Partition Points Research Articles

A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation

A nonlinear loaded differential equation with a parameter on a finite interval is studied. The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, we obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, we compose a system of nonlinear algebraic equations in parameters. A method of solving the boundary value problem with a parameter is proposed. The method is based on finding the solution to the system of nonlinear algebraic equations composed.

Spatiotemporal characterization and regional contributions of O3 and NO2: An investigation of two years of monitoring data in Henan, China

To investigate the characteristics of ground level ozone (O3) for Henan Province, the ambient air quality monitoring data of 2015 and 2016 were analyzed. The result showed that the 8 h-max-O3 concentrations displayed a distinct seasonality, where the maximum and minimum values, averaging 140.41, 54.19 μg/m3, occurred in summer and winter, respectively. There is a significant correlation between meteorological factors and O3 concentration. The Voronoi neighborhood averaging analysis indicated that O3, temperature, and ultraviolet radiation in Henan province were decreased from northwest to southeast, while relative humidity and precipitation displayed the opposite trend. Besides meteorological factors, the chemical processes play an essential role in ozone formation. Reactions of NO, NO2 and O3 form a closed system, and the partitioning point of the OX-component (O3 + NO2) was at 40 and 80 μg/m3 for nitrogen oxide (NOX) in winter and summer, respectively, with NO2 dominating at higher NOx pollution and O3 being the major component at lower levels. The relationship between oxidant (OX = O3+NO2) and NOx concentrations were evaluated to understand the regional and local contribution of OX. It showed that high regional contribution occurred in the spring, whereas the highest local contribution lead to the summer peak of O3 observed in Zhengzhou. This present study reveals important environment impacts from the photochemical process and the meteorological conditions in the region with better understanding on the O3 characterization.

Approximation methods for second kind weakly singular Volterra integral equations

Projection methods are applied to obtain the convergence rates for Volterra integral equations with weakly singular kernels. We consider Galerkin and multi Galerkin methods and their iterated versions to solve Volterra integral equations with weakly singular kernels, in the space of piecewise polynomials subspaces based on graded mesh. We will show that the iterated multi-Galerkin method improves over iterated Galerkin method. In fact, we show that iterated multi-Galerkin solution converges with the convergence rates O(n−3m) and O(n−3m(logn)2), for algebraic and logarithmic type kernels, respectively. We prove that iterated Galerkin method, for algebraic kernel, converges with the convergence rate O(n−2m) and for logarithmic type kernel converges with the convergence rate O(n−2mlogn), where n denotes the number of partition points and m is the highest order of the polynomials employed in the approximations. Theoretical results are justified by the numerical results.

Boomerang: On-Demand Cooperative Deep Neural Network Inference for Edge Intelligence on the Industrial Internet of Things

With the revolution of smart industry, more and more Industrial Internet of Things (IIoT) devices as well as AI algorithms are deployed to achieve industrial intelligence. While applying computation-intensive deep learning on IIoT devices, however, it is challenging to meet the critical latency requirement for industrial manufacturing. Traditional wisdom resorts to the cloud-centric paradigm but still works either inefficiently or ineffectively due to the heavy transmission latency overhead. To address this challenge, we propose Boomerang, an on-demand cooperative DNN inference framework for edge intelligence under the IIoT environment. Boomerang exploits DNN right-sizing and DNN partition to execute DNN inference tasks with low latency as well as high accuracy. DNN right-sizing reshapes the amount of DNN computation via the early-exit mechanism so as to reduce the total runtime of DNN inference. DNN partition adaptively segments DNN computation between the IoT devices and the edge server in order to leverage hybrid computation resources to achieve DNN inference immediacy. Combining these two keys, Boomerang carefully selects the partition point and the exit point to maximize the performance while promising the efficiency requirement. To further reduce the manual overhead of model profiling at the install phase, we develop an advanced version of Boomerang with the DRL model, achieving end-to-end automatic DNN inference plan generation. The prototype implementation and evaluations demonstrate the effectiveness of Boomerang on both versions in achieving efficient edge intelligence for IIoT.

