Fast multiscale regularization methods for high-order numerical differentiation

2015 ◽  
Vol 36 (3) ◽  
pp. 1432-1451 ◽  
Author(s):  
Bin Wu ◽  
Qinghui Zhang
Keyword(s):  
Numerical Differentiation ◽  
High Order ◽  
Regularization Methods

scholarly journals On accurate high-order numerical derivatives computations for quantum chemistry purposes

2018 ◽  
Keyword(s):  
Quantum Chemistry ◽  
Finite Field ◽  
Numerical Differentiation ◽  
Field Method ◽  
High Order ◽  
Parameter Choice ◽  
Derivatives Of ◽  
Finite Field Method ◽  
Account Electron

Various molecular parameters in quantum chemistry could be computed as derivatives of energy over different arguments. Unfortunately, it is quite complicated to obtain analytical expression for characteristics that are of interest in the framework of methods that account electron correlation. Especially it relates to the coupled cluster (CC) theory. In such cases, numerical differentiation comes to rescue. This approach, like any other numerical method has empirical parameters and restrictions that require investigation. Current work is called to clarify the details of Finite-Field method usage for high-order derivatives calculation in CC approaches. General approach to the parameter choice and corresponding recommendations about numerical steadiness verification are proposed. As an example of Finite-Field approach implementation characterization of optical properties of fullerene passing process through the aperture of carbon nanotorus is given.

scholarly journals Inverse differential quadrature method: mathematical formulation and error analysis

2021 ◽  
Vol 477 (2248) ◽  
Author(s):  
Saheed O. Ojo ◽  
Luan C. Trinh ◽  
Hasan M. Khalid ◽  
Paul M. Weaver
Keyword(s):  
Differential Equations ◽  
Numerical Stability ◽  
Differential Quadrature Method ◽  
Mathematical Formulation ◽  
High Sensitivity ◽  
Numerical Differentiation ◽  
High Order ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Engineering Systems

Engineering systems are typically governed by systems of high-order differential equations which require efficient numerical methods to provide reliable solutions, subject to imposed constraints. The conventional approach by direct approximation of system variables can potentially incur considerable error due to high sensitivity of high-order numerical differentiation to noise, thus necessitating improved techniques which can better satisfy the requirements of numerical accuracy desirable in solution of high-order systems. To this end, a novel inverse differential quadrature method (iDQM) is proposed for approximation of engineering systems. A detailed formulation of iDQM based on integration and DQM inversion is developed separately for approximation of arbitrary low-order functions from higher derivatives. Error formulation is further developed to evaluate the performance of the proposed method, whereas the accuracy through convergence, robustness and numerical stability is presented through articulation of two unique concepts of the iDQM scheme, known as Mixed iDQM and Full iDQM. By benchmarking iDQM solutions of high-order differential equations of linear and nonlinear systems drawn from heat transfer and mechanics problems against exact and DQM solutions, it is demonstrated that iDQM approximation is robust to furnish accurate solutions without losing computational efficiency, and offer superior numerical stability over DQM solutions.

Linear Regression to Minimize the Total Error of the Numerical Differentiation

2017 ◽  
Vol 7 (4) ◽  
pp. 810-826
Author(s):  
Jengnan Tzeng
Keyword(s):  
Truncation Error ◽  
Total Error ◽  
Numerical Differentiation ◽  
High Order ◽  
The Other ◽  
Unknown Coefficient ◽  
Rounding Error ◽  
Step Size ◽  
Rounding Errors ◽  
Numerical Derivative

AbstractIt is well known that numerical derivative contains two types of errors. One is truncation error and the other is rounding error. By evaluating variables with rounding error, together with step size and the unknown coefficient of the truncation error, the total error can be determined. We also know that the step size affects the truncation error very much, especially when the step size is large. On the other hand, rounding error will dominate numerical error when the step size is too small. Thus, to choose a suitable step size is an important task in computing the numerical differentiation. If we want to reach an accuracy result of the numerical difference, we had better estimate the best step size. We can use Taylor Expression to analyze the order of truncation error, which is usually expressed by the big O notation, that is, E(h) = Chk. Since the leading coefficient C contains the factor f(k)(ζ) for high order k and unknown ζ, the truncation error is often estimated by a roughly upper bound. If we try to estimate the high order difference f(k)(ζ), this term usually contains larger error. Hence, the uncertainty of ζ and the rounding errors hinder a possible accurate numerical derivative.We will introduce the statistical process into the traditional numerical difference. The new method estimates truncation error and rounding error at the same time for a given step size. When we estimate these two types of error successfully, we can reach much better modified results. We also propose a genetic approach to reach a confident numerical derivative.

scholarly journals Wavelets and high order numerical differentiation

2010 ◽  
Vol 34 (10) ◽  
pp. 3008-3021 ◽  
Author(s):  
Chu-Li Fu ◽  
Xiao-Li Feng ◽  
Zhi Qian
Keyword(s):  
Numerical Differentiation ◽  
High Order

A wavelet-Galerkin method for high order numerical differentiation

2010 ◽  
Vol 215 (10) ◽  
pp. 3702-3712 ◽  
Author(s):  
Fang-Fang Dou ◽  
Chu-Li Fu ◽  
Yun-Jie Ma
Keyword(s):  
Galerkin Method ◽  
Numerical Differentiation ◽  
High Order ◽  
Wavelet Galerkin Method

Numerical Modeling of Film Condensation in Horizontal Mini- and Macrocircular Tubes

2018 ◽  
Vol 140 (12) ◽  
Author(s):  
Jun-De Li
Keyword(s):  
Heat Transfer ◽  
Film Thickness ◽  
Numerical Differentiation ◽  
Vapor Condensation ◽  
High Order ◽  
Film Condensation ◽  
Thickness Variation ◽  
Condenser Tube ◽  
Transfer Coefficients ◽  
Heat Transfer Coefficients

A partial differential–integral equation has been derived to connect vapor condensation and the development of condensate film thickness in both the tangential and axial directions in a horizontal circular condenser tube. A high-order explicit numerical scheme is used to solve the strongly nonlinear equation. A simple strategy is applied to avoid possible large errors from high-order numerical differentiation when the condensate becomes stratified. A set of empirical friction factor and Nusselt number correlations covering both laminar and turbulent film condensation have been incorporated to realistically predict film thickness variation and concurrently allow for the predictions of local heat transfer coefficients. The predicted heat-transfer coefficients of film condensation for refrigerant R134a and water vapor in horizontal circular mini- and macrotubes, respectively, have been compared with the results from experiments and the results from the simulations of film condensation using computational fluid dynamics (CFD), and very good agreements have been found. Some of the predicted film condensations are well into the strong stratification regime, and the results show that, in general, the condensate is close to annular near the inlet of the condenser tube and becomes gradually stratified as the condensate travels further away from the inlet for all the simulated conditions. The results also show that the condensate in the minitubes becomes stratified much earlier than that in the macrotubes.

scholarly journals Heuristic regularization methods for numerical differentiation

2012 ◽  
Vol 63 (4) ◽  
pp. 816-826 ◽  
Author(s):  
Dinh Nho Hào ◽  
La Huu Chuong ◽  
D. Lesnic
Keyword(s):  
Numerical Differentiation ◽  
Regularization Methods
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