One Subspace, Many Residuals: Columnwise Stopping and Rank Deflation for Block Projected Iterated Tikhonov Regularization

Source abstract

Discrete ill-posed problems with several right-hand sides and a common forward operator arise in multichannel imaging and in repeated or parameter-dependent experiments. A block Golub–Kahan reduction builds one solution subspace for all right-hand sides, and iterated Tikhonov regularization can then be applied entirely inside the reduced problem. Combining the two, however, creates a difficulty that neither ingredient has on its own: a single iteration index must serve s residuals whose noise levels and effective ranks differ, so the index selected by a Frobenius discrepancy principle is generally not the index any individual column needs. Using the elementary fact that the reduced iterated-Tikhonov recursion decouples exactly over the columns of the reduced data, per-column stopping indices are available without additional operator applications, and we prove a dichotomy: unless all columns share a stopping index, the common Frobenius index either violates at least one column’s discrepancy condition or iterates other columns past their own, and the columnwise indices are the unique componentwise-minimal feasible choice. These statements concern discrepancy feasibility, not reconstruction error. We also give a noise-aware deflation rule under which the initial block size contracts to the numerical rank of the data at the noise level, quantify the per-column risk it creates and give a noise-whitened variant that bounds the discarded part of every column by a fixed fraction of that column’s own noise level, and prove that the deflated block sizes are nonincreasing. Locating each column’s discrepancy index requires care: a fixed iteration budget can silently misreport a column as unresolved when its true crossing index is merely large rather than infinite, so we locate it by bisection on the closed-form filter factors, followed by an integer certification step, instead of a truncated grid search. On five standard test problems, columnwise stopping reduces the reconstruction error by 10.6% to 24.7% over the common index (by 13% to 40% on average over twelve solution-and-noise realizations, with noise levels spanning two decades) with no additional operator applications, and deflation reduces the operator count by up to a factor of eight when the right-hand sides span a low-dimensional subspace. The columnwise principle is not specific to the iteration: it applies equally to one-step Tikhonov regularization with per-column parameters, and in our tests one-step Tikhonov with a discrepancy-selected parameter was at least as accurate as iterated Tikhonov at a fixed a priori parameter. A deliberately controlled comparison also corrects a claim that is easy to make and that we could not sustain: once each method is stopped by the same columnwise discrepancy principle, sharing a block subspace confers no accuracy advantage over independent solves without deflation, both reach the same error floor. The advantage is one of operator count: with matched stopping rules the block subspace reaches the accuracy of the independent solves with 3.0 to 5.0 times fewer operator applications on all five problems. With the common Frobenius index instead, the block method misses that accuracy on two of the five problems, so an unmatched comparison confounds the effect of the subspace with that of the stopping rule.