Sensitivity Problems

Types of Sensitivity Problems in Fortuna.jl

Fortuna.jl districts two types of sensitivity problems:

Defining and Solving Sensitivity Problems of Type I

In general, 4 main "items" are always needed to fully define a sensitivity problem of type I and successfully solve it to find the associated sensitivity vectors of probability of failure $\vec{\nabla}_{\vec{\Theta}_{g}} P_{f}$ and reliability index $\vec{\nabla}_{\vec{\Theta}_{g}} \beta$ w.r.t. to the parameters of the limit state function $\vec{\Theta}_{g}$:

Fortuna.jl package uses these 4 "items" to fully define sensitivity problems of type I using SensitivityProblem() type as shown in the example below.

# Define the random vector:
M_1 = randomvariable("Normal",  "M", [250,   250   * 0.3])
M_2 = randomvariable("Normal",  "M", [125,   125   * 0.3])
P   = randomvariable("Gumbel",  "M", [2500,  2500  * 0.2])
Y   = randomvariable("Weibull", "M", [40000, 40000 * 0.1])
X   = [M_1, M_2, P, Y]

# Define the correlation matrix:
ρ_X = [
    1.0 0.5 0.3 0.0
    0.5 1.0 0.3 0.0
    0.3 0.3 1.0 0.0
    0.0 0.0 0.0 1.0]

# Define the limit state function:
g(x::Vector, θ::Vector) = 1 - x[1] / (θ[1] * x[4]) - x[2] / (θ[2] * x[4]) - (x[3] / (θ[3] * x[4])) ^ 2

# Define parameters of the limit state function:
s_1 = 0.030
s_2 = 0.015
a   = 0.190
Θ   = [s_1, s_2, a]

# Define a sensitivity problem:
problem = SensitivityProblemTypeI(X, ρ_X, g, Θ)

After defining a sensitivity problem of type I, Fortuna.jl allows to easily perform sensitivity analysis using a single solve() function as shown in the example below.

# Perform the sensitivity analysis:
solution = solve(problem)
println("∇β   = $(solution.∇β)")
println("∇PoF = $(solution.∇PoF)")

∇β   = [36.772639417904365, 73.54527883580873, 9.257376773457299]
∇PoF = [-0.7013033431414412, -1.4026066862828823, -0.17655053819129737]

Defining and Solving Sensitivity Problems of Type II

Similar to sensitivity problem of type I, 4 main "items" are needed to fully define a sensitivity problem of type II and successfully solve it to find the associated sensitivity vectors of probability of failure $\vec{\nabla}_{\vec{\Theta}_{f}} P_{f}$ and reliability index $\vec{\nabla}_{\vec{\Theta}_{f}} \beta$ w.r.t. to the parameters and/or moments of the random vector $\vec{\Theta}_{f}$:

Fortuna.jl package uses these 4 "items" to fully define sensitivity problems of type I using SensitivityProblem() type as shown in the example below.

# Define the random vector as a function of its parameters and moments:
function X(Θ::Vector)
    M_1 = randomvariable("Normal",  "M", [Θ[1], Θ[2]])
    M_2 = randomvariable("Normal",  "M", [Θ[3], Θ[4]])
    P   = randomvariable("Gumbel",  "M", [Θ[5], Θ[6]])
    Y   = randomvariable("Weibull", "M", [Θ[7], Θ[8]])

    return [M_1, M_2, P, Y]
end

# Define the correlation matrix:
ρ_X = [
    1.0 0.5 0.3 0.0
    0.5 1.0 0.3 0.0
    0.3 0.3 1.0 0.0
    0.0 0.0 0.0 1.0]

# Define the parameters and moments of the random vector:
Θ = [
      250,   250 * 0.30,
      125,   125 * 0.30,
     2500,  2500 * 0.20,
    40000, 40000 * 0.10]

# Define the limit state function:
a   = 0.190
s_1 = 0.030
s_2 = 0.015
g(x::Vector) = 1 - x[1] / (s_1 * x[4]) - x[2] / (s_2 * x[4]) - (x[3] / (a * x[4])) ^ 2

# Define a sensitivity problem:
problem  = SensitivityProblemTypeII(X, ρ_X, g, Θ)

Similar to sensitivity problems of type I, sensitivity problems of type II are solved using the same solve() function as shown in the example below.

# Perform the sensitivity analysis:
solution = solve(problem)
println("∇β   = $(solution.∇β)")
println("∇PoF = $(solution.∇PoF)")

∇β   = [-0.003237735324247002, -0.0039166045667242445, -0.006475470671641298, -0.007833209139706684, -0.000545833565960213, -0.0007886353761454874, 0.00012366075320732482, -0.0002452926491760409]
∇PoF = [6.174793659211753e-5, 7.469487966831622e-5, 0.00012349587362568486, 0.00014938975945598458, 1.0409775057384956e-5, 1.5040329836676089e-5, -2.358375711194132e-6, 4.678058405332213e-6]

API

solve(Problem::SensitivityProblemTypeI; backend = AutoForwardDiff())

solve(Problem::SensitivityProblemTypeII; backend = AutoForwardDiff())

SensitivityProblemTypeI <: AbstractReliabilityProblem

SensitivityProblemTypeII <: AbstractReliabilityProblem

SensitivityProblemCache