Summary
This paper considers a discrete sampling scheme for the approximate recovery of initial data for one dimensional parabolic initial boundary value problems on a bounded interval. To obtain a given approximate, data is sampled at a single time and at a finite number of spatial points. The significance of this inversion scheme is the ability to accurately predict the error in approximation subject to choice of sample time and spatial sensor locations. The method is based on a discrete analogy of the continuous orthogonality for Sturm-Liouville systems. This property, which is of independent mathematical interest, is the notion of discrete orthogonal systems, which loosely speaking provides an exact (or approximate) Gauss-type quadrature for the continuous biorthogonality conditions.
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Supported in part by NSF Grant #DMS8905-344. Texas Advanced Research Program Grant #0219-44-5195 and AFOSR Grant #88-0309
Visiting at Texas Tech University, Fall 1989
Supported in part by NSF Grant #DMS8905-344, NSA grant #MDA904-85-H009 and NASA Grant #NAQ2-89
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Gilliam, D.S., Lund, J.R. & Martin, C.F. Inverse parabolic problems and discrete orthogonality. Numer. Math. 59, 361–383 (1991). https://doi.org/10.1007/BF01385786
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DOI: https://doi.org/10.1007/BF01385786