On optimal recovery of unbounded operators from inaccurate data
Oleg Davydov, Sergei Solodky
TL;DR
The paper addresses the problem of optimally recovering $Af$ from noisy data $f^ abla$ for unbounded self-adjoint operators $A$ on a real Hilbert space, with solution smoothness encoded by the weighted space $W$. It introduces a truncation-based regularization, derives sharp lower bounds via information-complexity arguments, and proves that the truncation method achieves order-optimal recovery under mild monotonicity conditions on $ ho_k/\xi_k$, with explicit error formulas $igl(igl[ ho_k^2/\xi_k^2igr]_{k o N} + abla^2 ho_{N-1}^2igr)^{1/2}$. The results extend to Banach-space metrics $L_q$ and are illustrated on numerical differentiation and the backward parabolic equation, providing precise guidelines for discretization parameters $N_ abla$ in ill-posed settings. Overall, the work connects information-theoretic limits with practical regularization strategies, yielding order-optimal procedures for a broad class of ill-posed problems and their concrete numerical realizations.
Abstract
The problems of optimal recovery of unbounded operators are studied. Optimality means the highest possible accuracy and the minimal amount of discrete information involved. It is established that the truncation method, when certain conditions are met, realizes the optimal values of the studied quantities. As an illustration of the general results, problems of numerical differentiation and the backward parabolic equation are considered.