Integer Approximation Research Articles

A Priori Estimates for the Solution of an Initial Boundary Value Problem of Fluid Flow through Fractured Porous Media

The paper studies a model of fluid flow in a fractured porous medium in which fractures are distributed uniformly over the volume. This model includes a nonlinear equation containing several terms with fractional derivatives in the sense of Caputo of order belonging to the interval 1,2. The relevance of studying this problem is determined by its practical significance in the oil industry, since most of the world’s oil reserves are in these types of reservoirs. The uniqueness of the solution to the problem in a differential form and its dependence on the initial data and the right-hand side of the equation is proved. A numerical method is proposed based on the use of the finite difference approximation for integer and fractional time derivatives and the finite element method in the spatial direction. A change of variables is introduced to reduce the order of the fractional derivatives. Furthermore, the fractional derivative is approximated by using the L1-method. The stability and convergence of the proposed numerical method are rigorously proved. The theoretical order of convergence is confirmed by the results of numerical tests for a problem of fluid flow in fractured porous media with a known exact solution.

An efficient MPC algorithm for switched systems with minimum dwell time constraints

This paper presents an efficient suboptimal model predictive control (MPC) algorithm for nonlinear switched systems subject to minimum dwell time constraints (MTC). While MTC are required for most physical systems due to stability, power and mechanical restrictions, MPC optimization problems with MTC are challenging to solve. To efficiently solve such problems, the on-line MPC optimization problem is decomposed into a sequence of simpler problems, which include two nonlinear programs (NLP) and a rounding step, as typically done in mixed-integer optimal control (MIOC). Unlike the classical approach that embeds MTC in a mixed-integer linear program (MILP) with combinatorial constraints in the rounding step, our proposal is to embed the MTC in one of the NLPs using move blocking. Such a formulation can speedup on-line computations by employing recent move blocking algorithms for NLP problems and by using a simple sum-up-rounding (SUR) method for the rounding step. An explicit upper bound of the integer approximation error for the rounding step is given. In addition, a combined shrinking and receding horizon strategy is developed to satisfy closed-loop MTC. Recursive feasibility is proven using a l-step control invariant (l-CI) set, where l is the minimum dwell time step length. An algorithm to compute l-CI sets for switched linear systems off-line is also presented. Numerical studies show significant speed-up and comparable control performance of the proposed MPC algorithm against the classical approach, though at the cost of significant solution quality degradation especially when the MTC are large.

Error estimates of the numerical method for the filtration problem with Caputo-Fabrizio fractional derivatives

This paper investigates a model of fluid flow in a fractured porous medium under the assumption of a uniform distribution of fractures throughout the volume. This model is based on the use of a fractional differential analogue of Darcy's law, as well as on the assumption that the properties of rock and fluid depend on pressure and its fractional derivative. Unlike previous studies, this study uses a fractional derivative in the Caputo-Fabrizio sense with a non-singular kernel. In this paper, we propose a numerical method for solving this initial boundary value problem and theoretically investigate the order of its convergence. The formulation of a fully discrete scheme is based on application of the finite difference approximation for integer and fractional timederivatives, and the Galerkin method in the spatial variable. A second-order formula is used to approximate both integer derivative and the fractional derivative in the sense of Caputo-Fabrizio. A priori estimates are obtained for both semi-discrete and fully discrete schemes, which imply their second-order convergence in time and space variables. A number of computational experiments were carried out on the example of a model problem to validate the accuracy of the scheme. The results of the numerical tests fully confirm the outcome of the theoretical analysis.

Integer approximation of a segment

Integer approximation of a segment

Statistical H.264 Double Compression Detection Method Based on DCT Coefficients

With the 2019 Coronavirus pandemic, we have seen an increasing use of remote technologies such has remote identity verification. The authentication of the user identity is often performed through a biometric matching of a selfie and a video of an official identity document. In such a scenario, it is essential to verify the integrity of both the selfie and the video. In this article, we propose a method to detect double video compression in order to verify the video integrity. We will focus on the H.264 compression which is one of the mandatory video codecs in the WebRTC Requests For Comments. H.264 uses an integer approximation of the Discrete Cosine Transform (DCT). Our method focuses on the DCT coefficients to detect a double compression. The coefficients roughly follow a Laplacian distribution, we will show that the distribution parameters vary with respect to the quantisation parameter used to compress the video. We thus propose a statistical hypothesis test to determine whether or not a video has been compressed twice.

