Error analysis of a multisingular inverse coefficient problem for the Sturm–Liouville operator based on boundary measurement

摘要

In this paper we study the inverse problem of determining the unknown leading coefficient k=k(x) of the linear Sturm–Liouville operator Au=−(k(x)u′(x))′+q(x)u(x), x∈(0,1), from boundary measurements, when u′(x) or/and u′′(x) vanishes at several, called singular, points of the interval (0,1). As a result the considered inverse problem has simultaneously different types (moderate or severely) of ill-conditioned situations in different parts of the interval (0,1). The presented inverse polynomial method permits use of a priori information about singular points either to increase the order of the polynomial approximation in each subinterval or to obtain an artificial Cauchy data for the unknown coefficient. Error estimations for the polynomial approximations are presented for well-conditioned, as well as for ill-conditioned situations. The behaviour of the inverse problem solution with respect to both types (Dirichlet and Neumann) of noisy data is analyzed. The obtained results are illustrated by various numerical examples.