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The Ensemble Kalman Filter for Dynamic Inverse Problems

Simon Weissmann, Neil K. Chada, Xin T. Tong

TL;DR

This article introduces and justifies a new methodology called dynamic-EKI (DEKI), which is a particle-based method with a changing forward operator, and introduces and justifies a new methodology called dynamic-EKI (DEKI), which is a particle-based method with a changing forward operator.

Abstract

In inverse problems, the goal is to estimate unknown model parameters from noisy observational data. Traditionally, inverse problems are solved under the assumption of a fixed forward operator describing the observation model. In this article, we consider the extension of this approach to situations where we have a dynamic forward model, motivated by applications in scientific computation and engineering. We specifically consider this extension for a derivative-free optimizer, the ensemble Kalman inversion (EKI). We introduce and justify a new methodology called dynamic-EKI, which is a particle-based method with a changing forward operator. We analyze our new method, presenting results related to the control of our particle system through its covariance structure. This analysis includes moment bounds and an ensemble collapse, which are essential for demonstrating a convergence result. We establish convergence in expectation and validate our theoretical findings through experiments with dynamic-EKI applied to a 2D Darcy flow partial differential equation.

The Ensemble Kalman Filter for Dynamic Inverse Problems

TL;DR

This article introduces and justifies a new methodology called dynamic-EKI (DEKI), which is a particle-based method with a changing forward operator, and introduces and justifies a new methodology called dynamic-EKI (DEKI), which is a particle-based method with a changing forward operator.

Abstract

In inverse problems, the goal is to estimate unknown model parameters from noisy observational data. Traditionally, inverse problems are solved under the assumption of a fixed forward operator describing the observation model. In this article, we consider the extension of this approach to situations where we have a dynamic forward model, motivated by applications in scientific computation and engineering. We specifically consider this extension for a derivative-free optimizer, the ensemble Kalman inversion (EKI). We introduce and justify a new methodology called dynamic-EKI, which is a particle-based method with a changing forward operator. We analyze our new method, presenting results related to the control of our particle system through its covariance structure. This analysis includes moment bounds and an ensemble collapse, which are essential for demonstrating a convergence result. We establish convergence in expectation and validate our theoretical findings through experiments with dynamic-EKI applied to a 2D Darcy flow partial differential equation.
Paper Structure (20 sections, 6 theorems, 92 equations, 4 figures, 1 algorithm)

This paper contains 20 sections, 6 theorems, 92 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem formulation
  4. Model assumptions
  5. Our Contributions:
  6. Outline
  7. Ensemble Kalman inversion with dynamic forward operator
  8. Main result
  9. Ensemble collapse and moment bounds
  10. Convergence analysis - proof of the main results
  11. Ergodic data convergence proof
  12. Periodic data convergence proof
  13. Numerical experiments
  14. Periodic experiment: Dynamic observation operator
  15. Independent and identically distributed(i.i.d.) observation model
  16. ...and 5 more sections

Key Result

Theorem 2.1

Let $(\{z_t^{(j)}\}_{j=1}^J, t\in\mathbb{N})$ be generated by eq:unpert_EKI with fixed $\eta_t=h>0$ and initial ensemble $\{z_0^{(j)}\}_{j=1}^J$, such that $C_0^{zz}\succ \sigma_l I$ almost surely for some $\sigma_l>0$ with $\lambda= \sigma_l q\in(0,1)$ and $q=2\mu$. Moreover, let where $E_0$ is defined as $E_0=\frac{1}{J}\sum_{j=1}^J \|e_0^{(j)}\|^2$, from eq:spread. Under Assumption assum:corre

Figures (4)

  • Figure 1: Convergence plot for EKI using periodic (left) and i.i.d. (right) data. We plot the error w.r.t. the reference solution $z_{\mathrm{ref}}$ vs time $T$.
  • Figure 2: left: Plot of our random field representation of $z^\ast$ based on the KLE \ref{['eq:kle']}. right: Solution of the PDE \ref{['eq:darcy_flow']}.
  • Figure 3: Convergence plot for EKI using ergodic data. We plot the error w.r.t. the reference solution $z_{\mathrm{ref}}$ vs time $T$.
  • Figure 4: Convergence plot for EKI using i.i.d., periodic and ergodic data. We plot the error w.r.t. the ground truth $z^\ast$ vs time $T$.

Theorems & Definitions (15)

  • Example 1.1: Darcy flow
  • Remark 1.3
  • Theorem 2.1: ergodic data
  • Corollary 2.2: i.i.d. data
  • Theorem 2.3: periodic data
  • Remark 2.4
  • Lemma 3.1: subspace property
  • proof
  • Lemma 3.2: Ensemble collapse
  • proof
  • ...and 5 more