scholarly journals Rational Approximation for Solving an Implicitly Given Colebrook Flow Friction Equation

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 26 ◽  
Author(s):  
Pavel Praks ◽  
Dejan Brkić
Keyword(s):  
Rational Approximation ◽  
Numerical Experiments ◽  
The Novel ◽  
Computationally Efficient ◽  
Flow Friction ◽  
Quasi Monte Carlo ◽  
Transcendental Functions ◽  
Processor Unit ◽  
Computer Processor ◽  
Rational Expressions

The empirical logarithmic Colebrook equation for hydraulic resistance in pipes implicitly considers the unknown flow friction factor. Its explicit approximations, used to avoid iterative computations, should be accurate but also computationally efficient. We present a rational approximate procedure that completely avoids the use of transcendental functions, such as logarithm or non-integer power, which require execution of the additional number of floating-point operations in computer processor units. Instead of these, we use only rational expressions that are executed directly in the processor unit. The rational approximation was found using a combination of a Padé approximant and artificial intelligence (symbolic regression). Numerical experiments in Matlab using 2 million quasi-Monte Carlo samples indicate that the relative error of this new rational approximation does not exceed 0.866%. Moreover, these numerical experiments show that the novel rational approximation is approximately two times faster than the exact solution given by the Wright omega function.

scholarly journals Colebrook’s Flow Friction Explicit Approximations Based on Fixed-Point Iterative Cycles and Symbolic Regression

Computation ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 48 ◽  
Author(s):  
Dejan Brkić ◽  
Pavel Praks
Keyword(s):  
Fixed Point ◽  
Relative Error ◽  
Symbolic Regression ◽  
Logarithmic Function ◽  
Flow Friction ◽  
First Order ◽  
Quasi Monte Carlo ◽  
Initial Stage ◽  
Starting Point ◽  
Transcendental Functions

The logarithmic Colebrook flow friction equation is implicitly given in respect to an unknown flow friction factor. Traditionally, an explicit approximation of the Colebrook equation requires evaluation of computationally demanding transcendental functions, such as logarithmic, exponential, non-integer power, Lambert W and Wright Ω functions. Conversely, we herein present several computationally cheap explicit approximations of the Colebrook equation that require only one logarithmic function in the initial stage, whilst for the remaining iterations the cheap Padé approximant of the first order is used instead. Moreover, symbolic regression was used for the development of a novel starting point, which significantly reduces the error of internal iterations compared with the fixed value staring point. Despite the starting point using a simple rational function, it reduces the relative error of the approximation with one internal cycle from 1.81% to 0.156% (i.e., by a factor of 11.6), whereas the relative error of the approximation with two internal cycles is reduced from 0.317% to 0.0259% (i.e., by a factor of 12.24). This error analysis uses a sample with 2 million quasi-Monte Carlo points and the Sobol sequence.

scholarly journals Review of New Flow Friction Equations: Constructing Colebrook’s Explicit Correlations Accurately

2020 ◽  
Author(s):  
Pavel Praks ◽  
Dejan Brkić
Keyword(s):  
Monte Carlo ◽  
Series Expansion ◽  
Relative Error ◽  
Asymptotic Series ◽  
Computationally Efficient ◽  
Arithmetic Operations ◽  
Flow Friction ◽  
Quasi Monte Carlo ◽  
Computationally Expensive ◽  
Maximal Relative Error

Using only a limited number of computationally expensive functions, we show a way how to construct accurate and computationally efficient approximations of the Colebrook equation for flow friction. The presented approximations are based on the asymptotic series expansion of the Wright ω-function and symbolic regression. The results are verified with 8 million of Quasi-Monte Carlo points covering the domain of interest for engineers. In comparison with the built-in “wrightOmega” feature of Matlab R2016a, the herein introduced related approximations of the Wright ω-function significantly accelerate the computation. With only two logarithms and several basic arithmetic operations used, the presented approximations are not only computationally efficient but also extremely accurate. The maximal relative error of the most promising approximation which is given in the form suitable for engineers’ use is limited to 0.0012%, while for a little bit more complex variant is limited to 0.000024%.

scholarly journals Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function: Reply to Discussion

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 410 ◽  
Author(s):  
Dejan Brkić ◽  
Pavel Praks
Keyword(s):  
Monte Carlo ◽  
Original Equation ◽  
Computationally Efficient ◽  
Flow Friction ◽  
Quasi Monte Carlo

This reply gives two corrections of typographical errors in respect to the commented article, and then provides few comments in respect to the discussion and one improved version of the approximation of the Colebrook equation for flow friction, based on the Wright ω-function. Finally, this reply gives an exact explicit version of the Colebrook equation expressed through the Wright ω-function, which does not introduce any additional errors in respect to the original equation. All mentioned approximations are computationally efficient and also very accurate. Results are verified using more than 2 million of Quasi Monte-Carlo samples.

