Uniqueness and stability of parameter identification in elliptic boundary value problem
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Abstract
This paper concerns the uniqueness and stability of an inverse problem
in PDE. Our problem consists of identifying two parameters b(x)b(x) and c(x)c(x) in the following boundary-value problem
{Lu:=−b(x)u′′(x)+c(x)u′(x)=f(x),u(0)=u(1)=0,{Lu:=−b(x)u″(x)+c(x)u′(x)=f(x),u(0)=u(1)=0,
from distributed observations u1u1 (resp. u2u2) associated with the source f1f1 (resp. f2f2). For one observation, the solution is not unique. However, we prove, under some conditions, the uniqueness of the solution p=(b,c)p=(b,c) in the case of two observations. Furthermore, we derive a H\"older-type stability result. The algorithm of reconstruction uses the least squares method. Finally, we present some numerical examples with exact and noisy data to illustrate our method.
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References
G. Chavent, Nonlinear Least Squares for Inverse Problems: Theoretical Foundations and Step by-Step Guide for Applications, Series Scientific Computation, Springer Dordrecht, New York, 2010.
H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, 1996.
M. Guidici, A result concerning identifiability of the inverse problem of groundwater hydrology. Vol. 5, L31-L36,IOP Publishing, 1989.
M. Kern, Numerical Methods for Inverse Problems, John Wiley & Sons, Inc., 2016.
J. J. Moré, The Levenberg-Marquardt algorithm: implementation and theory, In: Watson, G.A. (eds) Numerical Analysis. Lecture Notes in Mathematics, vol. 630, Springer, Berlin, Heidelberg. 1978.
J. Zou, Numerical methods for elliptic inverse problems, Int. J. Comput. Math., 70(2) (1998), 211-232