## 3 Methodology
### The Well Sequence  
Well-equidistributed long-period linear (WELL) sequence was proposed by [52]. Initially, it carried out an updated version of the Mersenne twister algorithm. The Algorithm 2 for generating WELL distribution is given as:  
```
WELL (0: \(t_{0}=\left(m_{x}\&v_{k,r-1}\right)+\left(m_{x}\&v_{k,r-2}\right)\) \(t_{1}=\left(A_{0}v_{k,0}\right)+\left(A_{1}v_{k,m_{1}}\right)\) \(t_{2}=\left(A_{2}v_{k,m_{2}}\right)+\left



### The Knuth Sequence

As discussed above, inbuilt library function is used, \(Knuth(x_{min,}x_{max})\) to generate Knuth sequences of random points. Knuth sequence is designed and was proposed by the authors in [53]. The pseudocode to generate Knuth sequences is as follows:

for i from 0 to n - 2 do

j \(\leftarrow\) random integer such that i \(\leq\) j \(<\) n

exchange a[i] and a[j]

### The Torus Sequence

Torus is a geometric term that was firstly used by the authors in [54] to generate a Torus mesh for a geometric coordinate system. In-game development Torus mesh is commonly used and can be created using a left-hand coordinate system or right-hand coordinate system. The shape for the Torus at 1D, 2D, and 3D is a circle and 2D rectangle, respectively. The Torus in 3D can be represented by the following Equations (6)-(8). \[a(\theta,\delta)=(D+r\cos\theta)\cos\delta \tag{6}\]

\[b(\theta,\delta)=(D+r\cos\theta)\sin\delta \tag{7}\]

\[c(\theta,\delta)=r\sin\delta \tag{8}\]

where their angles of circles are \(\theta,\delta\) and \(D\) is the distance from tube center to Torus center \(r\) denotes to the radius of circle. Inspired by this mesh with Torus, a low-discrepancy sequences have been generated that being initialize with the prime series as Torus effect. Equation (9) shows the mathematical notation for Torus series. \[a_{k}=(f(k\sqrt{s_{1}}),\ldots,f(k\sqrt{s_{d}})) \tag{9}\]

where \(s_{1}\) denotes the series of \(i\)th prime number, and \(f\) is a fraction which can be calculated by \(f=\) a \(-\) floor(a). Due to the prime constraints, the dimension for the Torus is limited to the 100,000, only if we use parameter prime in Torus function. For more than 100,000 dimensions the number must be provided manually. In Figures 1-6 random points following the Uniform, Sobol, Halton, WELL, Knuth, and Torus distribution are shown by bubble plot in which the \(y\)-axis represents the random values and the \(x\)-axis shows the relevant index of the concerned point in the table. Figure 2: Population initialization using Sobol distribution. Figure 4: Population initialization using WELL distribution. Figure 5: Population initialization using Knuth distribution. Figure 3: Population initialization using Halton distribution. Figure 6: Sample data generated using Torus distribution. We have made our first significant contribution to this study by introducing three novel methods of initialization of population WE-PSO, KN-PSO, and TO-PSO. Algorithm 3 shows the proposed distribution-based PSO initialization:

```
1. Initialize the swarm
2. Set epoch count \(I=0\), population size \(N_{z}\), Dimension of the problem \(D_{z}\), \(w_{max}\) and \(w_{min}\)
3. For each particle \(P_{z}\)
4. Initialize \(x_{z}\), as \(x_{z}=\) WELL, Knuth, Torus \((xmin,xmax)\)
5. Initialize the Particle velocity as, \(v_{z}=Rand(xmin,xmax)\)
6. Compute the fitness score \(f_{z}\)
7. Set global best position \(\delta_{z}^{best}\) as \(max(f_{1},f_{2},f_{3}\ldots f_{z})\) where \(f_{z}\in globally\ optimal\ fitness\)
8. Set local best position \(p_{z}^{best}\) as \(max(f_{1},f_{2},f_{3}\ldots f_{z})\) where \(f_{z}\in locally\ optimal\ fitness\)
9. Compare the current particle's fitness score \(x_{z}\) in the swarm and its old local best location \(p_{z}^{best}\) If the current fitness score \(x_{z}\) is greater than \(p_{z}^{best}\), then substitute \(p_{z}^{best}\), with \(x_{z}\); else retain the \(x_{z}\) unchanged
10.