What are Monte Carlo methods?

Monte Carlo methods encompass various computational algorithms that depend on repetitive random sampling to derive numerical outcomes. These methods utilize randomness to solve complex problems where deterministic solutions are challenging or impossible to solve. They find applications in physics, mathematics, and various other fields.

Steps involved in Monte Carlo

In Monte Carlo methods, specific rules and procedures are implemented to enhance the obtained results. The steps involved in Montel Carlo methods are:

1. Problem formulation using a mathematical model: Develop a mathematical equation that links the input and output variables.

2. Random sampling: We generate random samples from a specific probability distribution or use random number generators. These samples represent different possible scenarios or inputs for the problem.

3. Model evaluation: The generated random samples serve as inputs to our mathematical model, where calculations are performed.

4. Iteration: Repeat steps 2 to 4 to enhance estimation accuracy, with a greater number of samples leading to closer approximations of true values

5. Convergence: Convergence occurs when results stabilize within an acceptable level of uncertainty.

Estimation of π using Monte Carlo

Through the Monte Carlo method, we can approximate the value of $\pi$ by utilizing a random number generator. Prior to initiating the estimation process, it is essential to establish the mathematical model and equations involved to achieve the most accurate estimation of $\pi$.

Mathematical model and random sampling

One approach is to estimate$\pi$by using the circumference and diameter of a circle. However, this approach may not yield highly accurate results due to the potential for human errors and limitations in measurement devices, which can introduce uncertainties and impact the precision of the outcomes.

Circumference of a circle: C = $\pi d$ and therefore $\pi = \frac{d}{C}$

We can use another method where we can use a Cartesian planeThe Cartesian plane is a two-dimensional coordinate system representing points using two perpendicular axes. to estimate the value of $\pi$. The diagram below shows a circle with the equation $y^{2} + x^{2} = 1$which is inscribed in a square that has length 'l' = 2.

Based on our mathematical model, we can estimate the value of the $\pi$.

Where $n_{circle}$ refers to the number of points located within the circle, and $N_{total}$represents the total number of points within both the square and the circle.

Model evaluation and iteration

Once the mathematical model is developed, we can proceed to calculate the value of $\pi$ by using large random sampling across the entire space. Our approach involves generating random data points within the range of [-1, 1] along the X-axis and Y-axis. By doing so, we can distribute these points both inside the circle and outside the circle, while ensuring that all points fall within the boundaries of the square.

Convergence

By repeatedly generating larger sets of points, we enhance the accuracy of our estimation through iterative steps. Therefore as we expand the random sampling range within the Cartesian plane, specifically within the interval of [-1, 1], the error continuously diminishes. By observation, we see that as data points increase, the value of $\pi$ will fluctuate from its original value either above it or below it. It will approach its true value i.e. $\pi = 3.141592653589$when total random data points approach infinity i.e. $N \rightarrow \infty$.

Why do we use Monte Carlo methods?

Monte Carlo methods have applications across various domains. Here are a few illustrative examples.

• Business: Decision-makers utilize it to see the feasibility and profitability of choices, such as increasing advertising budget or subscription fees.

• Finance: Financial analysts employ it to make long-term forecasts and develop investment strategies considering market fluctuations.

• Engineering: Engineers rely on it to estimate product reliability, such as predicting failure rates of mechanical systems under different operating conditions.

Conclusion

Random sampling and Monte Carlo methods demonstrate their effectiveness in solving simple problems, but complex problems require additional considerations. Although random sampling and Monte Carlo methods offer intuitive solutions for diverse applications, they can be limited by challenges such as dealing with high dimensionality and meeting computational requirements.