The inverse Stefan problem as a problem of nonlinear approximation theory

Control of stefan problems by means of linear-quadratic defect minimization

“…2.1) a set of restrictions. This has been investigated in [8,9]: These results deal mainly with the solution of the minimization problem for a fixed dimension and for the most part omit the asymptotic quality of the approximate solutions. The general idea for an indirect approach on the other hand is not to deal with the non-linear operator equation ( by appropriate defect minimization procedures.…”

The numerical solution of the inverse Stefan problem

“…Given the initial distribution f, the associated problem of optimal control is to find, to a prescribed interface 3, a time dependent heat flux g which generates 0029-599X/80/0034/0411/$03.80 a free boundary s, which optimal fits ~ in a certain norm. This so-called inverse Stefan problem (ISP) has recently been treated by the author [14] by methods of nonlinear approximation theory, which were leading to an existence theorem and a characterization of an optimal boundary s* by an alternation criterion well known from Tchebychev approximation theory [16]. The essential argument for this point of view is the development of an effective procedure for the solution of nonlinear approximation problems by Osborne and Watson [19] and Cromme [11], which very well applies to the ISP.…”

“…We shall treat the norms II" IIo~, II'lt2 and, shortly, t[" IL 1. As the dimension of V, will be fixed from now on, we shall suppress the subscript n. In case [1" II = I1" Jl o~ we shall shortly summarize the properties of (2.12) which have been investigated in [14] and which will be needed in Sect. 3.…”

“…a) Since S is one-to-one [14], the optimal control g* is unique, too. b) In [14] also the existence of an optimal g* satisfying (2.12) with respect to /~ has been proved. But we did not succeed in characterizing an s*= S(g*) with g*e/~\R.…”

The Numerical Solution of an Inverse Two-Phase Stefan Problem

“…Jochum considers the inverse Stefan problem as a problem of nonlinear approximation theory(see [3,4]). In paper [5],the author use Adomian decomposition method to solve this problem .In paper [6],for solution of one phase two-dimensional problems, authors use a complete family of solutions to the heat equation to minimize the maximal defect in the initialboundary data.…”