Regularization Methods in Inverse Problem Solving

Inverse problems involve the recovery of unknown parameters or signals from indirect and often noisy measurements. Such problems are typically ill-posed in the sense that small perturbations in the data can lead to large errors in the solution, or the solution may not be unique or may not depend continuously on the data. Regularisation introduces additional information or constraints—such as smoothness, sparsity or convexity—to restore well-posedness and ensure stable reconstructions. Classical approaches include Tikhonov regularisation, which penalises the norm of the solution, and variational methods that employ more general convex penalties such as total variation or Besov norms. Iterative regularisation methods embed the regularisation within an optimisation loop, allowing for a natural stopping criterion to control noise amplification. Modern developments have extended these ideas to data-driven and projection-based schemes that learn structure from training pairs, as well as acceleration techniques inspired by advanced optimisation theory. Parameter-choice strategies, including discrepancy principles and a priori rules, remain central to balancing fidelity to measurements against enforcement of prior assumptions. Together, these methods underpin applications in medical imaging, seismic tomography, remote sensing and beyond, offering a rigorous framework for stable recovery in the presence of uncertainty.

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A new strategy for analysing classical regularisation has shed light on the fundamental mechanisms of Tikhonov regularisation, revealing that approximation of the exact solution and noise-free and noisy data can all be treated within a unified source-space framework. This work clarifies the interplay between convergence rates, parameter choice and saturation effects, and demonstrates the applicability of the approach to iterative schemes such as Landweber iteration.

Advances in optimisation-based acceleration have shown that Nesterov’s accelerated gradient method can serve as an optimal-order iterative regularisation technique for linear inverse problems. By selecting the acceleration parameter in accordance with solution smoothness, this method achieves provably optimal convergence rates under both a priori stopping rules and discrepancy principles, uniting classical source conditions with modern polynomial spectral analysis.

Data-driven projection methods now permit regularisation without explicit knowledge of the forward operator. By formulating projection and variational penalties directly from training pairs, these schemes reconstruct linear operators—such as the Radon transform—through purely empirical means. Theoretical results establish convergence and stability, while numerical experiments confirm the capacity to learn inverse mappings in practice.

Regularization Methods in Inverse Problem Solving publication trend

The graph below shows the total number of articles in regularization methods in inverse problem solving across all publications each year (not limited to Nature Index journals).

Technical terms

Inverse problem: A problem of estimating unknown causes from observed effects, often modelled by an operator equation linking parameters to measurements.

Regularisation: A strategy for stabilising ill-posed problems by incorporating prior information or constraints into the solution process.

Tikhonov regularisation: A penalised least-squares approach that adds a term proportional to the norm of the solution to control smoothness and stability.

Variational regularisation: A general class of methods that minimise a sum of a data-fidelity term and a convex penalty, allowing for non-quadratic norms or functionals.

Iterative regularisation: Techniques that embed regularisation within an iterative solver, using early stopping or adaptive rules to prevent overfitting to noise.

Source condition: An assumption on the smoothness or structure of the true solution, often expressed in terms of membership in the range of an operator or its adjoint.

Discrepancy principle: A parameter-choice method that selects the regularisation strength so that the model misfit matches the estimated noise level.

References

1. A new interpretation of (Tikhonov) regularization. Inverse Problems (2021).
2. Optimal-order convergence of Nesterov acceleration for linear ill-posed problems* *Dedicated to A Neubauer on the occasion of his 60th birthday.. Inverse Problems (2021).
3. Data driven regularization by projection. Inverse Problems (2020).