On the optimal control of the free boundary problems for the
second order parabolic equations. I. Well-posedness and convergence of the method of lines

Poggi

2020

Calc. Var.

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.

Section: Introductionmentioning

confidence: 99%

“…The new variational approach developed in [1,2] is not applicable to the multiphase Stefan problem. The reason is that the Stefan condition on the phase transition boundary includes the flux calculated from both phases.…”

Section: Introductionmentioning

confidence: 99%

Optimal Stefan problem

Poggi

2020

Calc. Var.

“…The goal of this paper is to implement and analyze gradient method in Besov spaces framework for the numerical solution of the optimal control problem introduced recently as a variational formulation of the inverse Stefan problem (ISP) in [1,2]. Consider the general one-phase Stefan problem:…”

Section: Introductionmentioning

confidence: 99%

“…Lu := (a(x, t)u x ) x + b(x, t)u x + c(x, t)u − u t = f (x, t), in Ω (1) u(x, 0) = φ(x), 0 ≤ x ≤ s(0) = s 0 (2) a(0, t)u x (0, t) = g(t), 0 ≤ t ≤ T…”

Section: Introductionmentioning

confidence: 99%

“…ISP is not well posed in the sense of Hadamard: the solution may not exist; if it exists, it may not be unique, and in general it does not exhibit continuous dependence on the data. The goal of this paper is to pursue numerical analysis of the gradient method in Besov-Sobolev spaces based on the Fréchet differential and necessary condition for optimality ( [3,4]) in the optimal control problem introduced recently as a variational formulation of the inverse Stefan problem (ISP) in [1,2].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Gradient method in Hilbert–Besov spaces for the optimal control of parabolic free boundary problems

Journal of Computational and Applied Mathematics

Bukshtynov

Hagverdiyev

2019

This paper presents computational analysis of the inverse Stefan type free boundary problem, where information on the boundary heat flux is missing and must be found along with the temperature and the free boundary. We pursue optimal control framework introduced in U.G. Abdulla, Inverse Problems and Imaging, 7, 2(2013), 307-340; 10, 4(2016), 869-898, where boundary heat flux and free boundary are components of the control vector, and optimality criteria consist of the minimization of the quadratic declinations from the available measurements of the temperature distribution at the final moment, phase transition temperature on the free boundary, and the final position of the free boundary. We develop gradient descent algorithm based on Frechet differentiability in Hilbert-Besov spaces complemented with preconditioning or increase of regularity of the Frechet gradient through implementation of the Riesz representation theorem. Three model examples with various levels of complexity are considered. Extensive comparative analysis through implementation of preconditioning and Tikhonov regularization, calibration of preconditioning and regularization parameters, effect of noisy data, comparison of simultaneous identification of control parameters vs. nested optimization is pursued.

Optimal Control of the Multiphase Stefan Problem

Poggi

2018

Appl Math Optim

We consider the inverse multiphase Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundaries. Optimal control framework is pursued, where boundary heat flux is the control, and the optimality criteria consist of the minimization of the L 2 -norm declination of the trace of the solution to the Stefan problem from the temperature measurement on the fixed right boundary. The state vector solves multiphase Stefan problem in a weak formulation, which is equivalent to Neumann problem for the quasilinear parabolic PDE with discontinuous coefficient. Full discretization through finite differences is implemented and discrete optimal control problem is introduced. We prove wellposedness in a Sobolev space framework and convergence of discrete optimal control problems to the original problem both with respect to the cost functional and control. Along the way, the convergence of the method of finite differences for the weak solution of the multiphase Stefan problem is proved. The proof is based on achieving a uniform L ∞ bound, and W 1,1 2 -energy estimate for the discrete multiphase Stefan problem.Key words: Inverse multiphase Stefan problem, quasilinear parabolic PDE with discontinuous coefficent, optimal control, Sobolev spaces, method of finite differences, discrete optimal control problem, energy estimate, embedding theorems, weak compactness, convergence in functional, convergence in control.AMS subject classifications: 35R30, 35R35, 35K20, 35Q93, 65M06, 65M12, 65M32, 65N21. * This research is funded by NSF grant #1359074Inverse Multiphase Stefan Problem (IMSP). Find the functions u(x, t), ξ j (t), j = 1, J, and the boundary heat flux g(t) satisfying (1)-(6),(8).Motivation for the IMSP arose from the modeling of bioengineering problems on the laser ablation of biological tissues through a multiphase Stefan problem (1)-(6). Laser ablation creates three phases -solid (skin), fluid (melted skin) and gas (evaporated skin). Free boundaries ξ 1 (t) and ξ 2 (t) are measuring ablation depth separating solid/fluid and fluid/air regions at the moment t. GR±ε J (g), ε > 0. Then lim ε→0 J * (ε) = J * = lim ε→0 J * (−ε).Proof. The proof of this lemma is very similar to the analogous lemma from [1]. If 0 < ε 1 < ε 2 , then