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ERA-MS

IPI

We develop a new variational formulation of the inverse Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundary. We employ optimal control framework, where boundary heat flux and free boundary are components of the control vector, and optimality criteria consists of the minimization of the sum of $L_2$-norm declinations from the available measurement of the temperature flux on the fixed boundary and available
information on the phase transition temperature on the free boundary. This approach allows
one to tackle situations when the phase transition temperature is not known explicitly, and is available through measurement with possible error. It also allows for the development of iterative numerical methods of least computational cost due to the fact that for every given control vector, the parabolic PDE is solved in a fixed region instead of full free boundary problem. We prove well-posedness in Sobolev spaces framework and
convergence of discrete optimal control problems to the original problem both with respect to cost functional and control.

DCDS

This paper introduces a notion of regularity (or irregularity)
of the point at infinity ($\infty$) for the unbounded open set $\Omega\subset {\mathbb R}^{N}$ concerning second order uniformly elliptic equations with bounded and measurable coefficients, according as whether the ${\mathcal A}$- harmonic measure of $\infty$ is zero (or positive). A necessary and sufficient condition for the existence of a unique bounded solution
to the Dirichlet problem in an arbitrary open set of ${\mathbb R}^{N}, N\ge 3$ is established
in terms of the Wiener test for the regularity of $\infty$. It coincides with the Wiener
test for the regularity of $\infty$ in the case of Laplace equation.
From the topological point of view, the Wiener test at $\infty$ presents thinness criteria
of sets near $\infty$ in fine topology. Precisely, the open set is a deleted neigborhood
of $\infty$ in fine topology if and only if $\infty$ is irregular.

keywords:
Dirichlet problem
,
fine topology
,
measurable coefficients
,
Uniformly elliptic equations
,
regularity (or irregularity) of $\infty$
,
differential generator.
,
${\mathcal A}$-thinness
,
characteristic Markov process
,
${\mathcal A}$-harmonic measure
,
${\mathcal A}$-super- or subharmonicity
,
Brownian motion
,
Newtonian capacity
,
PWB solution
,
Wiener test
,
metric compactification of $R^{N+1}$

IPI

We consider a variational formulation of the inverse Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundary. We employ optimal control framework, where boundary heat flux and free boundary are components of the control vector, and optimality criteria consist of the minimization of the sum of $L_2$-norm declinations from the available measurement of the temperature flux on the fixed boundary and available
information on the phase transition temperature on the free boundary. This approach allows
one to tackle situations when the phase transition temperature is not known explicitly, and is available through measurement with possible error. It also allows for the development of iterative numerical methods of least computational cost due to the fact that for every given control vector, the parabolic PDE is solved in a fixed region instead of full free boundary problem. In

*Inverse Problems and Imaging, 7, 2(2013), 307-340*we proved well-posedness in Sobolev spaces framework and convergence of time-discretized optimal control problems. In this paper we perform full discretization and prove convergence of the discrete optimal control problems to the original problem both with respect to cost functional and control.## Year of publication

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## Related Keywords

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