Strain Transformation and Rosette Gage Theory

Nick Whiteley

Description

AE3145 Strain Transformation and Rosette Gage Theory Page 1 Strain Transformation and Rosette Gage Theory It is often desired to measure the full state of strain on the surface of a part, that is to measure not only the two extensional strains, εx and εy, but also the shear strain, γxy, with respect to some given xy axis system. It should be clear from the previous discussion of the electrical resistance strain gage that a single gage is capable only of measuring the extensional strain in the direction that the gage is oriented.

• rectangular rosette
• ae3145 strain transformation
• circle diameter
• orientation with respect to the rosette gage
• cba bc accbba cba
• gage
• circle
• surface

Sujets

• Strain
• Circle
• Rosette
• Respect
• Diameter
• Surface

Informations

• Publié par: idwe0
• Nombre de lectures: 95
• Langue: English

Extrait

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Nick Whiteley

Lecture 7: The Metropolis-Hastings Algorithm Nick Whiteley

Algorithm

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7:

Lecture

Metropolis-Hastings

The

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X j ( t ) ∼ f X j | X − j ( ∙| X ( 1 t ) , . . . , X j ( t − ) 1 , X j ( t + − 1 1) , . . . , X p ( t − 1) )

Requires that it is possible / easy to sample from the full conditionals. Can yields a slowly mixing chain if (some of) the components of ( X 1 , . . . , X p ) are highly correlated.

Two drawbacks:

Samples ( X (0) , X (1) , . . . ) form a Markov chain (like the Gibbs sampler).

What we will see today: Metropolis-Hastings algorithm

Key idea: Generate a Markov chain by updating the component of ( X 1 , . . . , X p ) in turn by drawing from the full conditionals:

Key idea: Use rejection mechanism, with a “local proposal”: We let the newly proposed X depend on the previous state of the chain X ( t − 1) .

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Lecture 7: The Nick Whiteley

5.1

Algorithm

Metropolis-Hastings

Algorithm

5.3

&

5.4

RW

5.1 Algorithm 5.2 Convergence properties opolis & Choice of proposal

Metropolis & Choice of proposal

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The Metropolis-Hastings algorithm

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3. With probability α ( X | X ( t − 1) ) set X ( t ) = X , otherwise set X ( t ) = X ( t − 1) .

Algorithm 5.1: Metropolis-Hastings Starting with X (0) := ( X 1 (0) , . . . , X p (0) ) iterate for t = 1 , 2 , . . . 1. Draw X ∼ q ( ∙| X ( t − 1) ) . 2. Compute

1) | X ) α ( X | X ( t − 1) ) = min ( 1 , f ( f X ( ( t X − ) 1) ∙ ) q ∙ ( q X ( ( t X − | X ( t − 1) ) ) .

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Illustration

of

the

Metropolis-Hastings

Lecture 7: The Metropolis-Hastings Algorithm Nick Whiteley

method

5.1 Algorithm 5.2 Convergence properties 5.3 & 5.4 RW Metropolis & Choice of proposal

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The probability of remaining in state X ( t − 1) is

The probability of acceptance does not depend on the normalisation constant: If f ( x ) = C ∙ π ( x ) , then

α ( X | X ( t − 1) ) = min ( 1 , π ( π X ( ( t X − ) 1) ∙ ) q ∙ ( q X ( ( t X − | 1 X ) | ( t X − ) 1) ) )

Basic properties of the Metropolis-Hastings algorithm

The probability that a newly proposed value is accepted given X ( t − 1) = x ( t − 1) is a ( x ( t − 1) ) = Z α ( x | x ( t − 1) ) q ( x | x ( t − 1) ) d x .

P ( X ( t ) = X ( t − 1) | X ( t − 1) = x ( t − 1) ) = 1 − a ( x ( t − 1) ) .

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Lemma 5.1 The transition kernel of the Metropolis-Hastings algorithm is

K ( x ( t − 1) , x ( t ) ) = α ( x ( t ) | x ( t − 1) ) q ( x ( t ) | x ( t − 1) ) +(1 − a ( x ( t − 1) )) δ x ( t − 1) ( x ( t ) ) ,

where δ x ( t − 1) ( ∙ ) denotes Dirac-mass on { x ( t − 1) } .

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The Metropolis-Hastings Transition Kernel

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Lecture 7: The Nick Whiteley

5.2

Convergence

Metropolis-Hastings Algorithm

properties

5.3 &

5.4

RW

5.1 Algorithm 5.2 Convergence properties Metropolis & Choice of proposal

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Theoretical properties

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Thus f ( x ) is the invariant distribution of the Markov chain ( X (0) , X (1) , . . . ) . Furthermore the Markov chain is reversible.

K ( x ( t − 1) , x ( t ) ) f ( x ( t − 1) ) = K ( x ( t ) , x ( t − 1) ) f ( x ( t ) ) .

Proposition 5.1 The Metropolis-Hastings kernel satisfies the detailed balance condition

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and the proposal distribution q ( ∙| x ( t − 1) ) :

δ

X | X ( t − 1) = x ( t − 1) ∼ U [ x ( t − 1) − δ, x ( t − 1) + δ ]

δ

q ( ∙| x ( t − 1) )

f ( ∙ )

1 / (2 δ ) 1 / 2

f ( x ) = ( I [0 , 1] ( x ) + I [2 , 3] ( x )) / 2 .

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Example 5.1: Reducible Metropolis-Hastings

Consider the target distribution

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x ( t − 1) 1 2 3 Reducible if δ ≤ 1 : the chain stays either in [0 , 1] or [2 , 3] .

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Further theoretical properties

The Markov chain ( X (0) , X (1) , . . . ) is irreducible if q ( x | x ( t − 1) ) > 0 for all x , x ( t − 1) ∈ supp( f ) : every state can be reached in a single step. (less strict conditions can be obtained, see e.g.(see Roberts & Tweedie, 1996) The chain is aperiodic, if there is positive probability that the chain remains in the current state, i.e. P ( X ( t ) = X ( t − 1) ) > 0 ,

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