QMC integration errors and quasi-asymptotics

2020 ◽  
Vol 26 (3) ◽  
pp. 171-176
Author(s):  
Ilya M. Sobol ◽  
Boris V. Shukhman
Keyword(s):  
Monte Carlo ◽  
Error Estimate ◽  
Numerical Experiments ◽  
Probable Error ◽  
Quasi Monte Carlo ◽  
Integration Error ◽  
Asymptotic Error ◽  
Dimensional Cube ◽  
Integration Errors ◽  
Mc Method

AbstractA crude Monte Carlo (MC) method allows to calculate integrals over a d-dimensional cube. As the number N of integration nodes becomes large, the rate of probable error of the MC method decreases as {O(1/\sqrt{N})}. The use of quasi-random points instead of random points in the MC algorithm converts it to the quasi-Monte Carlo (QMC) method. The asymptotic error estimate of QMC integration of d-dimensional functions contains a multiplier {1/N}. However, the multiplier {(\ln N)^{d}} is also a part of the error estimate, which makes it virtually useless. We have proved that, in the general case, the QMC error estimate is not limited to the factor {1/N}. However, our numerical experiments show that using quasi-random points of Sobol sequences with {N=2^{m}} with natural m makes the integration error approximately proportional to {1/N}. In our numerical experiments, {d\leq 15}, and we used {N\leq 2^{40}} points generated by the SOBOLSEQ16384 code published in 2011. In this code, {d\leq 2^{14}} and {N\leq 2^{63}}.

scholarly journals Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function: Reply to the Discussion by Majid Niazkar

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 796
Author(s):  
Pavel Praks ◽  
Dejan Brkić
Keyword(s):  
Monte Carlo ◽  
Friction Factor ◽  
Large Scale ◽  
Optimization Procedure ◽  
Flow Friction ◽  
Quasi Monte Carlo

In this reply, we present updated approximations to the Colebrook equation for flow friction. The equations are equally computational simple, but with increased accuracy thanks to the optimization procedure, which was proposed by the discusser, Dr. Majid Niazkar. Our large-scale quasi-Monte Carlo verifications confirm that the here presented novel optimized numerical parameters further significantly increase accuracy of the estimated flow friction factor.

scholarly journals Prediction of Oilfield-Increased Production Using Adaptive Neurofuzzy Inference System with Smoothing Treatment

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Lin Chen ◽  
Zhibin Liu ◽  
Nannan Ma ◽  
Yi Wang
Keyword(s):  
Initial Data ◽  
Numerical Experiments ◽  
The Novel ◽  
Inference System ◽  
Feasible Method ◽  
Prediction Result ◽  
Well Stimulation ◽  
Adaptive Neurofuzzy Inference System ◽  
Oilfield Development ◽  
Better Than

A novel modified adaptive neurofuzzy inference system with smoothing treatment (MANFIS) is proposed. The MANFIS model considered the smoothing treatment of initial data basing on the adaptive neurofuzzy inference system, and we used it to predict oilfield-increased production under the well stimulation. Numerical experiments show the prediction result of the novel considering smoothing treatment is better than that without smoothing treatment. This study provides a novel and feasible method for prediction of oilfield-increased production under well stimulation, and it can be helpful in the further study of oilfield development measure planning.

scholarly journals Performance Evaluation of Hexagonal Array Adaptive Beamforming Algorithm Based on Inverse QR-RLS for Active Sonar Application

2009 ◽  
Vol 3 (3) ◽  
pp. 343-362 ◽  
Author(s):  
Q. Wang ◽  
X. Pan
Keyword(s):  
Spatial Filter ◽  
Adaptive Beamforming ◽  
Recursive Least Squares ◽  
Limited Area ◽  
Computationally Efficient ◽  
Rls Algorithm ◽  
Active Sonar ◽  
Adaptive Beamformer ◽  
Computation Efficiency ◽  
Computer Processor