An efficient method for privacy-preserving trajectory data publishing based on data partitioning

Since Osman Abul et al. first proposed the k-anonymity-based privacy protection for trajectory data, researchers have proposed a variety of trajectory privacy-preserving methods. These methods mainly adopt a static anonymity algorithm, which only focusing on the trajectories in a specific time span, directly anonymizes data and publishes them without considering dynamic nature of trajectory data as the new time slice arriving. Furthermore, due to its correlation with time and position, the trajectory data is produced in large scale and many sensitive attributes; the traditional k-anonymity-based privacy-preserving models need to recalculate the last released trajectory data, which will increase the computing cost and reduce the availability of the released trajectories, are not fit for privacy protection in large-scale trajectory data. Therefore, this paper presents a method to dynamically publish the large-scale vehicle trajectory data with privacy protection under $$(k,\delta )$$ security constraints. According to the spatial and temporal characteristics of vehicle trajectory data, this paper first proposes a method to partition the trajectory data for storage and computation. We choose the sample point $$(x_{i},y_{i})$$ at time $$t_{i}$$ as partition points and store the partitions of the trajectory data according to the time sequence and location of the running vehicle. This results in the efficient trajectory scanning, clustering and privacy protection. We use $$(x_{i},y_{i},t_{m}-t_{n})$$ to represent the identifier of trajectory data to publish, use the generalize function to cluster trajectory data under the $$(k,\delta )$$ security constraints. Through this way, we can effectively process the trajectory in every data partition as time goes on and need not to recalculate the released trajectories, effectively reduce the computing cost. Through experiments on real trajectory data and Oldenburg trajectory data, confirming the data partitioning method in privacy-preserving large-scale trajectory data publishing under the security constraint of $$(k,\delta )$$, and the l-diversity. By the experimental comparison, our method maintains a least level of computing cost and higher data availability.

Impedance-Based Analysis of Interconnected Power Electronics Systems: Impedance Network Modeling and Comparative Studies of Stability Criteria

Impedance is an intuitive and effective way for dynamical representation of power electronics devices [e.g., voltage source converters (VSCs)]. One of its strengths toward others is the natural association with circuits. However, impedances of VSCs are locally evaluated via linearization, a process dependent on the angle of the reference frame; thus, the reference frame transformation (i.e., rotation) is required before connecting them in circuits for the purpose of network analysis. Although this issue was properly treated in the state-space modeling, a counterpart for the impedance-based analysis, particularly the stability impacts of this rotation, has not been thoroughly discussed and worth being clarified. On the other hand, there are fundamental differences in applying the impedance-based stability criteria of a single-VSC system to an interconnected one. Several restrictions as revealed (e.g., sensitivity to partition points of the Nyquist-based analysis), if not properly considered, may lead to inaccurate stability assessments. In this respect, a clarification of three commonly employed impedance-based stability criteria is achieved. At last, the capability of the Nyquist-based analysis in identifying the system’s weak point and in facilitating better network design and planning is presented. All the models and analyses are verified by frequency-scanning and time-domain simulations in PSCAD/EMTDC.

NeuPart: Using Analytical Models to Drive Energy-Efficient Partitioning of CNN Computations on Cloud-Connected Mobile Clients

Data processing on convolutional neural networks (CNNs) places a heavy burden on energy-constrained mobile platforms. This work optimizes energy on a mobile client by partitioning CNN computations between in situ processing on the client and offloaded computations in the cloud. A new analytical CNN energy model is formulated, capturing all major components of the in situ computation, for ASIC-based deep learning accelerators. The model is benchmarked against measured silicon data. The analytical framework is used to determine the optimal energy partition point between the client and the cloud at runtime. On standard CNN topologies, partitioned computation is demonstrated to provide significant energy savings on the client over fully cloud-based or fully in situ computation. For example, at 80 Mbps effective bit rate and 0.78 W transmission power, the optimal partition for AlexNet [SqueezeNet] saves up to 52.4% [73.4%] energy over a fully cloud-based computation, and 27.3% [28.8%] energy over a fully in situ computation.

OPTiC: Optimizing Collaborative CPU–GPU Computing on Mobile Devices With Thermal Constraints

The CPU-graphic processing unit (GPU) co-execution of computation kernels on heterogeneous multiprocessor system-on-chip can significantly boost performance compared to the execution on either the CPU or the GPU alone. However, engaging multiple on-chip compute elements concurrently at the highest frequency may not provide the optimal performance in a mobile system with stringent thermal constraints. The system may repeatedly exceed the temperature threshold necessitating frequency throttling and hence performance degradation. We present OPTiC, an analytical framework that given a computation kernel can automatically select the partitioning point and the operating frequencies for optimal CPU–GPU co-execution under thermal constraints. OPTiC estimates, through modeling, CPU and GPU power, performance at different frequency points as well as the performance impact of thermal throttling and memory contention. Experimental evaluation on a commercial mobile platform shows that OPTiC achieves an average 13.68% performance improvement over existing schemes that enable co-execution without thermal considerations.

