Math & Numerical Methods
Monte Carlo & Uncertainty Simulators
A focused Math & Numerical Methods hub for monte carlo & uncertainty tools, keeping related formulas, assumptions, and engineering checks together.
20 simulators
Adjacent categories
Simulator list
Bayesian Calibration via MCMC (Metropolis–Hastings) Simulator
Monte Carlo & Uncertainty
Calibrate a CAE model parameter the Bayesian way: feed in a prior, a likelihood and observed data.
Bootstrap Confidence Interval Simulator
Monte Carlo & Uncertainty
Compute 95% bootstrap confidence intervals for the mean and median in real time. Resample with replacement and vary N and B to watch the distribution converge.
Brownian Motion & Random Walk Simulator
Monte Carlo & Uncertainty
Simulate Brownian motion and random walks in real time. Adjust particles, step size, and trails to verify Einstein's diffusion formula and explore parameters.
Buffon's Needle Simulator — Estimating Pi by Probability
Monte Carlo & Uncertainty
The classical experiment: drop needles at random on a floor of equally spaced parallel lines and estimate pi from the crossing probability. Change the count, length, spa…
Control Variates Simulator
Monte Carlo & Uncertainty
Experience the control variates method, which boosts Monte Carlo accuracy almost for free. Using an auxiliary variable strongly correlated with your target, watch how mu…
FFT Spectrum Analyzer — Time-Domain Signal to Frequency Spectrum
Monte Carlo & Uncertainty
Analyze time-series signals with the Fast Fourier Transform. Tweak window functions, sampling frequency and signal components to feel aliasing and leakage.
Gibbs Sampling Simulator
Monte Carlo & Uncertainty
Draw samples from a bivariate normal distribution with Gibbs sampling, the classic MCMC (Markov chain Monte Carlo) method.
Importance Sampling — Efficient Tail Probability Estimation
Monte Carlo & Uncertainty
Estimate the tail probability P(X>t) of the standard normal in parallel with crude Monte Carlo and importance sampling. Tune the proposal mean and standard deviation and…
Latin Hypercube Sampling — LHS vs Plain Monte Carlo
Monte Carlo & Uncertainty
Side-by-side 2D Latin Hypercube and plain Monte Carlo sampling. See how placing exactly one point per row and column changes the integration accuracy at the same sample …
MCMC Metropolis-Hastings Sampler — Acceptance & Autocorrelation
Monte Carlo & Uncertainty
Draw samples from an arbitrary 1D distribution with Metropolis-Hastings MCMC. Tune the proposal variance sigma and watch the trace, histogram, and autocorrelation update…
Monte Carlo Pi Estimator
Monte Carlo & Uncertainty
Estimate π using random numbers! This Monte Carlo simulation drops points into a square to reveal the value of Pi through probability and error analysis.
Monte Carlo Statistics Simulator — π, Integration, CLT
Monte Carlo & Uncertainty
Explore Monte Carlo methods: estimate π, test the Central Limit Theorem, perform integration, and simulate random processes with this interactive statistics simulator.
Morris Elementary Effects Method Simulator
Monte Carlo & Uncertainty
Run the cheapest global sensitivity method — the Morris Elementary Effects (EE) screening — directly in your browser.
Polynomial Chaos Expansion (PCE) Simulator
Monte Carlo & Uncertainty
For a Gaussian input X ~ N(μ, σ²) and the model Y = a₁X + a₂X², this tool builds a Hermite Polynomial Chaos Expansion in real time and reads off the mean, variance and S…
Quasi-Monte Carlo Simulator — Sobol Sequence
Monte Carlo & Uncertainty
Integrate numerically with low-discrepancy sequences such as the Sobol and Halton sequences. Change the sample count, sequence type and integrand to watch how determinis…
2D Random Walk Simulator — MSD & Diffusion Coefficient
Monte Carlo & Uncertainty
Simulate 2D random walks in real time. Visualize diffusion laws with MSD plots and switch between lattice, Gaussian, and Lévy flight modes.
Rejection Sampling Simulator — Monte Carlo Sample Generation
Monte Carlo & Uncertainty
When direct sampling from a target p(x) is hard, cover it with a proposal q(x) and an envelope M*q(x) >= p(x), then accept each candidate with probability p(x)/(M*q(x)).…
Sobol Sensitivity Indices Simulator
Monte Carlo & Uncertainty
A hands-on global sensitivity analysis tool that decomposes how input uncertainty drives output variance into main effects and interactions.
Central Limit Theorem Simulator — Sampling Distribution Visualization
Monte Carlo & Uncertainty
Repeatedly sample from uniform, exponential, bimodal, or skewed distributions. Vary sample size n and observe how sample means converge to a normal distribution regardle…
Stratified Sampling Simulator
Monte Carlo & Uncertainty
A tool for experiencing stratified sampling, a way to boost the accuracy of Monte Carlo integration for free. Split the interval [0,1] into strata and draw samples evenl…
How to Use
- Select a distribution type (Normal, Uniform, Lognormal, Triangular) for each input parameter from the cat-filter dropdown
- Enter mean/mode values and standard deviations or bounds based on your uncertainty source (measurement tolerance, material variability, process drift)
- Set sample count (typically 5,000–50,000 iterations) and click Run to execute the Monte Carlo sampling
- Inspect output histograms, percentile tables, and sensitivity indices to identify dominant uncertainty contributors
Worked Example
Concrete beam deflection analysis: L=4m simply-supported steel beam, E=200GPa (±2% variability), load P=25kN (±5% sensor uncertainty), moment of inertia I=150cm⁴ (±3% section tolerance). Assign Normal distributions to each. With 10,000 Monte Carlo samples, mean deflection δ=8.42mm, 95th percentile=9.28mm, 5th percentile=7.61mm. Sensitivity analysis shows load contributes 65% of variance, material properties 28%, geometry 7%.
Practical Notes
- Use Lognormal distributions for fatigue strength and crack growth rates where lower tail asymmetry dominates failure risk
- Validate input bounds against tolerance stacks: if assembly clearance is ±0.5mm, use Uniform[−0.5, +0.5] rather than Normal to avoid false tail probability
- Always compare 95th percentile output against design limits; Monte Carlo reveals tail risks missed by nominal stress analysis
- For brittle materials (ceramics, concrete), increase sample count to 50,000+ to resolve low-probability fracture events accurately