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C. H. Hesse

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

Keywords: heat equation, unknown initial conditions, time series, noisy data, nonparametric estimation

The Cauchy problem for the one-dimensional heat equation asks for solutions u_f(x,t) of ∂ u / ∂ t = ∂²u / ∂ x² 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.

Statistics & Decisions, Oldenbourg Wissenschaftsverlag

Print ISSN: 0721-2631
Volume: 23, 04/2005
Pages: 317 - 329

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