On determining sinusoidal components of source terms in elliptic equations from terminal data
Keywords:
ill-posed problems, Poisson operator, truncation method, Sobolev embeddingsAbstract
In this study, we investigate an inverse source problem for an elliptic-in-space PDE with the Poisson operator, driven by a time-dependent separable source. The reconstruction is based on terminal data, a setting that is ill-posed in the sense of Hadamard. We first establish a Lipschitz continuity (stability) estimate for the recovered source in \(L^p(\Omega), p \ge 2\), with respect to the frequency parameter \(\omega\). Our second contribution develops a regularization for the final-value problem with separable sources and noisy measurements in \(L^2(\Omega)\). A truncation-based scheme is analyzed to regularize the problem, and an \(L^2\) error estimate is derived.References
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