Abstract
We develop a mathematical framework and efficient computational schemes to obtain an approximate solution of partial differential equations (PDEs) via sampled data. Recently, DeVore and Zuazua revisited the classical problem of inverse heat conduction, and they investigated how to recover the initial temperature distribution of a finite body from temperature measurements made at a fixed number of later times. In this paper, we consider a Laplace equation and a variable coefficient wave equation. We show that only one sensor employed at a crucial location at multiple time instances leads to a sequence of approximate solutions, which converges to the exact solution of these PDEs. This framework can be viewed as an extension of the novel, dynamical sampling techniques.
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References
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Acknowledgements
This material is based upon work supported by the National Security Agency under Grant No. H98230-18-0144 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during Summer 2018. The authors thank the referees for their valuable comments.
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Aceska, R., Kim, Y.H. (2021). Time-Variant System Approximation via Later-Time Samples. In: Fasshauer, G.E., Neamtu, M., Schumaker, L.L. (eds) Approximation Theory XVI. AT 2019. Springer Proceedings in Mathematics & Statistics, vol 336. Springer, Cham. https://doi.org/10.1007/978-3-030-57464-2_1
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DOI: https://doi.org/10.1007/978-3-030-57464-2_1
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