Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nonparametric estimation of homogenized invariant measures from multiscale data via Hermite expansion

Jaroslav I. Borodavka, Max Hirsch, Sebastian Krumscheid, Andrea Zanoni

TL;DR

This work develops a nonparametric, spectrally based estimator for the invariant density $\rho$ of the homogenized one-dimensional Langevin diffusion, using a truncated Fourier expansion in Hermite functions and time-average coefficients from a single multiscale trajectory. A rigorous convergence analysis combines a Gaussian-mixture extension for general potentials with an $\varepsilon$-dependent mean ergodic theorem to control stochastic error, showing $\mathbb{E}\|\widehat{\rho}_N^{T,\varepsilon}-\rho\|_{L^2}^2 \to 0$ as $\varepsilon\to0$ under prescribed growth of the number of modes $N(\varepsilon)$ and observation time $T(\varepsilon)$. Numerically, the estimator demonstrates robustness to model misspecification, accurately recovers the homogenized invariant density, and enables practical wavelength (i.e., scale) inference from spectral data; it also extends naturally to two dimensions via tensor-product Hermite bases. The results advance nonparametric homogenization by enabling invariant-density estimation from multiscale data without explicit knowledge of the homogenized model and point to future work in CLTs, higher-dimensional extensions, and drift/potential estimation.

Abstract

We consider the problem of density estimation in the context of multiscale Langevin diffusion processes, where a single-scale homogenized surrogate model can be derived. In particular, our aim is to learn the density of the invariant measure of the homogenized dynamics from a continuous-time trajectory generated by the full multiscale system. We propose a spectral method based on a truncated Fourier expansion with Hermite functions as orthonormal basis. The Fourier coefficients are computed directly from the data owing to the ergodic theorem. We prove that the resulting density estimator is robust and converges to the invariant density of the homogenized model as the scale separation parameter vanishes, provided the time horizon and the number of Fourier modes are suitably chosen in relation to the multiscale parameter. The accuracy and reliability of this methodology is further demonstrated through a series of numerical experiments.

Nonparametric estimation of homogenized invariant measures from multiscale data via Hermite expansion

TL;DR

This work develops a nonparametric, spectrally based estimator for the invariant density of the homogenized one-dimensional Langevin diffusion, using a truncated Fourier expansion in Hermite functions and time-average coefficients from a single multiscale trajectory. A rigorous convergence analysis combines a Gaussian-mixture extension for general potentials with an -dependent mean ergodic theorem to control stochastic error, showing as under prescribed growth of the number of modes and observation time . Numerically, the estimator demonstrates robustness to model misspecification, accurately recovers the homogenized invariant density, and enables practical wavelength (i.e., scale) inference from spectral data; it also extends naturally to two dimensions via tensor-product Hermite bases. The results advance nonparametric homogenization by enabling invariant-density estimation from multiscale data without explicit knowledge of the homogenized model and point to future work in CLTs, higher-dimensional extensions, and drift/potential estimation.

Abstract

We consider the problem of density estimation in the context of multiscale Langevin diffusion processes, where a single-scale homogenized surrogate model can be derived. In particular, our aim is to learn the density of the invariant measure of the homogenized dynamics from a continuous-time trajectory generated by the full multiscale system. We propose a spectral method based on a truncated Fourier expansion with Hermite functions as orthonormal basis. The Fourier coefficients are computed directly from the data owing to the ergodic theorem. We prove that the resulting density estimator is robust and converges to the invariant density of the homogenized model as the scale separation parameter vanishes, provided the time horizon and the number of Fourier modes are suitably chosen in relation to the multiscale parameter. The accuracy and reliability of this methodology is further demonstrated through a series of numerical experiments.

Paper Structure

This paper contains 17 sections, 19 theorems, 234 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Outline.
  3. Problem setting
  4. Nonparametric estimation of the invariant density
  5. Numerical illustration
  6. Density estimator
  7. Inference of the scale separation parameter
  8. A two-dimensional example
  9. Convergence analysis
  10. Analysis of the deterministic component $\mathfrak q_N^\varepsilon$
  11. Properties of Hermite polynomials and Hermite functions
  12. Quadratic potential and the Gaussian setting
  13. Extension to a general potential
  14. Analysis of the stochastic component $\mathcal{Q}_N^{T,\varepsilon}$
  15. An $\varepsilon$-dependent mean ergodic theorem with rates
  16. ...and 2 more sections

Key Result

Theorem 2.3

Let $\widehat{\rho}_N^{T,\varepsilon}$ be the estimator defined in equation eq:estimator_density, and let $\rho$ denote the invariant density given in equation eq:invariant_density. Let as:potentials hold and assume that the number of Fourier modes and the final time of observation scale with $\vare respectively, for some $\kappa > 0$, where the parameters $\gamma$ and $\zeta$ satisfy with Then,

Figures (3)

  • Figure 1: Performance of the estimator $\widehat{\rho}_N^{T,\varepsilon}$ across varying values of $T = 50, 500, 5000$ and $N = 4, 16, 64$ with fixed $\varepsilon = 0.1$, for the double-well potential.
  • Figure 2: Top: magnitude of the Fourier transform of the density estimator $\mathcal{F}(\widehat{\rho}_N^{T,\varepsilon})$ across varying values of $\varepsilon = 0.075, 0.1, 0.125$ and $N = 30, 60, 90$ with fixed $T = 1000$, for the double-well potential. Bottom: inference of the scale separation parameter $\varepsilon$ from the dominant frequency $\bar{\xi} \neq 0$.
  • Figure 3: Performance of the estimator $\widehat{\rho}_N^{T,\varepsilon}$ for the two-dimensional test case.

Theorems & Definitions (45)

  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Lemma 4.1
  • proof
  • Remark 4.2
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • proof
  • ...and 35 more