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On gradient stability in nonlinear PDE models and inference in interacting particle systems

Aurélien Castre, Richard Nickl

TL;DR

An approach based on a Banach space version of the implicit function theorem to verify the gradient stability condition of Nickl&Wang (JEMS 2024) for the underlying non-linear inverse problem, providing also injectivity estimates and corresponding statistical identifiability results.

Abstract

We consider general parameter to solution maps $θ\mapsto \mathcal G(θ)$ of non-linear partial differential equations and describe an approach based on a Banach space version of the implicit function theorem to verify the gradient stability condition of Nickl&Wang (JEMS 2024) for the underlying non-linear inverse problem, providing also injectivity estimates and corresponding statistical identifiability results. We illustrate our methods in two examples involving a non-linear reaction diffusion system as well as a McKean--Vlasov interacting particle model, both with periodic boundary conditions. We apply our results to prove the polynomial time convergence of a Langevin-type algorithm sampling the posterior measure of the interaction potential arising from a discrete aggregate measurement of the interacting particle system.

On gradient stability in nonlinear PDE models and inference in interacting particle systems

TL;DR

An approach based on a Banach space version of the implicit function theorem to verify the gradient stability condition of Nickl&Wang (JEMS 2024) for the underlying non-linear inverse problem, providing also injectivity estimates and corresponding statistical identifiability results.

Abstract

We consider general parameter to solution maps of non-linear partial differential equations and describe an approach based on a Banach space version of the implicit function theorem to verify the gradient stability condition of Nickl&Wang (JEMS 2024) for the underlying non-linear inverse problem, providing also injectivity estimates and corresponding statistical identifiability results. We illustrate our methods in two examples involving a non-linear reaction diffusion system as well as a McKean--Vlasov interacting particle model, both with periodic boundary conditions. We apply our results to prove the polynomial time convergence of a Langevin-type algorithm sampling the posterior measure of the interaction potential arising from a discrete aggregate measurement of the interacting particle system.
Paper Structure (24 sections, 22 theorems, 201 equations)

This paper contains 24 sections, 22 theorems, 201 equations.

Table of Contents

  1. Introduction
  2. General framework
  3. Main results
  4. Notation
  5. Computing gradients in nonlinear PDE models
  6. Reaction-diffusion equation
  7. McKean--Vlasov equation
  8. Properties of the map f
  9. Global stability estimates and pseudo-linearisation identity
  10. Linearisation and higher derivatives
  11. Polynomial-time posterior sampling via Langevin-type algorithms
  12. Setting
  13. Approximation spaces
  14. Derivatives of the likelihood function
  15. Local properties of the likelihood function
  16. ...and 9 more sections

Key Result

Theorem 2.1

Let $d \leq 3$. The map $\mathcal{G}: C^{2}_b(\mathbb R) \to L^{2}([0,T];H^{2})\cap H^{1}([0,T];L^{2})$ defined by $\mathcal{G}(R) = u_R$ solving eq:reaction-diffusion is $C^1$ in the Fréchet sense. Moreover, for any $R\in C^{2}_b(\mathbb R)$, its Fréchet derivative at $R$, is given by the linear map $D\mathcal{G}(R)[H] \eqqcolon i_H$ where $i=i_H\in L^{2}([0,T];H^{2}) \cap H^{1}([0,T];L^{2})

Theorems & Definitions (57)

  • Claim
  • Claim : Derivative of reaction-diffusion forward map
  • Claim : Gradient of McKean--Vlasov forward map
  • Claim : Polynomial mixing of ULA
  • Theorem 2.1: Linearisation of the reaction-diffusion equation
  • proof : Proof of Theorem \ref{['thm:linearisation-reaction-diffusion']}
  • Remark 2.2: Stability of the gradient
  • Lemma 2.4
  • proof
  • Proposition 2.5: Derivatives of $f$
  • ...and 47 more