Telescoping Methods for Inverse Problems and Operator Identification A Regularization-Preserving Framework

We develop a telescoping approximation framework for inverse problems, where unknown parameters,operators, or distributed sources must be inferred from indirect, incomplete, or noisyobservations. Rather than treating the inverse map as an atomic black box, we systematicallyembed telescoping structure into forward operators and analyze how incremental refinementpropagates through regularized inversion procedures, including Tikhonov regularization, iterativemethods, and variational schemes.We establish that when forward operators admit telescoping approximants with controlledincremental decay of order k, the associated inverse reconstructions inherit stable telescopingbehavior under standard regularization schemes, with explicit convergence rates. This yieldsrigorous bias–variance tradeoffs, controlled error propagation through the inverse mapping, andconvergence guarantees for inverse problems governed by differential equations, integral equations,evolution operators, and nonlinear parameter identification.The framework encompasses:• Abstract inverse problem theory with telescoping forward operators• Tikhonov regularization with parameter selection strategies• Source conditions and convergence rates under varying regularity• Iterative methods (Landweber, conjugate gradients) with telescoping• Inverse problems for parabolic and hyperbolic PDEs• Fredholm integral equations and deconvolution• Nonlinear inverse problems and Fr´echet differentiability• Operator identification from time-series and dynamical data• Bayesian inverse problems with telescoping forward models• Numerical algorithms and computational complexity analysisApplications include parameter identification in PDEs, coefficient recovery in elliptic andparabolic equations, inverse scattering, deconvolution in signal processing, inverse heat conduction,system identification in control theory, and data-driven operator recovery from trajectoryobservations.