Iterated Lavrentiev regularization under random noise: variational source conditions and rates in expectation
Abstract
The convergence theory of iterated Lavrentiev regularization for nonlinear monotone ill-posed equations is deterministic: the data error is bounded by a known level that the discrepancy principle uses explicitly. Under Gaussian noise no such bound exists. For a discretized two-dimensional nonlinear inverse source problem for the Poisson equation, we analyse convergence in a finite-energy Gaussian noise model, keeping the linearized and nonlinear cases apart. For the linearized iteration the noisy and noise-free iterates differ by one resolvent applied to the noise; this gives an exact singular-value variance formula, a priori rates in expectation, and the main result: for a family of discretizations whose distance functions are bounded uniformly in the mesh, the rate is governed by that decay, which relaxes the classical range condition. The rate degrades gracefully, down to logarithmic, as the underlying continuum error loses smoothness, and a balancing rule attains it without knowing the smoothness. For the full nonlinear iteration under the order-one source condition, a truncated discrepancy principle fed only the noise level attains σE1/2 in expectation, in the energy noise level σEasympσ/h, the rare event of large noise being controlled by concentration. The rough nonlinear regime and the general-exponent nonlinear rate are left open and supported numerically for kink and jump errors. We separate the deterministic worst-case, Gaussian, and white-noise settings, prove a worst-case lower bound, and describe where the method is statistically inefficient relative to spectral cut-off. Heterogeneous media and a cost comparison with an adjoint-based scheme are given as supplementary material.
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BibTeX
@misc{singh2026iterated,
title = {{Iterated Lavrentiev regularization under random noise: variational source conditions and rates in expectation}},
author = {Ankit Singh and Hemant Sharma and Kiran Davuluri},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26965},
eprint = {2610.26965},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26965},
}
RIS
TY - UNPB
TI - Iterated Lavrentiev regularization under random noise: variational source conditions and rates in expectation
AU - Ankit Singh
AU - Hemant Sharma
AU - Kiran Davuluri
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.26965
UR - https://arxiv.org/abs/2610.26965
AB - The convergence theory of iterated Lavrentiev regularization for nonlinear monotone ill-posed equations is deterministic: the data error is bounded by a known level that the discrepancy principle uses explicitly. Under Gaussian noise no such bound exists. For a discretized two-dimensional nonlinear inverse source problem for the Poisson equation, we analyse convergence in a finite-energy Gaussian noise model, keeping the linearized and nonlinear cases apart. For the linearized iteration the noisy and noise-free iterates differ by one resolvent applied to the noise; this gives an exact singular-value variance formula, a priori rates in expectation, and the main result: for a family of discretizations whose distance functions are bounded uniformly in the mesh, the rate is governed by that decay, which relaxes the classical range condition. The rate degrades gracefully, down to logarithmic, as the underlying continuum error loses smoothness, and a balancing rule attains it without knowing the smoothness. For the full nonlinear iteration under the order-one source condition, a truncated discrepancy principle fed only the noise level attains σ_E^1/2 in expectation, in the energy noise level σ_Easympσ/h, the rare event of large noise being controlled by concentration. The rough nonlinear regime and the general-exponent nonlinear rate are left open and supported numerically for kink and jump errors. We separate the deterministic worst-case, Gaussian, and white-noise settings, prove a worst-case lower bound, and describe where the method is statistically inefficient relative to spectral cut-off. Heterogeneous media and a cost comparison with an adjoint-based scheme are given as supplementary material.
ER -