An inverse problem for an operator equation Az = u is considered. The exact operator A and the exact right-hand side u are unknown. Only their upper and lower estimates are available. A technique of finding upper and lower estimates for the exact solution is proposed under the assumption that this solution is positive and bounded. A posterior error estimate is obtained for approximate solutions. The approximate solutions with an optimal posterior error estimate are discussed. Various prior information on the exact solution is used (for example, its monotonicity and convexity). This work was supported by the Russian Foundation for Basic Research (projects 11-01-00040a and 09-01-00586a) and by Visby Program, Swedish Institute, Stockholm.

Keywords: linear ill-posed problems, error estimation, partially ordered sets

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