Compliant Conditions for Polynomial Time Approximation of Operator Counts

In this brief abstract, we develop a computationally simpler version of the operator count heuristic for a particular class of domains. The contribution of this abstract is thus threefold, we (1) propose an efficient closed form approximation to the operator count heuristic; (2) leverage compressed sensing techniques to obtain an integer approximation in polynomial time; and (3) discuss the relationship of the proposed formulation to existing heuristics and investigate properties of domains where such approaches are useful.

On construction of almost periodic sequences and applications to some discrete population models

In this work, we develop a novel approximation strategy for building almost periodic sequences in the theory of almost periodic functions. Here, we create a different perspective for the argument of Dirichlet in the theory of numbers and design an integer approximation strategy in this regard. The idea behind the strategy comes from Kronecker's theorem and it is proven that for given an almost periodic function, it is possible to design its corresponding almost periodic sequence. Moreover, we provide two population models in both continuous and discrete cases where almost periodic sequence solutions are designed under suitable circumstances.

Some number-theoretic methods for solving partial derivatives

In this paper, a new method is constructed for solving partial differential equations using a sequence of nested generalized parallelepiped grids. This method is a generalization and development of the V. S. Ryaben’kii and N. M. Korobov method for the approximate solution of partial differential equations for the case of using arbitrary generalized parallelepiped grids for integer lattices. The error of this method was also found. In the case of using an infinite sequence of nested generalized parallelepiped grids, a fairly fast convergence will take place. In addition, a variant of constructing optimal grids in the two-dimensional case is proposed. It is based on the integer approximation of algebraic lattices. In the two-dimensional case, the grids constructed in this way will always give generalized parallelepiped grids. Moreover, there are simple ways to assess the quality of the resulting meshes. One such method, based on the use of a hyperbolic parameter, is considered in this paper.

Optimal control of a multi-supplier and multi-buyer supply chain system with JIT delivery

A just-in-time centred production-facility supply chain consists of raw material suppliers, manufacturer, and retailers. This paper considers the concept where the production of finished goods follows continuous production cycles. In this scenario, it is assumed that the inventory build-up during production cycles of the concurrent cycle overlap the pure demand consumption cycle to reduce the idle time suggested by previous researchers. Considering this situation, a supply chain inventory models for raw materials, and finished goods supply are developed for multiple suppliers and multiple buyers. In addition, this paper considered that different suppliers deliver the raw materials in instantaneous replenishments supply, and the finished goods are delivered to multiple buyers in a fixed amount after a fixed interval of time (known as just-in-time delivery) according to buyers' demand. The problem in this research is formulated as an integer nonlinear programming problem and heuristic solutions are developed to solve it with the help of integer approximations and divide-and-conquer technique. The solution methodologies suggested lead to estimates of optimum production quantity and minimum total system cost. The solutions are verified through numerical examples and illustrated the effectiveness of the method with sensitivity analyses. [Submitted: 27 September 2019; Accepted: 3 January 2020]

Fractional Order Pole Placement for a buck converter based on commensurable transfer function

In this work, a novel methodology for tuning fractional order controller, so-called fractional order pole placement (FOPP) is proposed. The proposed methodology extends classical (integer order) pole placement technique, using commensurable transfer functions for representing fractional order controllers and placing the fractional dominant poles within an extended stability region based on three terms fractional transfer functions. The designed fractional order controller can be digitally implemented by using the Oustaloup integer approximation of the fractional order dynamics with Hankel model reduction. The proposed FOPP is used to design fractional order controllers for a DC/DC buck converter. The experimental tests include comparisons between the proposed FOPP and other tuning methodologies for integer and fractional order controllers for the DC/DC buck converter that is subject to load, parametric, and load variations. Integral indices are computed to aid the assessment of the control strategies with respect to the robustness, performance specification compliance, and control effort. The results show that the proposed method outperforms the other approaches when occurs the variations.

Optimized reconstruction of the crystallographic orientation density function based on a reduced set of orientations.

