Probabilistic constraints

[mohdibntarek]

Handling probabilistic constraints can be done in a few ways, e.g.:

1. Quadrature integration or expectation-based methods
2. Reliability analysis based method, e.g. first order reliability method and second order reliability method
3. Sampling based method

[mohdibntarek]

Some notes from a meeting with @LucasMSpereira.

Reliability analysis
Reliability-based design optimisation RBDO - sizing optimisation
Reliability-based topology optimisation RBTO

First order reliability method (FORM)
	RIA
	PMA
Second order reliability method (SORM)
	RIA
	PMA

x p

f(x; p)

f(x; p) ~= f(x; p = p0) + df(x; p = p0)/dp * (p - p0)

E_p(f(x; p)) = E_p(f(x; p = p0)) + E_p(df(x; p = p0)/dp * (p - p0))
		     = E_p(f(x; p = p0)) + E_p(df(x; p = p0)/dp * p) - E_p(df(x; p = p0)/dp * p0)
		     = f(x; p = p0) + df(x; p = p0)/dp * mu_p - df(x; p = p0)/dp * p0

mu(x) = f(x; p = p0(x)) + df(x; p = p0(x))/dp * mu_p - df(x; p = p0(x))/dp * p0
cov(x) = (df(x; p = p0)/dp) * cov_p * (df(x; p = p0)/dp)'
MvNormal(mu(x), cov(x))

P(f(x; p) <= 1.0) > 0.95

Reliability index approach (RIA)

p0(x) = argmin_p (p - mu_p)' * cov_p^-1 * (p - mu_p)
        s.t. f(x; p) = 1.0

Performance measure approach (PMA)

p0(x) = argmin_p f(x; p)
		(p - mu_p)' * cov_p^-1 * (p - mu_p) = alpha # target reliability index


p ~ MvNormal(mu_p, cov_p)
f(x; p) ~ MvNormal(mu(x), cov(x))
  
rf = RandomFunction(f, MvNormal(zeros(3), I(3)), method = FORM(RIA(g)))
rf = RandomFunction(f, MvNormal(zeros(3), I(3)), method = FORM(PMA(alpha)))
rf = RandomFunction(f, MvNormal(zeros(3), I(3)), method = ...)


rf(x) -> Normal(mu(x), cov(x))

function obj(x)
	dist = rf(x)
	mean(dist) + 2 * std(dist)
end