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TAS > Chapter 11 > Flashcards

Chapter 11 Flashcards

1
Q

What are some sources of uncertainty in design? (4)

A

 Structural/material properties

 Geometric properties and dimensions

 Load conditions, operating conditions

 Human factors

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2
Q

How do we deal with uncertainty in design?

A

 Treat uncertain parameters as random variables with a probability distribution

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3
Q

What are Monte Carlo Methods?

A
  • 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
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4
Q

What is the basic process for MC?

A
  1. Use stochastic methods to generate a (large) random sample from a population
  2. Use randomly generated numbers as input to the considered system to generate a range of possible outcomes for an uncertain event
  3. Compute the likelihood of a particular outcome occurring
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5
Q

What are the main components of MC? (6)

A
  • Probability distribution function
  • Random number generator
  • Sampling rule
  • Scoring
  • Error estimation
  • Variance reduction techniques
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6
Q

What are some important sampling methods?

A
  • Inverse Transform
  • Rejection
  • Importance
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7
Q

What is a Markov chain?

A

A joint probability over a sequence of random variables is a Markov chain.

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8
Q

What is Markov chain monte Carlo?

A
  • 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

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9
Q

What are two important MCMC methods?

A

 Metropolis-Hastings
 Gibbs sampling

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10
Q

What is the Monte Carlo Estimator formula?

A

\hat{\theta} = 1/N \sum_{i=1}^{\infty} f(x_i)

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11
Q

What are the steps for utilizing the MC Estimator?

A
  1. Specify probability domain
  2. Generate random samples from probability distribution p(x)
  3. Compute system response for each data point x_i
  4. Obtain and analyze results using MC estimator
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12
Q

What are three general uses for MC and one specific for engineering?

A

General:
- Numerical Integration
- Numerical Simulation
- Optimization

Engineering:
- Probabilistic Design

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13
Q

How can MC be useful in optimization?

A

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.

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14
Q

Is MC biased or unbiased?

A

Unbiased

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15
Q

What are some properties of MC error?

A
  • O(N^{-1/2})
  • Reduces more slowly for increasing N
  • Independent of dimension
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16
Q

What are the three kinds of randomness?

A
  • True randomness
  • Pseudo-randomness
  • Pseudo random sequences
17
Q

What are some properties of Random number generators?

A
  • Pseudo random
  • Deterministic
18
Q

What is MC inverse transform sampling?

A

Concept: apply random number generator on a uniform (or other very simple) distribution, then transform to the distribution we need

19
Q

What is MC rejection sampling?

A

Idea: find a probability distribution that is close to the original distribution but easier to sample from

20
Q

What is MC importance sampling?

A

Method to approximate expectations for a complex distribution p using a different “proposal distribution” q

21
Q

What are the 5 Markov chain states?

A
  1. Absorbing
  2. Recurrent and transient
  3. Ergodic
  4. Periodic
  5. Aperiodic
22
Q

When is a Markov Chain Irreducable?

A

When there is a path from any given chain to another chain.

23
Q

When is a Markov Chain considered stationary?

A

When the transition properties do not change over time

24
Q

What is a recurrent state in a Markov chain.

A

A state that if reached will certainly be revisited

25
Q
A

TAS (12 decks)

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