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Convergence analysis of multi-step one-shot methods for linear inverse problems

Marcella Bonazzoli, Houssem Haddar, Tuan Anh Vu

TL;DR

This work addresses linear inverse problems where the forward model is solved by fixed-point iterations and the inverse update is performed simultaneously via multi-step one-shot schemes. The authors conduct a rigorous eigenvalue-based convergence analysis of two variants (k-step and shifted k-step), deriving explicit bounds on the descent step $\tau$ that depend on the number of inner iterations $k$ and on operator norms of the forward/adjoint operators, ensuring all eigenvalues lie inside the unit disk. The analysis covers both real and complex eigenvalues, extends to complex forward problems via a real-imaginary decomposition, and provides scalar-case results that are necessary and sufficient for convergence. Numerical experiments on a toy Helmholtz inverse problem illustrate that very few inner iterations suffice for good convergence and that moderate $k$ can even accelerate convergence relative to classical gradient descent. The findings offer practical guidance for choosing inner iterations and step sizes in large-scale linear inverse problems and lay groundwork for extension to nonlinear and time-dependent settings.

Abstract

In this work we are interested in general linear inverse problems where the corresponding forward problem is solved iteratively using fixed point methods. Then one-shot methods, which iterate at the same time on the forward problem solution and on the inverse problem unknown, can be applied. We analyze two variants of the so-called multi-step one-shot methods and establish sufficient conditions on the descent step for their convergence, by studying the eigenvalues of the block matrix of the coupled iterations. Several numerical experiments are provided to illustrate the convergence of these methods in comparison with the classical usual and shifted gradient descent. In particular, we observe that very few inner iterations on the forward problem are enough to guarantee good convergence of the inversion algorithm.

Convergence analysis of multi-step one-shot methods for linear inverse problems

TL;DR

This work addresses linear inverse problems where the forward model is solved by fixed-point iterations and the inverse update is performed simultaneously via multi-step one-shot schemes. The authors conduct a rigorous eigenvalue-based convergence analysis of two variants (k-step and shifted k-step), deriving explicit bounds on the descent step that depend on the number of inner iterations and on operator norms of the forward/adjoint operators, ensuring all eigenvalues lie inside the unit disk. The analysis covers both real and complex eigenvalues, extends to complex forward problems via a real-imaginary decomposition, and provides scalar-case results that are necessary and sufficient for convergence. Numerical experiments on a toy Helmholtz inverse problem illustrate that very few inner iterations suffice for good convergence and that moderate can even accelerate convergence relative to classical gradient descent. The findings offer practical guidance for choosing inner iterations and step sizes in large-scale linear inverse problems and lay groundwork for extension to nonlinear and time-dependent settings.

Abstract

In this work we are interested in general linear inverse problems where the corresponding forward problem is solved iteratively using fixed point methods. Then one-shot methods, which iterate at the same time on the forward problem solution and on the inverse problem unknown, can be applied. We analyze two variants of the so-called multi-step one-shot methods and establish sufficient conditions on the descent step for their convergence, by studying the eigenvalues of the block matrix of the coupled iterations. Several numerical experiments are provided to illustrate the convergence of these methods in comparison with the classical usual and shifted gradient descent. In particular, we observe that very few inner iterations on the forward problem are enough to guarantee good convergence of the inversion algorithm.
Paper Structure (26 sections, 47 theorems, 258 equations, 5 figures)

This paper contains 26 sections, 47 theorems, 258 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Multi-step one-shot inversion methods
  3. Convergence of one-step one-shot methods ($k=1$)
  4. Block iteration matrices and eigenvalue equations
  5. Real eigenvalues
  6. Complex eigenvalues
  7. Final result ($k=1$)
  8. Convergence of multi-step one-shot methods ($k\ge 2$)
  9. Block iteration matrices and eigenvalue equations
  10. Real eigenvalues
  11. Complex eigenvalues
  12. Final result ($k\ge 2$)
  13. Inverse problem with complex forward problem and real parameter
  14. Numerical experiments
  15. Conclusion
  16. ...and 11 more sections

Key Result

Proposition 3.1

Assume that $\lambda\in\mathbb{C}$ is an eigenvalue of the iteration matrix in 1-shot-itermat n.

Figures (5)

  • Figure 1: Domain with six source points for the numerical experiments. The unknown $\sigma$ is supported on the three squares.
  • Figure 2: Convergence curves of usual gradient descent and $k$-step one-shot.
  • Figure 3: Convergence curves of shifted gradient descent and shifted $k$-step one-shot.
  • Figure 4: Comparison of usual gradient descent and $k$-step one-shot with shifted gradient descent and shifted $k$-step one-shot.
  • Figure 5: Admissible $\tau$ in the scalar case as a function of $b$.

Theorems & Definitions (90)

  • Proposition 3.1: Eigenvalue equation for the shifted $1$-step one-shot method
  • Remark 3.2
  • proof
  • Proposition 3.3: Eigenvalue equation for the $1$-step one-shot method
  • proof
  • Proposition 3.4: shifted $1$-step one-shot method
  • proof
  • Proposition 3.5: $1$-step one-shot method
  • proof
  • Proposition 3.6: shifted $1$-step one-shot method
  • ...and 80 more