A Stability Analysis on Beck's Procedure for Inverse Heat Conduction Problems

Abstract

The sequential function specification method proposed first by Beck is considered as one of the most efficient methods for the inverse heat conduction problem (IHCP) which is extremely ill-posed and time-dependent. This method determines an “inverse solution” advancing in a sequential fashion in time. The values estimated at any given time depend on the solution obtained previously. The main question connected with this method is the stability;i.e.,the cumulative error in the solution must remain bounded at all time. Since the first paper of Beck in 1970, few theoretical stability analyses have been studied in the literature. The aim of this paper is to find the conditions under which this method is stable irrespective of the data measurements. For a 1D linear IHCP, we try to construct a sequenced (1)X1= 1,Xj= Σj−1l=1αj−l+1Xl,j≥ 2, such that the coefficients αiare independent of the data measured and the convergence of the series Σi=1 |Xi| guarantees the stability of the method. In other words, we need to find an adequate condition on αisuch that Σ∞i=1|Xi| is convergent, implying that the method is stable. The values of αidepend on the discretization sizehof the function to be determinedq(t) and the sliding time horizon (or future time interval) τ of the method. The range of values ofhand τ which give the values of αisuch that the series Σ∞i=1 |Xi| is convergent is established numerically. Under the stability condition, an error estimation of the Beck's method is derived. The approach presented could be also applied to multidimensional IHCPs, in which the coefficients αiandXiare no longer scalar but become square matrices.