What are some sources of uncertainty in design? (4)
Structural/material properties
Geometric properties and dimensions
Load conditions, operating conditions
Human factors
How do we deal with uncertainty in design?
Treat uncertain parameters as random variables with a probability distribution
What are Monte Carlo Methods?
- Monte Carlo (MC) methods: computational approach to simulate/analyse highly complex systems, using approximate solutions
- MC methods are based on repeated random sampling
- By using inferential statistics, we can estimate the value of an unknown quantity
▪ There is no unbiased estimator that could potentially converge faster than MC!
- used for problems where finding an analytical solution would be impossible
- allows high flexibility
- With MC, we get rough results fairly quickly, but it takes very long to get accurate results
What is the basic process for MC?
- Use stochastic methods to generate a (large) random sample from a population
- Use randomly generated numbers as input to the considered system to generate a range of possible outcomes for an uncertain event
- Compute the likelihood of a particular outcome occurring
What are the main components of MC? (6)
- Probability distribution function
- Random number generator
- Sampling rule
- Scoring
- Error estimation
- Variance reduction techniques
What are some important sampling methods?
- Inverse Transform
- Rejection
- Importance
What is a Markov chain?
A joint probability over a sequence of random variables is a Markov chain.
What is Markov chain monte Carlo?
- Combines Markov chain sampling with MC to approximate complex probability distributions.
Exploration by random walk through a Markov chain, whose stationary distribution equals the target distribution
Sample generation & computation through MC
What are two important MCMC methods?
Metropolis-Hastings
Gibbs sampling
What is the Monte Carlo Estimator formula?
\hat{\theta} = 1/N \sum_{i=1}^{\infty} f(x_i)
What are the steps for utilizing the MC Estimator?
- Specify probability domain
- Generate random samples from probability distribution p(x)
- Compute system response for each data point x_i
- Obtain and analyze results using MC estimator
What are three general uses for MC and one specific for engineering?
General:
- Numerical Integration
- Numerical Simulation
- Optimization
Engineering:
- Probabilistic Design
How can MC be useful in optimization?
MC methods tend to not get stuck at local minima by allowing random exits from a local minimum, thus allowing the solver to search for a different (potentially better) minimum.
Is MC biased or unbiased?
Unbiased
What are some properties of MC error?
- O(N^{-1/2})
- Reduces more slowly for increasing N
- Independent of dimension
What are the three kinds of randomness?
- True randomness
- Pseudo-randomness
- Pseudo random sequences
What are some properties of Random number generators?
- Pseudo random
- Deterministic
What is MC inverse transform sampling?
Concept: apply random number generator on a uniform (or other very simple) distribution, then transform to the distribution we need
What is MC rejection sampling?
Idea: find a probability distribution that is close to the original distribution but easier to sample from
What is MC importance sampling?
Method to approximate expectations for a complex distribution p using a different “proposal distribution” q
What are the 5 Markov chain states?
- Absorbing
- Recurrent and transient
- Ergodic
- Periodic
- Aperiodic
When is a Markov Chain Irreducable?
When there is a path from any given chain to another chain.
When is a Markov Chain considered stationary?
When the transition properties do not change over time
What is a recurrent state in a Markov chain.
A state that if reached will certainly be revisited
