Overview
Monte Carlo Integration estimates an integral from randomly sampled function values, which is especially useful in high-dimensional domains.
CoreSource checked
Monte Carlo Integration
Approximates integrals by averaging random samples.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Numerical Methods
- Difficulty
- Intermediate
- Family
- Stochastic Integration
- Tier
- Core
- Status
- Verified
- Review
- Source Checked
- Sources
- 1
- Formulas
- 1
- Relationships
- 2
- Code example
- Ready
- Visualization
- Ready
Core Concepts
random samplesample meandomain volumevariance
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(samples)
Average
O(samples)
Worst
O(samples)
Space
O(1)
When to Use
- Use when dimension is high or deterministic grids are too expensive.
- Use when high-dimensional integration is a central requirement.
When Not to Use
- Avoid when a smooth low-dimensional integral can be solved faster with quadrature.
- Avoid when roundoff, conditioning, or discontinuities dominate the result.
What It Does
- Draws random samples
- Evaluates the integrand
- Scales the sample average by domain volume
Use Cases
- High-dimensional integration
- Uncertainty estimates
- Rendering estimators
Inputs
- Function
- Sampling domain
- Sample count
Outputs
- Integral estimate
Key Formulas (1)
Monte Carlo integral
integral ≈ volume(domain) · mean(f(samplei))
integral ~= volume(domain) * mean(f(sample_i))Scales the average sampled value by the size of the domain.
Pseudocode
- 1.Draw random points from the integration domain
- 2.Evaluate the function at each point
- 3.Average the values
- 4.Multiply by domain volume
- 5.Report estimate and optional uncertainty
Canonical Example
- Input
- sample f(x)=x^2 on [0,1]
- Output
- estimate near 0.333
- Explanation
- The sample average approximates the area under the curve.
Implementation Notes
- Track sample variance along with the mean estimate.
- Compare against a small hand-checkable case before scaling up.
- Keep a tiny example with expected output beside the implementation.
Common Pitfalls
- Using biased samples without weighting gives the wrong integral.
- Ignoring conditioning can make a correct method return unreliable numbers.
- Not defining boundary, empty, or degenerate inputs before implementation.
Interactive Mechanism Walkthrough
Iterative Convergence & State TrajectoryShow random points under a curve contributing to an average area estimate.
Step 1 of 5: Draw random points from the integration domain
numeric-iterationNotes
Error decreases slowly, so variance reduction is often important.
Sources (1)
What each reference supports for this entry.
Explore Next
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Relationships
View full relationship pageSpecializesMonte Carlo Simulation
Monte Carlo integration estimates integrals with random samples.
Related Algorithms
Tags
Numerical MethodsSamplingStochasticEstimation