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An adaptive heavy ball method for ill-posed inverse problems

Qinian Jin, Qin Huang

TL;DR

We address ill-posed inverse problems $F(x)=y$ by developing an adaptive heavy-ball method that augments a Landweber-type update with a momentum term and employs a strongly convex regularizer ${\mathcal{R}}$ to enforce desirable solution features. Explicit, backtracking-free rules for the step-size $\alpha_n^\delta$ and momentum coefficient $\beta_n^\delta$ are derived, leveraging a computable surrogate and a dual formulation to relate to a heavy-ball update; the per-iteration cost remains close to a single Landweber step. The method is shown to possess a regularization property when termination is governed by the discrepancy principle, with weak convergence in general and strong convergence to the minimum-norm solution under additional assumptions. Numerical experiments across linear integral equations, computed tomography, and elliptic parameter identification demonstrate substantial reductions in iteration counts and computational time compared with Landweber, ν-method, and Nesterov acceleration, validating the practical acceleration potential of the approach.

Abstract

In this paper we consider ill-posed inverse problems, both linear and nonlinear, by a heavy ball method in which a strongly convex regularization function is incorporated to detect the feature of the sought solution. We develop ideas on how to adaptively choose the step-sizes and the momentum coefficients to achieve acceleration over the Landweber-type method. We then analyze the method and establish its regularization property when it is terminated by the discrepancy principle. Various numerical results are reported which demonstrate the superior performance of our method over the Landweber-type method by reducing substantially the required number of iterations and the computational time.

An adaptive heavy ball method for ill-posed inverse problems

TL;DR

We address ill-posed inverse problems by developing an adaptive heavy-ball method that augments a Landweber-type update with a momentum term and employs a strongly convex regularizer to enforce desirable solution features. Explicit, backtracking-free rules for the step-size and momentum coefficient are derived, leveraging a computable surrogate and a dual formulation to relate to a heavy-ball update; the per-iteration cost remains close to a single Landweber step. The method is shown to possess a regularization property when termination is governed by the discrepancy principle, with weak convergence in general and strong convergence to the minimum-norm solution under additional assumptions. Numerical experiments across linear integral equations, computed tomography, and elliptic parameter identification demonstrate substantial reductions in iteration counts and computational time compared with Landweber, ν-method, and Nesterov acceleration, validating the practical acceleration potential of the approach.

Abstract

In this paper we consider ill-posed inverse problems, both linear and nonlinear, by a heavy ball method in which a strongly convex regularization function is incorporated to detect the feature of the sought solution. We develop ideas on how to adaptively choose the step-sizes and the momentum coefficients to achieve acceleration over the Landweber-type method. We then analyze the method and establish its regularization property when it is terminated by the discrepancy principle. Various numerical results are reported which demonstrate the superior performance of our method over the Landweber-type method by reducing substantially the required number of iterations and the computational time.
Paper Structure (7 sections, 10 theorems, 114 equations, 5 figures, 3 tables, 2 algorithms)

This paper contains 7 sections, 10 theorems, 114 equations, 5 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. The adaptive heavy-ball method
  3. Assumptions
  4. The method
  5. Convergence analysis
  6. Numerical results
  7. Conclusion

Key Result

Lemma 2.1

Let Assumption ass0 and Assumption ass1 hold. Consider Algorithm alg:AHB and let $k\ge 0$ be an integer such that $x_n^\delta \in B_{2\rho}(x_0)$ for all $0\le n < k$. Define $\gamma_n^\delta$ by (AHB.3) for $0\le n\le k$. Then $\gamma_n^\delta \le \tilde{\gamma}_n^\delta$ for all $0\le n \le k$.

Figures (5)

  • Figure 1: Relative error versus the number of iterations for Example \ref{['ex1']}.
  • Figure 2: The reconstruction results for computed tomography. (a) True phantom; (b) Landweber; (c) AHB with $\beta = 0.99$; (d) AHB with $\beta = \infty$.
  • Figure 3: Computed tomography: relative errors versus the number of iterations.
  • Figure 4: The reconstruction results for parameter identification in elliptic equation. (a) True parameter; (b) Landweber; (c) AHB with $\beta = 0.99$; (d) AHB with $\beta = \infty$.
  • Figure 5: relative errors versus the number of iterations.

Theorems & Definitions (22)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 12 more