From 56f1f3e809b038f336871d79662101ad61600fed Mon Sep 17 00:00:00 2001
From: MRIDUL JAIN <105979087+Spinachboul@users.noreply.github.com>
Date: Sat, 16 Dec 2023 00:27:54 +0530
Subject: [PATCH] Create gekpls_lhs_sampling.md

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+# Latin Hypercube Sampling (LHS) for Surrogate Modeling in Julia
+
+Latin Hypercube Sampling (LHS) is a method used for generating representative samples across multiple dimensions of an input space. This README demonstrates how to use LHS for surrogate modeling in Julia, specifically replacing Sobol Sampling in the context of GEKPLS (Gradient Enhanced Kriging with Partial Least Squares) surrogate modeling.
+
+### Requirements
+- Julia programming language
+- LatinHypercubeSampling.jl package
+
+### Installation
+Make sure you have the LatinHypercubeSampling.jl package installed in your Julia environment:
+
+```julia
+using Pkg
+Pkg.add("LatinHypercubeSampling")
+```
+
+# Usage Example
+Consider a scenario where you want to create a surrogate model using GEKPLS for a given function, **water_flow**, with an input space defined by lower bounds **lb** and upper bounds **ub**. Here's how to use LHS instead of Sobol Sampling:
+
+```
+using Surrogates
+using LatinHypercubeSampling
+
+# Function definition for water_flow
+function water_flow(x)
+    r_w = x[1]
+    r = x[2]
+    T_u = x[3]
+    H_u = x[4]
+    T_l = x[5]
+    H_l = x[6]
+    L = x[7]
+    K_w = x[8]
+    log_val = log(r/r_w)
+    return (2*pi*T_u*(H_u - H_l))/ ( log_val*(1 + (2*L*T_u/(log_val*r_w^2*K_w)) + T_u/T_l))
+end
+
+n_samples = 1000  # Number of LHS samples
+lb = [0.05, 100, 63070, 990, 63.1, 700, 1120, 9855]
+ub = [0.15, 50000, 115600, 1110, 116, 820, 1680, 12045]
+
+# Generating Latin Hypercube Samples
+lhs_samples = lhsdesign(n_samples, length(lb))
+x = [(ub - lb) .* lhs_samples[i, :] .+ lb for i in 1:n_samples]
+```
+# Surrogate Optimization
+
+```
+using Surrogates
+using Zygote
+
+function sphere_function(x)
+    return sum(x .^ 2)
+end
+
+lb = [-5.0, -5.0, -5.0]
+ub = [5.0, 5.0, 5.0]
+n_comp = 2
+delta_x = 0.0001
+extra_points = 2
+initial_theta = [0.01 for i in 1:n_comp]
+n = 100
+x = sample(n, lb, ub, SobolSample())
+grads = gradient.(sphere_function, x)
+y = sphere_function.(x)
+g = GEKPLS(x, y, grads, n_comp, delta_x, lb, ub, extra_points, initial_theta)
+x_point, minima = surrogate_optimize(sphere_function, SRBF(), lb, ub, g,
+                                        RandomSample(); maxiters = 20,
+                                        num_new_samples = 20, needs_gradient = true)
+println(minima)
+```
+
+
+# Conclusion
+
+Latin Hypercube Sampling (LHS) provides a more evenly distributed set of samples across the input space compared to purely random methods. Experiment with different sampling sizes and assess the surrogate model's performance to find the optimal sampling strategy for your problem.
+
+For more information, refer to this link: 
+https://github.com/MrUrq/LatinHypercubeSampling.jl
+
+