A note on an ill-posed problem for the heat equation

Talenti, Giorgio; Vessella, Sergio

doi:10.1017/s1446788700024915

Cited by 38 publications

(20 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Later development of the inverse Stefan problem proceeded along two lines: inverse Stefan problems with given phase boundaries in [9,10,6,11,12,13,14,15,16,17], or inverse problems with unknown phase boundaries in [18,19,15,20,21,22,23,24,25,26,27,28,29,30,31]. We refer to the monograph [15] for a complete list of references for both types of inverse Stefan problem, both for linear and quasilinear parabolic equations.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Frechet differentiability in Besov spaces in the optimal control of parabolic free boundary problems

Abdulla

Goldfarb

2017

Journal of Inverse and Ill-Posed Problems

View full text Add to dashboard Cite

Abstract. We consider the inverse Stefan type free boundary problem, where information on the boundary heat flux and density of the sources are missing and must be found along with the temperature and the free boundary. We pursue optimal control framework where boundary heat flux, density of sources, and free boundary are components of the control vector. The optimality criteria consists of the minimization of the L2-norm declinations of the temperature measurements at the final moment, phase transition temperature, and final position of the free boundary. We prove the Frechet differentiability in Besov spaces, and derive the formula for the Frechet differential under minimal regularity assumptions on the data. The result implies a necessary condition for optimal control and opens the way to the application of projective gradient methods in Besov spaces for the numerical solution of the inverse Stefan problem.

show abstract

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Frechet differentiability in Besov spaces in the optimal control of parabolic free boundary problems

Abdulla

Goldfarb

2017

Journal of Inverse and Ill-Posed Problems

View full text Add to dashboard Cite

show abstract

“…A more mathematical background dealing with uniqueness and establishing the problem as an inverse problem is given by J. R. Cannon in [8,9] and in numerous other papers by the same author. Continuing the analytical approach we have the theoretical stability analysis of papers [16,17,18,19,20,21]. Examples of regularization strategies to battle against the ill-posedness are found in [10,11,12,13,14,15].…”

Section: Literature Reviewmentioning

confidence: 99%

Detection of time-varying heat sources using an analytic forward model

Tamminen¹

2020

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

“…Most of the papers on ISP are in the one-dimensional case. Inverse Stefan problems with given phase boundaries were considered in [7,9,11,12,13,14,15,16,17,18,23,39,21]; optimal control of Stefan problems, or equivalently inverse problems with unknown phase boundaries were investigated in [8,19,24,25,26,27,29,31,35,33,37,38,40,21,22,41,43].…”

Section: Introductionmentioning

confidence: 99%

Optimal Stefan problem

Abdulla

Poggi

2020

Calc. Var.

View full text Add to dashboard Cite

We consider the inverse multiphase Stefan problem with homogeneous Dirichlet boundary condition on a bounded Lipschitz domain, where the density of the heat source is unknown in addition to the temperature and the phase transition boundaries. The variational formulation is pursued in the optimal control framework, where the density of the heat source is a control parameter, and the criteria for optimality is the minimization of the L2−norm declination of the trace of the solution to the Stefan problem from a temperature measurement on the whole domain at the final time. The state vector solves the multiphase Stefan problem in a weak formulation, which is equivalent to Dirichlet problem for the quasilinear parabolic PDE with discontinuous coefficient. The optimal control problem is fully discretized using the method of finite differences. We prove the existence of the optimal control and the convergence of the discrete optimal control problems to the original problem both with respect to cost functional and control. In particular, the convergence of the method of finite differences for the weak solution of the multidimensional multiphase Stefan problem is proved. The proofs are based on achieving a uniform L∞ bound and W 1,1 2 energy estimate for the discrete multiphase Stefan problem.

show abstract

A note on an ill-posed problem for the heat equation

Cited by 38 publications

References 8 publications

Frechet differentiability in Besov spaces in the optimal control of parabolic free boundary problems

Frechet differentiability in Besov spaces in the optimal control of parabolic free boundary problems

Detection of time-varying heat sources using an analytic forward model

Optimal Stefan problem

Contact Info

Product

Resources

About