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.

Worked Example: Exact vs Approximation

Consider the specific integral I(λ, L) = ∫_-L^L exp(-λx²) dx. Here, f(x) = x² with its minimum at x = 0.

For λ = 4 and L = 1, the true integral is accurately calculated as 0.8820813907624215.
The leading Laplace approximation extends the integration limits to ±∞, yielding ∫_-∞^∞ exp(-λx²) dx = √(π/λ).
For λ = 4, the approximation is √(π/4) ≈ 0.8862269254527579.
The absolute error is the difference, roughly 0.004145534690336333, indicating that the concentration inside the interval [-1, 1] captures the majority of the area.

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

Set Concentration lambda to 1, and Half width L to 1. Notice the peak width is 1.0. The relative error is roughly 18%.
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.
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).

Limitations

The large-parameter approximation √(π/λ) assumes λ is strictly positive and sufficiently large relative to the bounds. At λ = 0, the integral I = 2L, but the asymptotic expression and peak width (1/√λ) are undefined or unavailable. Furthermore, the asymptotic expression is never exactly true for finite boundaries, and the error behaves like erfc(L√λ). This method requires the minimum to be strictly in the interior; boundary minima have a different asymptotic behavior.

References

The underlying properties and formulas for error functions and their asymptotics are available in the Digital Library of Mathematical Functions (DLMF).

DLMF 2.3: Asymptotic Approximations (Laplace's Method) - Provides theoretical foundation.
DLMF 7.2: Error Functions, Dawson’s and Fresnel Integrals - Details the definition and series representation of erf and erfc.

Laplace Method for Concentrating Integrals