Abstract
In this paper, on the one hand, we take the conventional quasi-reversibility method to obtain the error estimates of approximate solutions of the Cauchy problems for parabolic equations in a sub-domain of Q T with strong restrictions to the measured boundary data. On the other hand, weakening the conditions on the measured data, then combining the duality method in optimization with the quasi-reversibility method, we solve the Cauchy problems for parabolic equations in the presence of noisy data. Using this method, we can get the proper regularization parameter ɛ that we need in the quasi-reversibility method and obtain the convergence rate of approximate solutions as the noise of amplitude δ tends to zero.
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The first author is supported by National Natural Science Foundation of China (Grant No. 11226166) and Scientific Research Fund of Hu’nan Provincial Education Department (Grant No. 11C0052)
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Li, J., Guo, B.L. The quasi-reversibility method to solve the Cauchy problems for parabolic equations. Acta. Math. Sin.-English Ser. 29, 1617–1628 (2013). https://doi.org/10.1007/s10114-013-1735-x
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DOI: https://doi.org/10.1007/s10114-013-1735-x