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Bi-level iterative regularization for inverse problems in nonlinear PDEs

Tram Thi Ngoc Nguyen

TL;DR

The paper tackles the ill-posed problem of identifying spatially varying coefficients in nonlinear time-dependent PDEs by designing a bi-level Landweber method that embeds a lower-level state approximation into the outer parameter-recovery iteration. The outer (upper) level updates the parameter $θ$ using an approximate state $\widetilde{S}_{K(j)}$, while the inner (lower) level solves for the state via a Landweber scheme; convergence relies on coercivity of the PDE model and a tangential cone condition to control nonlinear effects. The work provides explicit error propagation analyses for the state-approximation error $ε_{K(j)}$, develops both a posteriori and a priori stopping rules, and proves convergence of the bi-level scheme as the data noise $δ$ tends to zero. It further discusses the applicability to nonlinear, time-dependent problems and illustrates the approach in magnetic particle imaging through the Landau-Lifshitz-Gilbert equation, highlighting practical relevance and potential for integration with reduced-order models. Overall, the bi-level framework blends reduced and all-at-once formulations to enable regularized identification without requiring exact PDE solves, offering a robust tool for inverse coefficient problems in complex PDE systems.

Abstract

We investigate the ill-posed inverse problem of recovering unknown spatially dependent parameters in nonlinear evolution PDEs. We propose a bi-level Landweber scheme, where the upper-level parameter reconstruction embeds a lower-level state approximation. This can be seen as combining the classical reduced setting and the newer all-at-once setting, allowing us to, respectively, utilize well-posedness of the parameter-to-state map, and to bypass having to solve nonlinear PDEs exactly. Using this, we derive stopping rules for lower- and upper-level iterations and convergence of the bi-level method. We discuss application to parameter identification for the Landau-Lifshitz-Gilbert equation in magnetic particle imaging.

Bi-level iterative regularization for inverse problems in nonlinear PDEs

TL;DR

The paper tackles the ill-posed problem of identifying spatially varying coefficients in nonlinear time-dependent PDEs by designing a bi-level Landweber method that embeds a lower-level state approximation into the outer parameter-recovery iteration. The outer (upper) level updates the parameter using an approximate state , while the inner (lower) level solves for the state via a Landweber scheme; convergence relies on coercivity of the PDE model and a tangential cone condition to control nonlinear effects. The work provides explicit error propagation analyses for the state-approximation error , develops both a posteriori and a priori stopping rules, and proves convergence of the bi-level scheme as the data noise tends to zero. It further discusses the applicability to nonlinear, time-dependent problems and illustrates the approach in magnetic particle imaging through the Landau-Lifshitz-Gilbert equation, highlighting practical relevance and potential for integration with reduced-order models. Overall, the bi-level framework blends reduced and all-at-once formulations to enable regularized identification without requiring exact PDE solves, offering a robust tool for inverse coefficient problems in complex PDE systems.

Abstract

We investigate the ill-posed inverse problem of recovering unknown spatially dependent parameters in nonlinear evolution PDEs. We propose a bi-level Landweber scheme, where the upper-level parameter reconstruction embeds a lower-level state approximation. This can be seen as combining the classical reduced setting and the newer all-at-once setting, allowing us to, respectively, utilize well-posedness of the parameter-to-state map, and to bypass having to solve nonlinear PDEs exactly. Using this, we derive stopping rules for lower- and upper-level iterations and convergence of the bi-level method. We discuss application to parameter identification for the Landau-Lifshitz-Gilbert equation in magnetic particle imaging.
Paper Structure (24 sections, 12 theorems, 121 equations, 1 figure, 1 algorithm)

This paper contains 24 sections, 12 theorems, 121 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Abstract.
  2. Parameter identification
  3. Introduction
  4. Regularization with state approximation error
  5. Introducing the lower-level
  6. Bi-level as a mixed approach
  7. Connection and contrast to other directions
  8. Contribution
  9. Structure
  10. Problem setting
  11. Bi-level iteration
  12. Bi-level algorithm
  13. Preliminary error analysis
  14. The lower-level iteration
  15. The upper-level iteration
  16. ...and 9 more sections

Key Result

Proposition 1

Let $D_U:U\to U^*$ and $I_X:X^*\to X$ be isomorphisms. Then, the Hilbert space adjoint of $G'$ with $G$ defined in original-red-forward-original-red-measure is given by where $z$ is the solution to the final value problem at source $D_Uv$ Here, the Banach space adjoints are

Figures (1)

  • Figure 1: Left: Lower-level, applied field $\textbf{h}(t)$ (left), initial state $\textbf{m}_0$ (middle), approximate trajectory $\textbf{m}(t)$ (right). Top right: Bi-level convergence via reconstruction. Bottom right: Number of iterations for bi-level. 250 total upper iterations (x-axis), corresponding number of lower-iterations (y-axis). These figures are taken from NguyenWald:2022.

Theorems & Definitions (34)

  • Proposition 1: Adjoint
  • proof
  • Remark 1
  • Lemma 1: System output error
  • proof
  • Remark 2
  • Lemma 2: Adjoint state error
  • proof
  • Theorem 1: Convergence and rate
  • Remark 3: On assumptions of Theorem \ref{['low-conv-rate']}
  • ...and 24 more