Core Concepts

The authors propose an adaptive heavy ball method to efficiently solve ill-posed inverse problems, both linear and nonlinear, by incorporating a strongly convex regularization function to detect the desired solution features. The method adaptively chooses the step-sizes and momentum coefficients to achieve acceleration over the standard Landweber-type method.

Abstract

The content discusses an ill-posed inverse problem of the form F(x) = y, where F could be a linear or nonlinear operator between Hilbert spaces X and Y. The authors consider the situation where the solution does not depend continuously on the data, which often occurs in practical applications due to noisy measurements.
The authors propose an adaptive heavy ball method (Algorithm 1) to solve this ill-posed inverse problem. The key aspects are:
Incorporating a strongly convex regularization function R to determine a solution with desired features (e.g., non-negativity, sparsity, piecewise constancy).
Adapting the step-sizes αδn and momentum coefficients βδn at each iteration to accelerate the convergence compared to the standard Landweber-type method.
Deriving the update formulas for αδn and βδn based on minimizing an upper bound of the Bregman distance between the current iterate and the true solution.
Proving that the proposed method is well-defined and the iteration terminates in finite steps using the discrepancy principle.
Establishing the weak and strong convergence of the method under suitable assumptions.
The authors also present Algorithm 2, which is the counterpart of Algorithm 1 using the exact data y, to facilitate the convergence analysis.
Extensive numerical results demonstrate the superior performance of the proposed adaptive heavy ball method over the Landweber-type method in terms of reducing the required number of iterations and computational time.

Stats

∥yδ - y∥ ≤ δ
∥L(x)∥ ≤ L for all x ∈ B2ρ(x0)
∥F(x) - F(x̄) - L(x̄)(x - x̄)∥ ≤ η∥F(x) - F(x̄)∥ for all x, x̄ ∈ B2ρ(x0)

Quotes

"In practical scenarios, data are acquired through experiments and thus the exact data may not be available; instead we only have measurement data corrupted by noise. Due to the ill-posed nature of the underlying problem, it is therefore important to develop algorithms to produce stable approximate solutions of (1) using noisy data."
"Extensive numerical simulations have demonstrated that the Landweber-type method (2) is a slowly convergent approach, often necessitating a large number of iteration steps before termination by the discrepancy principle. It is therefore important to develop strategies for accelerating the Landweber-type method (2) while maintaining its simple implementation feature."

Key Insights Distilled From

by Qinian Jin,Q... at **arxiv.org** 04-05-2024

Deeper Inquiries

To extend the proposed adaptive heavy ball method to handle other types of regularization functions beyond the strongly convex case, we can consider incorporating different types of regularization functions that are not strictly convex. One approach could be to modify the step-size and momentum coefficient selection criteria to accommodate the properties of the specific regularization function. For non-convex regularization functions, the optimization landscape may have multiple local minima, so the method may need to be adapted to navigate these challenges. Additionally, techniques such as stochastic approximation or adaptive learning rates could be employed to handle the non-convexity of the regularization function.

To strengthen the convergence analysis and establish quantitative rates of convergence for the adaptive heavy ball method, we can delve deeper into the properties of the optimization algorithm. By analyzing the behavior of the iterates, the convergence of the method can be studied in terms of the objective function, the regularization function, and the noise level. Techniques from optimization theory, such as Lyapunov functions, stability analysis, and convergence theorems, can be applied to derive quantitative convergence rates. By carefully examining the algorithm's dynamics and the properties of the problem, we can provide rigorous bounds on the convergence speed and accuracy of the method.

The adaptive heavy ball method has various potential applications in real-world ill-posed inverse problems across different domains such as image reconstruction, signal processing, medical imaging, and geophysics. In image reconstruction, the method can be tailored to handle challenges like noise reduction, artifact removal, and super-resolution imaging. For medical imaging, the method can assist in tasks like MRI reconstruction, CT image enhancement, and PET image reconstruction. In geophysics, the method can be used for seismic imaging, subsurface modeling, and inversion problems. To address the specific challenges in these applications, the method can be customized by incorporating domain-specific constraints, data characteristics, and regularization functions tailored to the problem at hand. By adapting the algorithm to the unique requirements of each application, the adaptive heavy ball method can offer efficient and accurate solutions to complex inverse problems.

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