scholarly journals Geometry Optimization Speedup Through a Geodesic Approach to Internal Coordinates

2021 ◽  
Author(s):  
Eric Hermes ◽  
Khachik Sargsyan ◽  
Habib Najm ◽  
Judit Zádor
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Displacement Vector ◽  
Geometry Optimization ◽  
Molecular Geometry ◽  
Internal Coordinates ◽  
Derivatives Of

We present a new geodesic-based method for geometry optimization in a basis of redundant internal coordinates. Our method updates the molecular geometry by following the geodesic generated by a displacement vector on the internal coordinate manifold, which dramatically reduces the number of steps required to reach convergence. Our method can be implemented in any existing optimization code, requiring only implementation of derivatives of the Wilson B-matrix and the ability to solve an ordinary differential equation.

scholarly journals Geometry Optimization Speedup Through a Geodesic Approach to Internal Coordinate Optimization

2021 ◽  
Author(s):  
Eric Hermes ◽  
Khachik Sargsyan ◽  
Habib Najm ◽  
Judit Zádor
Keyword(s):  
Molecular Structure ◽  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Newton Method ◽  
Displacement Vector ◽  
Geometry Optimization ◽  
Internal Coordinates ◽  
Derivatives Of ◽  
Method Approach

We present a new geodesic-based method for geometry optimization in a basis of redundant internal coordinates.<br>This method realizes displacements along internal coordinates by following the geodesic generated by the displacement vector on the internal coordinate manifold.<br>Compared to the traditional Newton method approach to taking displacements in internal coordinates, this geodesic approach substantially reduces the number of steps required to reach convergence on a molecular structure minimization benchmark.<br>This new geodesic method can in principle be implemented in any existing optimization code, and only requires the implementation of derivatives of the Wilson B-matrix and the ability to solve a relatively inexpensive ordinary differential equation.

scholarly journals Geometry Optimization Speedup Through a Geodesic Approach to Internal Coordinate Optimization

2021 ◽  
Author(s):  
Eric Hermes ◽  
Khachik Sargsyan ◽  
Habib Najm ◽  
Judit Zádor
Keyword(s):  
Molecular Structure ◽  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Newton Method ◽  
Displacement Vector ◽  
Geometry Optimization ◽  
Internal Coordinates ◽  
Derivatives Of ◽  
Method Approach

We present a new geodesic-based method for geometry optimization in a basis of redundant internal coordinates.<br>This method realizes displacements along internal coordinates by following the geodesic generated by the displacement vector on the internal coordinate manifold.<br>Compared to the traditional Newton method approach to taking displacements in internal coordinates, this geodesic approach substantially reduces the number of steps required to reach convergence on a molecular structure minimization benchmark.<br>This new geodesic method can in principle be implemented in any existing optimization code, and only requires the implementation of derivatives of the Wilson B-matrix and the ability to solve a relatively inexpensive ordinary differential equation.

scholarly journals On the link between finite difference and derivative of polynomials

2017 ◽  
Author(s):  
Kolosov Petro
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Finite Difference ◽  
Differential Calculus ◽  
Divided Difference ◽  
Applied Mathematics ◽  
Numerical Differentiation ◽  
Discrete Analog ◽  
Derivatives Of

The main aim of this paper to establish the relations between forward, backward and central finite (divided) differences (that is discrete analog of the derivative) and partial &amp; ordinary high-order derivatives of the polynomials.MSC 2010: 46G05, 30G25, 39-XXarXiv:1608.00801Keywords: Finite difference, Derivative, Divided difference, Ordinary differential equation, Partial differential equation, Partial derivative, Differential calculus, Difference Equations, Numerical Differentiation, Finite difference coefficient, Polynomial, Power function, Monomial, Exponential function, Exponentiation, arXiv, Preprint, Calculus, Mathematics, Mathematical analysis, Numerical methods, Applied Mathematics

scholarly journals Bounded Solutions of the Second Order Differential Equation x ?+f(x) x ?+g(x)=u(t)

2015 ◽  
Vol 12 (4) ◽  
pp. 822-825
Author(s):  
Baghdad Science Journal
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Order Differential Equation ◽  
Second Order ◽  
Bounded Solutions ◽  
Order Ordinary Differential Equation ◽  
Second Order Differential Equation ◽  
Derivatives Of

In this paper we prove the boundedness of the solutions and their derivatives of the second order ordinary differential equation x ?+f(x) x ?+g(x)=u(t), under certain conditions on f,g and u. Our results are generalization of those given in [1].

Non-Equidistant Numerical Methods for Ordinary Differential Equation with Periodical Boundary Value

2011 ◽  
Vol 467-469 ◽  
pp. 383-388
Author(s):  
Xin Cai
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Numerical Methods ◽  
Boundary Value ◽  
Singular Component ◽  
Thin Region ◽  
Smooth Component ◽  
Error Estimations ◽  
Transition Points ◽  
Derivatives Of

Ordinary differential equation with periodical boundary value and small parameter multiplied in the highest derivative was considered. The solution of the problem has boundary layers, which is thin region in the neighborhood of the boundary of the domain. Firstly, the properties of boundary layer were discussed. The solution was decomposed into the smooth component and the singular component. The derivatives of the smooth component and the singular component were estimated. Secondly, mesh partition techniques were presented according to one transition point method and multi-transition points method. Thirdly numerical methods based on non-equidistant mesh partition were presented to solve the problem. Finally error estimations were given for both computational methods.

scholarly journals Solving Differential Equation by Modified Genetic Algorithms

2018 ◽  
Vol 26 (10) ◽  
pp. 233-241
Author(s):  
Eman Ali Hussain ◽  
Yahya Mourad Abdul – Abbass
Keyword(s):  
Differential Equation ◽  
Genetic Algorithm ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Mathematical Equation ◽  
Individual Solution ◽  
Result Show ◽  
String Representation ◽  
Derivatives Of ◽  
Physical Quantities

   Differential equation is a mathematical equation which contains the derivatives of a variable, such as the equation which represent physical quantities, In this paper  we introduced modified on the method which proposes a polynomial to solve the ordinary differential equation (ODEs) of second order and by using the evolutionary algorithm to find the coefficients of the propose a polynomial [1] . Our method propose a polynomial to solve the ordinary differential equations (ODEs) of nth  order and partial differential equations(PDEs) of order two  by using the Genetic algorithm to find the coefficients of the propose a polynomial ,since Evolution Strategies (ESs) use  a string representation of the solution to some problem and attempt to evolve a good solution through a series of fitness –based evolutionary steps .unlike (GA)  ,an ES will typically not use a population of solution but instead will make a sequence of mutations of an individual solution ,using fitness as a guide[2] . A numerical example with good result show the accuracy of our method compared with some existing methods .and the best error of method it’s not much larger than the error in best of the numerical method solutions.

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