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

Hesse C. H.

Statistics & Risk Modeling, 2005, vol. 23, issue 4, 317-329

Abstract: The Cauchy problem for the one-dimensional heat equation asks for solutions uf(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 uf is unknown but that at times tj, j = 1, 2, …, n, noisy measurements are available from which an estimator ~fn of the initial condition may be obtained. The paper studies the asymptotics (as t → ∞ and n → ∞) of uf(xt1/2, t)–u~fn(xt1/2, t) in mean integrated squared error.

Keywords: heat equation; unknown initial conditions; time series; noisy data; nonparametric estimation; asymptotics (search for similar items in EconPapers)
Date: 2005
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DOI: 10.1524/stnd.2005.23.4.317

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