Surface temperature determination from borehole measurements: Regularization and error estimates (Q1895293)

Summary: The authors consider the problem of determining the temperature distribution \(u(x,t)\) on the half-line \(x = 0\), \(t > 0\), from measurements at an interior point, for all \(t > 0\). As is well-known, this is an ill-posed problem. Using the Tikhonov method, the authors give a regularized solution, and assuming the (unknown) exact solution is in \(H^\alpha (\mathbb{R})\), \(\alpha > 0\). They give an error estimate of the order \(1/(1n1/ \varepsilon)^\alpha\) for \(\varepsilon \to 0\), where \(\varepsilon > 0\) is a bound on the measurement error.