LASS

Local Averaged Stratified Sampling

Valentini, F., Silva, O.M., Torii, A.J. et al. Local averaged stratified sampling method. J Braz. Soc. Mech. Sci. Eng. 44, 294 (2022). https://doi.org/10.1007/s40430-022-03589-6

```julia using LASS using Distributions, Statistics

Lets load QuadGK to evaluate some reference values

using QuadGK

Lets define a function of z. The argument must z

be a vector

f(z::Vector,x=1.0) = (z[1]-5x)^2 + z[1]x;

and assume that z is N(5,1.0)

pz = Normal(5.0,1.0);

Lets obtain some reference values.

The expected value E[f] is

pE(z)=f([z])*pdf(pz,z); E = quadgk(pE,-Inf,Inf)[1];

And the variance is

pV(z)=(f([z])-E)^2*pdf(pz,z); Var = quadgk(pV,-Inf,Inf)[1];

Now lets use the Local Averaged Stratified Sampling Method

generate a large number of realizations. Each realization must

be stored in a column of a nz x n matriz.

n = 1000000; distrib = rand(pz,1,n);

Generate the bins

Nb = [50]; bins = Generate_bins(distrib,Nb);

Evaluate E and Var using only 50 evaluations

EL,VarL = Lass(bins,f);

println("E[f] $E - reference") println("E[f] $EL obtained with $(Nb[1]) evaluations") println() println("Var[f] $Var - reference") println("Var[f] $VarL obtained with $(Nb[1]) evaluations")

The same thing can be done to evaluate E[df] and Var[df]

where df is the derivative of f w.r.t a deterministic

design variable x

dfx(z::Vector,x=1.0) = -10(z[1]-5x) + z[1];

Lets obtain some reference values.

The derivative of the expected value E[f] is

pdE(z)=dfx([z])*pdf(pz,z); dE = quadgk(pdE,-Inf,Inf)[1];

And the derivative of the variance is

d(f-E)^2 ==> 2(f-E)*(df-dE)

pdV(z)=2(f([z])-E)(dfx([z])-dE)*pdf(pz,z); dVar = quadgk(pdV,-Inf,Inf)[1];

Evaluate dE and dVar using only 50 evaluations

dEL,dVarL = dLass(bins,f,dfx,1);

println("dE[f] $dE - reference") println("dE[f] $dEL obtained with $(Nb[1]) evaluations") println() println("dVar[f] $dVar - reference") println("dVar[f] $dVarL obtained with $(Nb[1]) evaluations") ``` The results for this problem are (they can vary a little bit, since z is random)

julia E[f] 6.000000000000621 - reference E[f] 5.997049691574353 obtained with 50 evaluations julia Var[f] 3.0000000000000417 - reference Var[f] 2.98478094281721 obtained with 50 evaluations and

julia dE[f] 4.999999999999871 - reference dE[f] [4.9985604747676415] obtained with 50 evaluations julia dVar[f] -17.99999999988275 - reference dVar[f] [-17.8981052338314] obtained with 50 evaluations