In shallow water areas, to enhance underwater targets detection performance improve computation efficiency of active sonar, a computationally efficient adaptive beamformer (spatial filter) based on inverse QR (IQR) and recursive least-squares (RLS) is developed under fast Fourier transform framework, for standard hexagonal receiving array implementation. The IQR-RLS algorithm has good numerical stability and can be mapped onto coordinate rotation digital computer processor-based systolic arrays, which is suitable for real time applications. Using the proposed scheme to construct beamformer, which reduces computational complexity significantly and offers better converge rate than conventional adaptive beamformer. The simulation and lake test results demonstrates the algorithm improves interference (reverberation) suppression ability. It improves SNR about 2dB of still bottom target detection in reverberation limited area.

scholarly journals solutions of the CW equation for flow friction

2017 ◽  
Author(s):  
Dejan Brkić
Keyword(s):  
Fluid Flow ◽  
Friction Factor ◽  
Explicit Form ◽  
Iterative Procedure ◽  
Lambert W Function ◽  
Explicit Formulas ◽  
Flow Friction ◽  
Transcendental Functions ◽  
Flow Through

The empirical Colebrook–White (CW) equation belongs to the group of transcendental functions. The CW function is used for the determination of hydraulic resistances associated with fluid flow through pipes, flow of rivers, etc. Since the CW equation is implicit in fluid flow friction factor, it has to be approximately solved using iterative procedure or using some of the approximate explicit formulas developed by many authors. Alternate mathematical equivalents to the original expression of the CW equation, but now in the explicit form developed using the Lambert W-function, are shown (with related solutions). The W-function is also transcendental, but it is used more general compared with the CW function. Hence, the solution to the W-function developed by mathematicians can be used effectively for the CW function which is of interest only for hydraulics.

An accelerated majorization-minimization algorithm with convergence guarantee for non-Lipschitz wavelet synthesis model *

2021 ◽  
Vol 38 (1) ◽  
pp. 015001
Author(s):  
Yanan Zhao ◽  
Chunlin Wu ◽  
Qiaoli Dong ◽  
Yufei Zhao
Keyword(s):  
Minimization Problem ◽  
Stationary Point ◽  
Numerical Experiments ◽  
Complete Convergence ◽  
Convergence Result ◽  
Image Process ◽  
Minimization Algorithm ◽  
Computationally Efficient ◽  
Special Case ◽  
Reconstruction Model

Abstract We consider a wavelet based image reconstruction model with the ℓ p (0 < p < 1) quasi-norm regularization, which is a non-convex and non-Lipschitz minimization problem. For solving this model, Figueiredo et al (2007 IEEE Trans. Image Process. 16 2980–2991) utilized the classical majorization-minimization framework and proposed the so-called Isoft algorithm. This algorithm is computationally efficient, but whether it converges or not has not been concluded yet. In this paper, we propose a new algorithm to accelerate the Isoft algorithm, which is based on Nesterov’s extrapolation technique. Furthermore, a complete convergence analysis for the new algorithm is established. We prove that the whole sequence generated by this algorithm converges to a stationary point of the objective function. This convergence result contains the convergence of Isoft algorithm as a special case. Numerical experiments demonstrate good performance of our new algorithm.

Message Embedded Chaotic Masking Synchronization Scheme Based on the Generalized Lorenz System and Its Security Analysis

2016 ◽  
Vol 26 (08) ◽  
pp. 1650140 ◽  
Author(s):  
Sergej Čelikovský ◽  
Volodymyr Lynnyk
Keyword(s):  
Numerical Experiments ◽  
Chaotic Dynamics ◽  
Lorenz System ◽  
Bounded Function ◽  
Security Analysis ◽  
Encryption Scheme ◽  
The Novel ◽  
Generalized Lorenz System ◽  
Synchronization Signal ◽  
Synchronization Scheme

This paper focuses on the design of the novel chaotic masking scheme via message embedded synchronization. A general class of the systems allowing the message embedded synchronization is presented here, moreover, it is shown that the generalized Lorenz system belongs to this class. Furthermore, the secure encryption scheme based on the message embedded synchronization is proposed. This scheme injects the embedded message into the dynamics of the transmitter as well, ensuring thereby synchronization with theoretically zero synchronization error. To ensure the security, the embedded message is a sum of the message and arbitrary bounded function of the internal transmitter states that is independent of the scalar synchronization signal. The hexadecimal alphabet will be used to form a ciphertext making chaotic dynamics of the transmitter even more complicated in comparison with the transmitter influenced just by the binary step-like function. All mentioned results and their security are tested and demonstrated by numerical experiments.

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