An efficient regularization method for a large scale ill-posed geothermal problem

Abstract

The inverse geothermal problem consists of estimating the temperature distribution below the earth's surface using measurements on the surface. The problem is important since temperature governs a variety of geologic processes, including the generation of magmas and the deforma

Solvability of a non-linear Cauchy problem for an elliptic equation

“…The main topic of discussion in this paper is the solvability of the operator equation (5). However, since the operator is non-linear calculating K(φ 0 , ψ 1 ) for a given set of boundary data (φ 0 , ψ 1 ) is also non-trivial.…”

“…In our previous work [5], we implemented a simple fixed point iteration as follows: Let T 0 ∈ H 1 per ( ) be a starting guess and iterate by solving…”

“…In our previous work we studied the inverse geothermal problem [5], where measurements at the surface level, are used to estimate the stationary temperature profile below the earths surface, see also [7], by solving a Cauchy problem for the heat equation.…”

“…Thus, we introduce a space H 1 per ( ) consisting of all functions in H 1 ( ) that satisfies the periodicity condition (3). In our previous work [5] we solved the Cauchy problem (1) by reformulating it as an operator equation. If we require the solution to be a function in H 1 per ( ) then we we have the following direct problem:…”

“…The main result is that the operator K, cf. (5), is well-defined if certain bounds for ∇ x k(x, T) holds, while the coefficient k(x, T) may be discontinuous with respect to T. As a part of the proof we develop a convergent iterative procedure that lets us solve the non-linear problem (4). This paper is organized as follows: In Subsection 2.1, we outline the function spaces used.…”