Numerical Integration
Approximating a definite integral or advancing a differential equation using discrete sums.
Definition
Numerical integration approximates the value of an integral (quadrature) or advances a differential equation in time using discrete evaluations. Quadrature rules estimate the area under a curve from sampled values.
In several dimensions, grid-based quadrature suffers the curse of dimensionality, its cost rising exponentially with dimension, which is exactly where Monte Carlo integration takes over. The crossover point is a practical reason to switch methods as problems grow.
One-dimensional quadrature is largely a solved problem, with Gaussian rules achieving high accuracy from few points on smooth integrands, but the difficulty grows sharply with dimension. Grid-based methods suffer the curse of dimensionality, their cost rising exponentially, which is exactly where Monte Carlo integration takes over. Choosing the method to match the integrand's dimension and smoothness is the practical decision, since each excels in a different regime.
Methods
- Trapezoidal and Simpson's rules for one-dimensional quadrature.
- Gaussian quadrature for high accuracy with few points.
- Monte Carlo integration in high dimensions.
- Runge-Kutta and similar for time integration.
Why it matters
Most integrals arising in science have no closed form, so numerical integration is indispensable. The choice of method balances accuracy against the number of function evaluations, which matters when each evaluation is expensive.
Fusion connection
Integrating physical quantities over the plasma volume, and advancing equations in time, are routine numerical-integration tasks throughout Kronos simulations.