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Title

Description

Modelling

Total Cards

Subject

Science

Level

Undergraduate 3

Created

05/31/2011

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Term

Abstract Visualization Object (AVO)

Definition

either looks like a real object or is completely abstract

based on derived information such as time, temperature, color, etc.

Term

Data fitting

Is this a good fit?

Definition

Occam's Razor-use as few degrees of freedom as possible

R² has little meaning

Maximum likelihood, chi squared, and neural networks help answer this question

Term

Chi Squared

Definition

"What level of error is consistent with this linear fit model? How well is the data fit?"

Chisquared converges to a normal distribution

X~1 good fit

X<1 overfit

X>1 underfit

Term

Max Likelihood

Definition

Used in hypothesis testing. Parameter estimation method that can limit model complexity

pick a model function [f=λexp(-λt)]

sum product of function
L(t1,t2...|λ)=Πf(ti)
L=λⁿexp[-λΣt_i]

3. Maximize
dL/dλ>>>λ=n/Σ_iⁿ(t_i)

Term

Overfitting

Why is this not advantage

Definition

Perhaps has too many degrees of freedom to fit a very finite bit of data therefore does not allow for any predictive power

(though, this may have a very high value of R²)

Term

What methods can be used to fit non linear data

Definition

deconvolution of gaussian fits to have a series of parabolic fits that sum together to show the overall image of the system

Term

Monte Carlo (MC) Methods

Examples

Definition

calculate the expected value of some mechanical quantity (thermodynamic quantity)

Buffon's Needle

Metropolis Algorithm

Microstructural evolution

Ising Model

Term

Metropolis Algorithm

Definition

State 1: p1, T1

State 2: p2, T2

There is a probability P* of going from state 1 to 2 (this is an ergodic perturbation)

if energy will decrease, P*=1

if energy will increase, P*=exp[-β(E₂-E₁)]

degrees of freedom must be considered (ie polymers can twist)

Term

Ising/Microstructural evolution

Definition

H=-εΣsisj-BΣsi

Start with perfectly ordered or perfectly random, pick a spin, flip it, recalcuate energy

exactly solvable and perfectly ergodic

Term

Molecular Dynamics

Types

Definition

Future state is determined by present state using force field

RDF

Fast Ion Conductors

Term

Principles of MD

Definition

F=f(dV,dx) gradient of velocities

F=d/dt(mv)=md²R/dt²

Verlet method: a way of advancing system by Σt

MD at constant T: Velocity scaling or Berendsen

VACF: decay of velocities based on collisions

Compute equilibrium or transport properties

Term

Velocity Scaling

Definition

An MD method of making the system at constant temperature

<KE>=1/2Σmivi2=3/2NkT

Solve for T

ΔT=T(λv)-T(v)

Term

Berendsen Method

Definition

dT/dt=1/τ(Tbath-Ti)

Integrate then plug into velocity scaling method to have a scale with only fixed variables

Term

Fast Ion Conductors

Definition

Model structure

diffusion model consider jumps of Q, qDE, d/2 etc. (different probability of jumps in either direction...biased)

Potential energy function used

RDF used to find the probaility of another particle r away

Thermal expansion

MSD as a function of t can be found then diffusivity derived

Term

Velocity Autocorrelation Function (VACF)

Definition

measure of how correlations in the velocities of particles in the simulation decay as a function of time. Provides a means of investigating collision processes in molecular systems and diffusion

if molecules did not interact VACFv.t-flat

""have weak forces-exp decay to zero

"" strong forces-oscillate about zero/damp

Term

What should be considered when building a model

Definition

target precision

sensitivity to precision w/ added data

desired outputs

available inputs

dimensional analysis

ease of implementation

Term

Benefits of ANN

Definition

Functional relationship does not have to be chosen before analysis

Relationship does not have to be linear

Fitted function can vary across input space

Term

How is complexity altered in ANN

Definition

increase the number of hidden layers/units

Term

Training v. Test Set

Definition

training refers to the manipulation of weights and bias at random until there is a best fit to eh data obtained experimentally

A training set is created and compared to a test data set obtained by actually carrying out experiment

Term

Why is the metropolis algorithm not the best to use?

Definition

unphysical perterbations are valid in this model therefore it is not always representative of the true dynamics of the system

Term

Verlet method

Definition

a method of defining the time step of the system

new positions of particles are calculated using existing positions and velocities (Newton's laws)

it is ideal for the time step to be about the same as the time between collisions in a system

~10^-15s

Term

Markov Process

Definition

present and future states are independent. These are not varying with time, they were snapshots at different periods

Markov processes are ergodic and satisfy the principle of detailed balance

The ratio of probabilities (going from state 1 to state 2 or vis versa) needs to be Boltzmann-like (aka there is time reversal-detailed balance principle)

Term

MD v. MC

Definition

(MD) Successive states are directly connected in time

MD has a kinetic energy contribution to total energy while MC total energy is determined solely by potential energy

MD samples from states at constant energy while MC samples from states at constant temperature

Term

Disadvantages of MD at Constant temperature

Definition

gives rise to fictious forces, larger simulations are required to generate accurate averages, artificially prolong temperature differences among different components of system

Term

Radial distribution function (RDF)

Definition

the probability density of finding a particle_i at some distance from another particle_j

Related to Velocity autocorrelation function and mean square displacement

G=Σδ(r-ri)

Term

Continuous v. Impulsive MD

Definition

Continuous: Derive Newton's laws, advance system by some time step (use Verlet method), recalculate

Adv: high precision, longer time step can be used, low drift

Disadv: v(t) not explicitly calculated, non self-starting, must assume a value for r(t-Dt) at first duration, not fast

Impulsive: particles interact by discontinuous potentials ie ball bearing/hard core spheres

Term

Integrator

Definition

a finite time step acquired mathematically

Term

Equilibration

Definition

the repeated application of Markov processes. Speed of equilibration can be increased by making moves in clusters

Term

Application of ANN

Definition

crack growth is based on many parameters

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