We consider an inverse problem for an operator equation Az = u. The exact operator A and the exact right-hand side u are unknown. Only their upper and lower estimations are available. We provide techniques of calculating upper and lower estimations for the exact solution belonging to a compact set in this case, as well as a posteriori error estimations. We obtain approximate solutions with an optimal a posteriori error estimate. We also make use of a priori information about the exact solution, e.g. its monotonicity and convexity. The developed software package was applied to solving practical ill-posed problems.