Laplace Method for Concentrating Integrals

The Laplace method is a general technique for approximating integrals of the form I(λ) = ∫ e^{-λ f(x)} dx for large values of the parameter λ. As λ grows, the integrand concentrates heavily around the global minimum of f(x). By approximating f(x) with its Taylor series around the minimum, the integral can be evaluated approximately using a Gaussian integral.

Experiment with Concentration

You can interact with these concepts using the module. Explore how λ affects the approximation error at:
https://riano.app/module/laplace-method-concentrating-integrals

1. Set Concentration lambda to 1, and Half width L to 1. Notice the peak width is 1.0. The relative error is roughly 18%.
2. Change Concentration lambda to 9. The peak width drops to 0.3333333333333333, and the integral concentrates heavily near 0. The relative error drops to roughly 2.2e-5.
3. Change Half width L to 0.5. At this narrower width, the approximation becomes worse again if λ is small (e.g. λ = 0.25).