This paper deals with the selection of an initial distribution in the first boundary-value problem for the heat equation in a given domain [0,θ]×Ωθ<∞ with zero values on its boundary S so that the deviation of the respective solution from a given distribution would not exceed a preassigned value γ>0. The result is formulated here in terms of the “theory of guaranteed estimation” for noninvertible evolutionary systems. It also allows an interpretation in terms of regularization methods for ill-posed inverse problems and in particular, in terms of the quasiinvertibility techniques of J.-L. Lions and R. Lattes.