md2011-11-notes

md2011-11-notes - Molecular Dynamics simulations Lecture...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Molecular Dynamics simulations Lecture 11: Energy Minimization Dr. Olli Pakarinen University of Helsinki Fall 2011 Original lecture notes by Dr. Jani Kotakoski, 2010 Optimization Problem I Optimization problems are in the heart of many physical applications I In short, optimization is about finding the set of parameters x , y , . . . for a given objective function f ( x , y , . . . ) which will yield either the minimum or maximum value I The extreme values searched for can be either local or global I In one-dimensional case: local minimum global minimum I The brute force method would be to map the complete configuration space (all possible parameters) and thereby find all possible values of the function I However, this is obviously not a very effective approach to a multi-dimensional problem I The computational approach is, as so often, dictated by the computational cost I This leads to a simple rule: evaluate f ( x , y , . . . ) as few times as possible I In general, it is a very difficult problem to effectively find the global extreme value I There are two traditional ways to do this: (1) Start minimization from several initial conditions and find the local extreme (2) Perturb a local extreme and optimize the function to see if you are returned to the initial extreme I Depending on the exact problem, some constraints may apply I In the context of this course, we are mostly interested in energy minimization (for other methods, see f.ex. [http://www.nrbook.com/c/] ) I This is important in two different regards: I Local energy minimization for structure relaxation, e.g., after a defect is created I Global energy minimization for finding minimum energy structures under certain constraints ( P , T ) Energy Minimization I The first of these is obviously a local minimization problem (without changing the lattice around the defect) whereas the second one is about finding the global minimum (under constraints) I Correspondingly, different methods are often used for these different problems I Local energy minimization is carried out using traditional optimization techniques I Global optimization, on the other hand, has exploded during the recent years with the introduction of evolutionary structure optimization algorithms (allowed by fast enough computers) I In any case, energy minimization is a surprisingly complicated task to solve efficiently I The set of parameters is the degrees of freedom of the system [i.e., three spatial coordinates for each of the particles x = ( x 1 , x , x 1 , y , x 1 , z , x 2 , x , x 2 , y , x 2 , z , . . . ) ] I (Energy minimization is a T = K problem) I Obviously, the quantity to minimize is the potential energy of the system U ( x ) I One limiting factor for the energy minimization techniques is the memory requirement; for a very large set of atoms, e.g., O ( N 2 ) methods can be impossible ( N is the number of parameters) Examples of Methods Used in Atomistic Energy Minimization (1) Monte Carlo simulation This can be carried out by setting the temperature...
View Full Document

This note was uploaded on 02/14/2012 for the course CSE 6590 taught by Professor Kotakoski during the Winter '12 term at York University.

Page1 / 16

md2011-11-notes - Molecular Dynamics simulations Lecture...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online