Crystallographic textures, as they develop for example during cold forming, can have a significant influence on the mechanical properties of metals, such as plastic anisotropy. Textures are typically characterized by a non-uniform distribution of crystallographic orientations that can be measured by diffraction experiments like electron backscatter diffraction (EBSD). Such experimental data usually contain a large number of data points, which must be significantly reduced to be used for numerical modeling. However, the challenge in such data reduction is to preserve the important characteristics of the experimental data, while reducing the volume and preserving the computational efficiency of the numerical model. For example, in micromechanical modeling, representative volume elements (RVEs) of the real microstructure are generated and the mechanical properties of these RVEs are studied by the crystal plasticity finite element method. In this work, a new method is developed for extracting a reduced set of orientations from EBSD data containing a large number of orientations. This approach is based on the established integer approximation method and it minimizes its shortcomings. Furthermore, the L 1 norm is applied as an error function; this is commonly used in texture analysis for quantitative assessment of the degree of approximation and can be used to control the convergence behavior. The method is tested on four experimental data sets to demonstrate its capabilities. This new method for the purposeful reduction of a set of orientations into equally weighted orientations is not only suitable for numerical simulation but also shows improvement in results in comparison with other available methods.

A novel and efficient 8-point DCT approximation for image compression

The Discrete Cosine Transform is widely used in the field of still image compression. Many integer approximations are given in the literature whereas the most of these transforms requires bit shift operations. This paper presents an efficient and low complexity integer approximation of the DCT for image compression. Our new approach involves replacing the bit shift elements of a variant of the Signed DCT transform by zeros, in order to eliminate the bit shift operations. As a result, all elements of the proposed transform are zeros and ± 1. Indeed, the proposed transform retains all the characteristics of its original transform, such as orthogonality and high energy compaction capabilities, while generating computing cost savings. Experiments show that the proposed transform has a good compromise performance-computational complexity as well as state-of the-art DCT approximations. Moreover, an efficient algorithm primarily involving a small amount of arithmetical computation is well developed as no multiplications and bit-shift operations are required, with only 16 additions being involved.

Hyperbolic parameter of approximation of quadratic algebraic lattices

The article considers a variant of the approximation of algebraic lattices by integer ones in the quadratic case, the set of their local minima is written out explicitly, and it is also shown that for these integer approximations of algebraic quadratic lattices it is possible to construct efficient algorithms for calculating the hyperbolic parameter.

A Bi-level Approach to MPC for Switching Nonlinear Systems

In this paper, a nonlinear model predictive control scheme for switching dynamical systems is presented. The controller comprises of two layers of optimization. The upper layer is based on the embedding transformation technique, hence it does not require prior knowledge of the switching sequence. In particular, it provides the optimal relaxed switching sequence together with the optimal regulating inputs and the corresponding trajectories of the states. Within the lower layer, the integrality constraints are restored and a switching solution is recomputed to minimize the error with respect to the trajectories given from the upper-level optimization. The scheme is presented and the bounds of the integer approximation errors are evaluated together with brief recursive feasibility analysis. Simulation results of a tracking and an economics optimizing nonlinear MPC for a supermarket refrigerator system show the applicability and efficiency of the proposed approach.

Dual incremental fuzzy schemes for frequent itemsets discovery in streaming numeric data

Discovering frequent itemsets is essential for finding association rules, yet too computational expensive using existing algorithms. It is even more challenging to find frequent itemsets upon streaming numeric data. The streaming characteristic leads to a challenge that streaming numeric data cannot be scanned repetitively. The numeric characteristic requires that streaming numeric data should be pre-processed into itemsets, e.g., fuzzy-set methods can transform numeric data into itemsets with non-integer membership values. This leads to a challenge that the frequency of itemsets are usually not integer. To overcome such challenges, fast methods and stream processing methods have been applied. However, the existing algorithms usually either still need to re-visit some previous data multiple times, or cannot count non-integer frequencies. Those existing algorithms re-visiting some previous data have to sacrifice large memory spaces to cache those previous data to avoid repetitive scanning. When dealing with big streaming data nowadays, such large-memory requirement often goes beyond the capacity of many computers. Those existing algorithms unable to count non-integer frequencies would be very inaccurate in estimating the non-integer frequencies of frequent itemsets if used with integer approximation of frequency-counting.To solve the aforementioned issues, in this paper we propose two incremental schemes for frequent itemsets discovery that are capable to work efficiently with streaming numeric data. In particular, they are able to count non-integer frequency without re-visiting any previous data. The key of our schemes to the benefits in efficiency is to extract essential statistics that would occupy much less memory than the raw data do for the ongoing streaming data. This grants the advantages of our schemes 1) allowing non-integer counting and thus natural integration with a fuzzy-set discretization method to boost robustness and anti-noise capability for numeric data, 2) enabling the design of a decay ratio for different data distributions, which can be adapted for three general stream models: landmark, damped and sliding windows, and 3) achieving highly-accurate fuzzy-item-sets discovery with efficient stream-processing.Experimental studies demonstrate the efficiency and effectiveness of our dual schemes with both synthetic and real-world datasets.