A new method for quantized sampled-data synchronization of delayed chaotic Lur’e systems

• The extended Wirtinger-inequality-based Lyapunov approach is newly proposed. • A zero equality is formulated, which can capture the information of system at the dynamic partitioning point. • A quantized sampled-data controller is designed. • Compared with some existing results, less conservative synchronization criteria are established for chaotic Lur’e systems. We design a quantized sampled-data controller for synchronization of delayed chaotic Lur’e systems. A new approach, extended Wirtinger-inequality-based Lyapunov–Krasovskii functional, is firstly proposed. This approach grasps more sampling information by introducing more free matrices in comparison with some existing methods. Using the system information at the dynamic partitioning point, a zero equality is formulated to fully utilize the inner sampling information. Based on the new approach and zero equality, some novel synchronization criteria are established. In the meantime, the desired quantized sampled-data control gain is obtained with larger sampling period than those in the existing works. Finally, two numerical examples illustrate the merits of the method.

Power Line Extraction From Mobile LiDAR Point Clouds

This paper proposes a three-step approach to extract power lines from components with mobile laser scanning (MLS) data. First, we use the maximum a posteriori estimation to partition point clouds into components. Each component contains points from one object only. The segmentation is optimized by the minimum-cost perfect matching globally and robust to Gaussian noise with the help of the proposed robust estimator. Then, we extract power lines from components based on the linear structure information. Finally, power line components are grouped into individual spans. Experiments show that our method succeeds to achieve the power line extraction from MLS data effectively and outperforms state-of-the-art approaches in terms of the accuracy and robustness.

Investigation of the influence of changing the boundary conditions along the side of the plate on its chaotic dynamics

In this article the effect of changing the boundary conditions along the side on the nonlinear dynamics of the plate are investigated. The mathematical model of the plate is constructed on the basis of the kinematic hypothesis of the first approximation. The governing PDEs are reduced to ODEs by the method of finite differences (FDM) of the second order. Results of convergence versus the number of partition points along the spatial co-ordinate and time steps are investigated. The obtained ODEs are solved by a fourth-order Runge-Kutta type method. The convergence of numerical solutions as a function of the time step is made. To take into account the change in the boundary conditions along the side of the plate, at the points of change of the boundary conditions the condition of matching is formulated in the work. Chaotic dynamics is investigated on the basis of Fourier power spectra, phase portraits, and Poincaré pseudo-mappings. To ensure the reliability of chaotic vibrations, the largest Lyapunov exponents (LLe) are calculated using three methods: Kantz, Wolf, and Rosenstein.

АРХІТЕКТУРА СИСТЕМИ ДЕДУБЛІКАЦІЇ ТА РОЗПОДІЛУ ДАНИХ У ХМАРНИХ СХОВИЩАХ ПІД ЧАС РЕЗЕРВНОГО КОПІЮВАННЯ

The conceptual model of the system is developed and described in detail. An intelligent system of deduplication and distribution of data in the cloud storage is developed, the description of the software is described, the stages of the user's work are considered. Testing of the projected system was carried out. Several control samples are described and results are analyzed. The purpose of the system is to deduplicate and distribute data in cloud repositories in such a way that the end result of the backup is to eliminate duplicate pieces of data using distributed computing and cloud repositories. By picking the right approach to distribute tasks and data during deduplication, you can harness the full potential of cloud-based distributed systems to increase backup speed and bandwidth. Analyzes (disadvantages and advantages of using different approaches) are analyzed and effective methods of solution are selected: hybrid block-level deduplication, splitting of data flow on the basis of Rabin's digital imprint, distribution of data based on hash values of blocks of deduplication and use of distributed index. Block-level deduplication involves two types of data flow splitting into blocks, a fixed-length, algorithm-based split. Fixed-length partitioning is rather trivial and fast with respect to the complexity of the algorithm, but the downside is that data is displaced at the beginning of the stream, since the blocks that will follow after the changes will be considered new. However, in the case of partitioning of blocks of variable length, the point of proper partitioning is determined by the algorithm. This algorithm should work with infinite data flows using the ring hash function. The algorithm absorbs each input byte of data from the stream, and as soon as the value of the annular hash function corresponds to the previously specified template, it also serves as a point of splitting the stream into blocks. Thus, if the data is changed or displaced by a couple of bytes, only the data block that covers the data will be considered new. However, in order to track changes and correctly set breakpoints, it is necessary to check the input data for a specific preset digital pattern - a hash value. It is a common practice to calculate a hash value every time an input byte is received in a data stream. The point of partition will be the moment when the resulting hash value matches the specified pattern. To do these calculations effectively, an algorithm has been devised for the ring hash. One of the most common ring hash algorithms is a digital Rabin imprint. During the analysis of the solutions, the Rust programming language for client-side writing, the Scala programming language for the server-side, the Akka distributed computing management tool, and Amazon S3 as the cloud repository were selected.

