase separation and interface motion in materials science, physics, and various other scientific domains. The presence of interfacial width (ϵ) between two stable phases, associated with a nonlinear term, is a small positive parameter which makes the problem more challenging to solve as ϵ approaches zero. This paper proposes a novel hybrid deep splitting method to efficiently and accurately solve the Allen–Cahn equation in a convex polygonal domain in Rd(d=1,2,3). The method combines the benefits of deep learning and splitting strategies, leveraging the strengths of both approaches. Essentially, a second-order splitting method is employed to split the Allen–Cahn equation into two simpler linear and non-linear sub-problems. While the nonlinear sub-problem can be solved analytically, the deep neural network is utilized to approximate the linear sub-problem. By integrating deep learning into the splitting strategy, we achieve a more efficient and accurate solution for the Allen–Cahn equation, demonstrating promising results. We also derive an error estimate for the proposed hybrid method. Modified space adaptivity and transform learning techniques are employed to enhance the efficiency of the neural network.

Accelerated iterative splitting methods on Hadamard manifolds

This paper aims to solve the monotone inclusion problem, minimization problem of multiple summands and the generalized Heron problem. We present an innovative approach, the modified normal S-iteration method, designed to approximate common fixed points of nearly nonexpansive sequences and families of operators via the property ( A ) . Some deductions of our results improve some existing results in the literature. To show the applicability of our result, we give application to the inclusion problem via forward–backward splitting method version of our algorithm and minimization problem via Douglas–Rachford splitting method version of our algorithm. To demonstrate the practical utility of the algorithm, we apply it to the generalized Heron problem.