Projected iterated Tikhonov regularization in low precision

TL;DR

The paper addresses solving linear inverse problems that are ill-posed in low-precision environments by introducingProjected Iterated Tikhonov (PIT) regularization built on Golub-Kahan bidiagonalization. It reframes PIT as a projected preconditioned Landweber method with a Tikhonov-type preconditioner, derives stationary and nonstationary (adaptive) filter factors, and uses a secant-discrepancy principle to update the regularization parameter per iteration. Through numerical experiments on 1D and 2D ill-posed problems, it demonstrates that PIT in low precision can achieve working accuracy comparable to fp64 when a sufficiently large Krylov subspace is formed and reorthogonalization is applied, highlighting practical viability for efficient, high-performance inverse problems. The results underscore the importance of basis orthogonality in low-precision Krylov methods and provide a robust framework for spectral filtering and adaptive regularization in reduced precision computing.

Abstract

We investigate the regularizing behavior of an iterative Krylov subspace method for the solution of linear inverse problems in precisions lower than double. Recent works have considered the projection of iterated Tikhonov methods using Krylov subspaces for both computational efficiency and an additional regularizing effect. To investigate the regularizing behavior of this projected algorithm applied to problems that are naturally severely ill-posed, we formulate the iterates as a filtered solution using the preconditioned Landweber method with a Tikhonov-type preconditioner in a Krylov subspace. Through numerical examples simulating multiple low precision choices, we showcase the filtering properties of the method and the achievement of comparable working accuracy applied to discrete inverse problems (i.e., to within a few decimal places in relative error) compared to results computed in traditional double precision.

Figures (4)

• Figure 1: Spectra example: effective filter factors for $25$ iterations of PIT in fp64 with a subspace of size $p=30$ and with $3\%$ noise. Each iteration represents $64$ filter factors whose values are denoted by $\omega_i^{(k)}$ and whose values may be read by the colorbar to the right or the value of the vertical axis.
• Figure 2: 2D image reconstruction examples: (A) true Hubble image ($256\times 256$ pixels), (B) Gauss PSF, (C) Defocus PSF, (D) Hubble image blurred by Gauss with $1\%$ noise ($256\times 256$ pixels), and (E) Hubble image blurred by Defocus with $1\%$ noise ($256\times 256$ pixels).
• Figure 3: Relative residual (top row) and relative error (bottom row) against iteration number for the PIT method (with $30$ steps of GKB) applied to the Gauss test case with $3\%$ noise, shown for the three precision levels: fp64, fp32, and fp16. The PIT method applied with reorthogonalization in the GKB process is represented by the solid magenta line and magenta circles. The PIT method without reorthogonalization is represented by the dashed blue line and blue asterisks. In the relative residual plots, the black dashed horizontal lines correspond to the discrepancy principle's breakout threshold. The red circles indicate the iteration at which the breakout criterion is satisfied.
• Figure 4: 2D image reconstructions for the PIT method applied to the Defocus example with $1\%$ noise, $35$ GKB steps, and precision levels fp64, fp32, and fp16.

Theorems & Definitions (5)

• Proposition 1
• proof
• Proposition 2
• proof
• Corollary 1