Numerical solution of a control problem for ordinary differential equations with multipoint integral condition

A linear boundary value problem with a parameter for ordinary differential equations with multipoint integral conditionis investigated.The method of parameterization is used for solving the considered problem. The linear boundary value problem with a parameter for ordinary differential equations with multipoint integral condition by introducing additional parameters at the partition points is reduced to equivalent boundary value problem with parameters. The equivalent boundary value problem with parameters consists of the Cauchy problem for the system of ordinary differential equations with parameters, multipoint integral condition and continuity conditions. The solution of the Cauchy problem for the system of ordinary differential equations with parameters is constructed using the fundamental matrix of differential equation. The system of linear algebraic equations with respect to the parameters are composed by substituting the values of the corresponding points in the built solutions to the multipoint integral condition and the continuity condition. Numerical method for finding solution of the problem is suggested, which based on the solving the constructed system and Runge-Kutta method of the 4-th order for solving Cauchy problem on the subintervals.

Phase Identification and Online Monitoring for the Uneven Batch Processes

In practice, the batch processes are usually uneven and show significantly different variables' characteristics in different sub-phases. Therefore, it is necessary to divide each batch into several sub-phases separately. In this paper, a moving window-based multiway information increment matrix (MWMIIM) algorithm for the uneven batch processes is proposed for single-batch phase identification and online monitoring by combining the moving window technique with an information increment matrix (IIM) algorithm. Similar to the IIM algorithm, the MWMIIM algorithm captures variables' correlation changes by calculating the increment matrix between adjacent covariance matrices, which does not need to extract feature from data or carry out complicated matrix decomposition, thus improving the computational efficiency. Besides, the influence of several vital parameters on the phase identification performance is discussed in detail. After phase identification, the partition points of each sub-phase need not to be strictly aligned. Furthermore, a batch process is divided into three types of regions, namely common region, transition region, and end region. Next, fine modeling and online monitoring strategies are adopted in different regions separately. The comparative experiment is conducted by the window-based stepwise sequential phase partition method for nonlinear uneven batch processes (WNSSPP-U). A practical application on batch processes, namely aircraft steering gear system fault diagnosis experiment, is given to confirm the feasibility and effectiveness of the proposed method.

A Data-Driven Parameter Adaptive Clustering Algorithm Based on Density Peak

Clustering is an important unsupervised machine learning method which can efficiently partition points without training data set. However, most of the existing clustering algorithms need to set parameters artificially, and the results of clustering are much influenced by these parameters, so optimizing clustering parameters is a key factor of improving clustering performance. In this paper, we propose a parameter adaptive clustering algorithm DDPA-DP which is based on density-peak algorithm. In DDPA-DP, all parameters can be adaptively adjusted based on the data-driven thought, and then the accuracy of clustering is highly improved, and the time complexity is not increased obviously. To prove the performance of DDPA-DP, a series of experiments are designed with some artificial data sets and a real application data set, and the clustering results of DDPA-DP are compared with some typical algorithms by these experiments. Based on these results, the accuracy of DDPA-DP has obvious advantage of all, and its time complexity is close to classical DP-Clust.

AN ALGORITHM FOR SOLVING A CONTROL PROBLEM FOR A DIFFERENTIAL EQUATION WITH A PARAMETER

On a finite interval, a control problem for a linear ordinary differential equations with a parameter is considered. By partitioning the interval and introducing additional parameters, considered problem is reduced to the equivalent multipoint boundary value problem with parameters. To find the parameters introduced, the continuity conditions of the solution at the interior points of partition and boundary condition are used. For the fixed values of the parameters, the Cauchy problems for ordinary differential equations are solved. By substituting the Cauchy problem’s solutions into the boundary condition and the continuity conditions of the solution, a system of linear algebraic equations with respect to parameters is constructed. The solvability of this system ensures the existence of a solution to the original control problem. The system of linear algebraic equations is composed by the solutions of the matrix and vector Cauchy problems for ordinary differential equations on the subintervals. A numerical method for solving the origin control problem is offered based on the Runge-Kutta method of the 4-th order for solving the Cauchy problem for ordinary differential equations. Key words: boundary value problem with parameter, differential equation, solvability, algorithm.