Lead and lag controller design in fractional-order control systems

This paper presents a controller design method using lead and lag controllers for fractional-order control systems. In the presented method, it is aimed to minimize the error in the control system and to obtain controller parameters parametrically. The error occurring in the system can be minimized by integral performance criteria. The lead and lag controllers have three parameters that need to be calculated. These parameters can be determined by the simulation model created in the Matlab environment. In this study, the fractional-order system in the model was performed using Matsuda’s fourth-order integer approximation. In the optimization model, the error is minimized by using the integral performance criteria, and the controller parameters are obtained for the minimum error values. The results show that the presented method gives good step responses for lead and lag controllers.

PI-PD Controllers Design Using Bode’s Ideal Transfer Function

In this paper, a method is presented to design PIPD controllers for control system with time delay. The presented method takes Bode’s ideal transfer function as reference model and thus PI-PD controller parameters can be obtained by optimization. Bode's ideal transfer function gives the desired unit step response by setting two parameters. PI-PD parameters corresponding to the minimum value of the error which is the difference between the reference model and system to be controlled are obtained by using integral performance criteria which are frequently used in evaluating the performance of control systems. Matsuda's 4th order integer approximation is used in the Bode’s ideal transfer function. The reason of using Bode’s ideal transfer function instead of a standard second order system is that a fractional order plant represents a real system better than an integer order plant. Finally, the success of the method is demonstrated in the given example.

PID Controller Design for a Fractional Order System using Bode’s Ideal Transfer Function

In this paper, an optimization method is proposed to design a fractional order system with PID controller by taking Bode’s ideal transfer function as reference model. In the study, PID controllers are preferred because of their simplicity, reliability and robustness. PID controller parameters can be obtained by optimization method. In order to obtain the desired time response, it is sufficient to set two parameters in the Bode’s ideal transfer function. The Bode's ideal transfer function was considered as the reference model and compared with the generated model. The error in the output signal is minimized by the integral performance criteria, and the PID controller parameters are optimized. Integral performance criteria are frequently used in evaluating the performance of control systems. In Simulink model, Matsuda's 4th order integer approximation model is used for fractional order control system. Finally, the success of the optimization method is seen in the given numerical example.

Investigating the hydrodynamics of airlift pumps by wavelet packet transform and the recurrence plot

The hydrodynamics of airlift pumps at various gas superficial velocities were investigated using the wavelet packet transform and recurrence plot due to their capabilities for extracting the micro structures and determining the whole complexity in airlift pumps. For this purpose, the pressure fluctuations of an airlift pump were measured at various gas superficial velocities. The wavelet packet transform was first used to extract the integer approximation and detailed information from the pressure signals. Then, the energy distributions in each frequency band were calculated and the corresponding dynamical characteristics were obtained using a recurrence plot. It was found that the highest pumping efficiency under the slug-churn flow condition is caused by the highest mean pressure. In addition, the energy of the slug-churn flow regime has a dramatic increase in the frequency band of 31.25–62.5 Hz, due to the burst of big Taylor bubbles. In the low-frequency band (0–31.25 Hz), the pressure shows a chaotic behavior due to the turbulence of liquid flow. In the frequency band of 31.25–62.5 Hz, the pressure executes a transition from chaotic fluctuation in slug flow to periodic fluctuation in slug-churn due to the linear motion of small bubbles.

Integer Matrix Approximation and Data Mining

Integer datasets frequently appear in many applications in science and engineering. To analyze these datasets, we consider an integer matrix approximation technique that can preserve the original dataset characteristics. Because integers are discrete in nature, to the best of our knowledge, no previously proposed technique developed for real numbers can be successfully applied. In this study, we first conduct a thorough review of current algorithms that can solve integer least squares problems, and then we develop an alternative least square method based on an integer least squares estimation to obtain the integer approximation of the integer matrices. We discuss numerical applications for the approximation of randomly generated integer matrices as well as studies of association rule mining, cluster analysis, and pattern extraction. Our computed results suggest that our proposed method can calculate a more accurate solution for discrete datasets than other existing methods.