Monte Carlo Methods

TL;DR

• Use random sampling to approximate solutions and estimate outcomes for mathematical problems and models.
• Commonly applied to numerical integration and to simulate complex systems with uncertain variables.
• Accuracy depends on sample count and distribution; they can be computationally intensive but are well-suited to parallel computation.

Definition

Monte Carlo methods are a class of computational algorithms that use random sampling to solve mathematical problems. These methods are named after the city of Monte Carlo in Monaco, where the first random sampling experiments were conducted in the early 20th century.

Explanation

Monte Carlo methods obtain approximate solutions by generating and evaluating random samples rather than enumerating all possible outcomes. They are applied both to numerical problems (for example, approximating integrals) and to modeling complex systems with many variables and uncertain outcomes. Advantages cited include relative speed and efficiency for problems where explicit calculation of all outcomes is impractical, applicability to complex non-linear relationships, and suitability for parallel computation. Limitations include potential inaccuracy depending on the number and distribution of samples, significant computational cost for very large or complex problems, and challenges in interpretation because results arise from random sampling rather than deterministic calculations.

Examples

Monte Carlo integration

The Monte Carlo integration technique is used to approximate the value of a definite integral. This technique works by dividing the region under the curve of a function into a number of small rectangles. A random sample of points is then generated within each rectangle, and the average value of the function is calculated at these points. The area of each rectangle is then multiplied by this average value, and the sum of these products is used as an approximation of the integral.

Monte Carlo simulation

Monte Carlo simulation is used to model complex systems with many variables and uncertain outcomes. This technique works by generating random samples of the variables that affect the system, and using these samples to calculate the likely outcomes of the system. For instance, a Monte Carlo simulation could be used to model the stock market, where random samples of stock prices and interest rates are used to calculate the likely return on investment for a given portfolio.

Use cases

• Finance
• Engineering
• Computer science

Notes or pitfalls

• Not always accurate: quality of approximation depends on the number and distribution of random samples.
• Can be computationally intensive and may be infeasible for very large or complex problems.
• Results can be difficult to interpret and explain because they rely on random sampling rather than deterministic calculations.

Related terms

• Monte Carlo integration
• Monte Carlo simulation
• Random sampling
• Parallel computation