On the convergence analysis of one-shot inversion methods

TL;DR

The paper analyzes convergence of one-shot and multi-step one-shot inversion schemes for linear inverse problems where the forward/adjoint problems are solved by fixed-point iterations. It derives eigenvalue-based sufficient conditions on the outer descent step $ au$, providing explicit bounds that depend on inner-iteration count $k$, operator norms of $B$, $M$, and $H$, and the regularization parameter $eta$, ensuring convergence even with incomplete inner solves. The analysis covers both real and complex eigenvalues and extends to semi-implicit schemes, with bounds that are explicit in $k$ and independent of problem dimension. Numerical experiments on a toy Helmholtz inverse problem show that a small number of inner iterations suffices to achieve convergence comparable to classical gradient descent, even in the presence of noise, highlighting the practical efficiency of these methods for large-scale inverse problems.

Abstract

When an inverse problem is solved by a gradient-based optimization algorithm, the corresponding forward and adjoint problems, which are introduced to compute the gradient, can be also solved iteratively. The idea of iterating at the same time on the inverse problem unknown and on the forward and adjoint problem solutions yields the concept of one-shot inversion methods. We are especially interested in the case where the inner iterations for the direct and adjoint problems are incomplete, that is, stopped before achieving a high accuracy on their solutions. Here, we focus on general linear inverse problems and generic fixed-point iterations for the associated forward problem. We analyze variants of the so-called multi-step one-shot methods, in particular semi-implicit schemes with a regularization parameter. We establish sufficient conditions on the descent step for 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 gradient descent, where the forward and adjoint problems are solved exactly by a direct solver instead. We observe that very few inner iterations are enough to guarantee good convergence of the inversion algorithm, even in the presence of noisy data.

Figures (11)

• Figure 1: Illustration of the regions in the complex plane associated with the cases 1, 2 and 3 indicated in the proof of Proposition \ref{['prop:seim-1-shot:tau:complex-lam']} for $\theta_0=\frac{\pi}{4}$. The center circle is the unit circle.
• Figure 2: The configuration for the experiment in Section \ref{['sec:exp1']}. The small circles outside the mesh for $u$ indicate the source locations $y_i$, $1\le i\le 6$.
• Figure 3: Convergence curves of gradient descent and $k$-step one-shot: dependence on the descent step $\tau$.
• Figure 4: Convergence curves of gradient descent and $k$-step one-shot: dependence on the number of inner iterations $k$.
• Figure 5: The configuration for the experiment in Section \ref{['sec:exp2']}. The small circles outside the mesh for $u$ indicate the source locations $y_i$, $1\le i\le 6$.

Theorems & Definitions (43)

• Remark 2.1
• Proposition 3.1
• proof
• Proposition 3.2: Real eigenvalues
• proof
• Proposition 3.3
• proof
• Lemma 3.4
• Proposition 3.5: Complex eigenvalues
• proof