The heat equation given a time series of initial data subject to error

Summary The Cauchy problem for the one-dimensional heat equation asks for solutions u f ( x,t ) of ∂ u /∂ t = ∂ 2 u / ∂ x 2 on R × [1, ∞) with u ( x , 1) = f ( x ) on R. Here we assume that the initial condition f ( x ), x ∊ R, and hence the solution u f is unknown but that at times t j , j = 1, 2, …, n , noisy measurements are available from which an estimator ~ f n of the initial condition may be obtained. The paper studies the asymptotics (as t → ∞ and n → ∞) of u f ( xt 1/2 , t )– u ~fn ( xt 1/2 , t ) in mean integrated squared error.