  We investigate a simple model using the numerical simulation in the complex
Langevin equation (CLE) and the analytical approximation with the Gaussian
Ansatz. We find that the Gaussian Ansatz captures the essential and even
quantitative features of the CLE results quite well including unwanted behavior
in the unstable region where the CLE converges to a wrong answer. The Gaussian
Ansatz is therefore useful for looking into this convergence problem and we
find that the exact answer in the unstable region is nicely reproduced by
another solution that is naively excluded from the stability condition. We
consider the Gaussian probability distributions corresponding to multiple
solutions along the Lefschetz thimble to discuss the stability and the
locality. Our results suggest a prescription to improve the convergence of the
CLE simulation to the exact answer.