A fast analytical approximation type-reduction method for a class of spiked concave type-2 fuzzy sets

In this paper, an analytical type-reduction method called concave analytical type-reduction method with spikes (CATRS) of two kinds of spiked concave type-2 fuzzy sets is proposed. The concave type-2 fuzzy sets which are considered in the paper include triangular type-2 fuzzy set, trapezoidal type-2 fuzzy set and its spiked versions. The analytical type-reduction method is free of the following steps, such as discourse partition and discourse refinement for algorithms' discrete implementation, thus its calculation complexity is reduced. To obtain the formulae of concave triangular type-2 fuzzy set, the method proposed by Starczewski is extended from the convex case to the concave one, secondary membership function with spikes is considered as a special form of triangular fuzzy set. Centroid set is formed by all the type-1 fuzzy sets that is derived from the convex part of the primary membership with or without spikes at a given discourse partition point. Moreover, the method proposed for concave triangular type-2 fuzzy set is extended to concave trapezoidal type-2 fuzzy set with a trapezoidal fuzzy number approximation operator. The proposed analytical type-reduction methods can address a more general class of type-2 fuzzy sets, it is more convenient for type-2 fuzzy modeling and inference, and the application of type-reduction method to variable universe of discourse controller design shows that the approximation method is applicable for the adaptive fuzzy controller design.

Differential bond energy algorithm for optimal vertical fragmentation of distributed databases

Distributed database systems are gaining importance due to the production in massive amount of data. The efficacy of such systems is highly dependent upon the design of the system. To increase the effectiveness and efficiency of distributed databases, two processes are mainly employed i.e. fragmentation and allocation. Fragmentations can be vertical or horizontal. This work focuses on vertical fragmentation design methods. In this paper, a novel differential bond energy (DBE) algorithm is proposed with objective to determine optimal partition point. The performance of proposed algorithm is compared with classical bond energy algorithm (BEA) on basis of global affinity measure (GAM) value. Results are depicted in form of line graphs. The mean difference in GAM values for both algorithms are also illustrated. The experimental results portrays that DBE is suitable for vertical fragmentation of high dimensional problems as it attain high GAM value as compared to BEA on various datasets.

On the contact interaction of a two-layer beam structure with clearance described by kinematic models of the first, second and third order approximation

Abstract In this article, the contact interaction of two geometrically non-linear beams with a small clearance is studied by employing various approximations of the kinematic beam models. Our investigation concerns two-case studies, i.e. when the upper beam is governed by the second order Timoshenko model (i) or by third order Pelekh-Sheremetev-Reddy-Levinson model (ii), whereas in both cases the lower beam is described by the kinematic (Bernoulli-Euler) model of the first approximation. The upper beam is subjected to the transversal, uniformly distributed harmonic load, whereas the beam interaction follows the classical Kantor’s model. The problem is highly non-linear due to occurrence of the geometric von Karman non-linearity and the contact interaction between beams (structural non-linearity). The governing PDEs are reduced to ODEs by the method of finite differences (FDM) of the second order. The obtained ODEs are solved by a few Runge-Kutta type methods of different orders. Results of convergence versus the number of partition points along the spatial co-ordinate and time steps are investigated. New non-linear phenomena of the studied structural package are detected, illustrated and discussed with emphasis on the “true” chaotic vibrations. In particular, three qualitatively different algorithms for computation of largest Lyapunov exponents are employed, and the differences between geometrically linear versus non-linear problems are reported.

On the numerical solution of high order multi-dimensional elliptic PDEs

Using the idea of differential quadrature, two new methods are constructed to approximate the solution of elliptic partial differential equations in higher dimensions. To obtain the weighting coefficients of the first method, a mixture of grid points and mid points of the uniform partition is used. The order of convergence obtained by the first algorithm is non-optimal, so we mixed the idea with spline collocation to obtain higher order approximations. Using the weighting coefficients of the non-optimal algorithm, some new weighting coefficients are obtained which are used to obtain higher accuracy. For the first time, a sixth order approximation is obtained to the solution of well-known high order multi-dimensional elliptic PDEs such as biharmonic and triharmonic problems. We found a way to increase the order of convergence and the accuracy without increasing the CPU runtime. The block form of the coefficients matrix for the linear case is presented to have a better vision on the system arised from the method. Some examples of biharmonic and triharmonic problems are solved to show the good performance and applicability of the proposed algorithms.