Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On inertial iterated Tikhonov methods for solving ill-posed problems

Joel C. Rabelo, Antonio Leitão, Alexandre L. Madureira

TL;DR

The paper addresses solving ill-posed linear equations $A x = y$ from noisy measurements by introducing the inertial iterated Tikhonov (iniT) method, an implicit two-point extension of the iterated Tikhonov approach. At each iteration, it extrapolates to $w_k = x_k + \alpha_k (x_k - x_{k-1})$ and computes $x_{k+1}$ by minimizing $\lambda_k \|A x - y^\delta\|^2 + \|x - w_k\|^2$, equivalently solving $(I + \lambda_k A^* A) x_{k+1} = w_k + \lambda_k A^* y^\delta$. The authors prove convergence for exact data under suitable conditions on $\alpha_k$ and $\lambda_k$, and establish stability and semi-convergence with a discrepancy principle for noisy data when $\alpha_k^\delta$ are chosen with summable $\theta_k$. Numerical experiments on a 2D Inverse Potential Problem and an Image Deblurring Problem show that iniT reduces the number of inner CG steps and overall iterations compared to iT, and remains competitive with explicit two-point schemes (Nesterov, FISTA). These results indicate that inertial two-point proximal updates are effective regularization tools for linear ill-posed problems, offering practical computational advantages in PDE- and integral-operator settings.

Abstract

In this manuscript we propose and analyze an implicit two-point type method (or inertial method) for obtaining stable approximate solutions to linear ill-posed operator equations. The method is based on the iterated Tikhonov (iT) scheme. We establish convergence for exact data, and stability and semi-convergence for noisy data. Regarding numerical experiments we consider: i) a 2D Inverse Potential Problem, ii) an Image Deblurring Problem; the computational efficiency of the method is compared with standard implementations of the iT method.

On inertial iterated Tikhonov methods for solving ill-posed problems

TL;DR

The paper addresses solving ill-posed linear equations from noisy measurements by introducing the inertial iterated Tikhonov (iniT) method, an implicit two-point extension of the iterated Tikhonov approach. At each iteration, it extrapolates to and computes by minimizing , equivalently solving . The authors prove convergence for exact data under suitable conditions on and , and establish stability and semi-convergence with a discrepancy principle for noisy data when are chosen with summable . Numerical experiments on a 2D Inverse Potential Problem and an Image Deblurring Problem show that iniT reduces the number of inner CG steps and overall iterations compared to iT, and remains competitive with explicit two-point schemes (Nesterov, FISTA). These results indicate that inertial two-point proximal updates are effective regularization tools for linear ill-posed problems, offering practical computational advantages in PDE- and integral-operator settings.

Abstract

In this manuscript we propose and analyze an implicit two-point type method (or inertial method) for obtaining stable approximate solutions to linear ill-posed operator equations. The method is based on the iterated Tikhonov (iT) scheme. We establish convergence for exact data, and stability and semi-convergence for noisy data. Regarding numerical experiments we consider: i) a 2D Inverse Potential Problem, ii) an Image Deblurring Problem; the computational efficiency of the method is compared with standard implementations of the iT method.
Paper Structure (9 sections, 9 theorems, 59 equations, 8 figures, 2 tables, 2 algorithms)

This paper contains 9 sections, 9 theorems, 59 equations, 8 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. The iteration
  3. Description of method
  4. The exact data case
  5. The noise data case
  6. Numerical experiments
  7. The Inverse Potential Problem
  8. The Image Deblurring Problem
  9. Final remarks and conclusions

Key Result

Lemma 2.2

Let $(x_k)$, $(w_k)$ be sequences generated by Algorithm alg:init-exact. Given $x \in X$, it holds

Figures (8)

  • Figure 1: IPP Noise level $0.1\%$. Results obtained by the iniT method (the stopping criterion is reached at $k^*(\delta) = 17$ steps). (TOP) Ground truth $x^\star$; (CENTER) Approximate solution $x_{17}^\delta$; (BOTTOM) Relative iteration error $|x_{17}^\delta-x^\star| / |x^\star|$.
  • Figure 2: IPP Noise level $0.1\%$. Comparison between iT and iniT methods. (TOP) Relative iteration error; (BOTTOM) Relative residual.
  • Figure 3: IPP Noise level $5.0\%$. Comparison between iT and iniT methods. (TOP) Relative iteration error; (BOTTOM) Relative residual.
  • Figure 4: IPP Noise level $0.1\%$. Comparison between iT method and iniT method with constant $\alpha_k$. (TOP) Relative iteration error; (BOTTOM) Relative residual.
  • Figure 5: IPP Noise level $5.0\%$. Comparison between iT method and iniT method with constant $\alpha_k$. (TOP) Relative iteration error; (BOTTOM) Relative residual.
  • ...and 3 more figures

Theorems & Definitions (24)

  • Remark 2.1
  • Lemma 2.2
  • proof
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Remark 2.7
  • Theorem 2.8: Convergence for exact data
  • proof
  • ...and 14 more