Optimal control of coefficients in parabolic free boundary problems modeling laser ablation

Abdulla, Ugur G.; Goldfarb, Jonathan; Hagverdiyev, Ali

doi:10.1016/j.cam.2020.112736

Cited by 8 publications

(4 citation statements)

References 31 publications

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“…Free boundary problems for parabolic partial differential equations have significant applications in various fields of engineering, physics, chemistry, Abdulla et al (2020), Broadbridge et al (1993), Chen and Feldman (2015), Huntul and Lesnic (2019a), Hussein et al (2016), Johansson et al (2011) to mention only a few. In particular, the Stefan problem is a moving free boundary problem that concerns the distribution of heat in a phase-change transforming medium.…”

Section: Introductionmentioning

confidence: 99%

Determination of the time-dependent convection coefficient in two-dimensional free boundary problems

Huntul

Lesnic

2021

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Purpose The purpose of the study is to solve numerically the inverse problem of determining the time-dependent convection coefficient and the free boundary, along with the temperature in the two-dimensional convection-diffusion equation with initial and boundary conditions supplemented by non-local integral observations. From the literature, there is already known that this inverse problem has a unique solution. However, the problem is still ill-posed by being unstable to noise in the input data. Design/methodology For the numerical discretization, this paper applies the alternating direction explicit finite-difference method along with the Tikhonov regularization to find a stable and accurate numerical solution. The resulting nonlinear minimization problem is solved computationally using the MATLAB routine lsqnonlin. Both exact and numerically simulated noisy input data are inverted. Findings The numerical results demonstrate that accurate and stable solutions are obtained. Originality/value The inverse problem presented in this paper was already showed to be locally uniquely solvable, but no numerical solution has been realized so far; hence, the main originality of this work is to attempt this task.

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Section: Introductionmentioning

confidence: 99%

Determination of the time-dependent convection coefficient in two-dimensional free boundary problems

Huntul

Lesnic

2021

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“…Furthermore, this optimal control problem will be referred to as Problem I. Motivation for the Problem I arises in many applications, such as a modeling and control of biomedical engineering problem about the laser ablation of biomedical tissues [9,6], preventing aerodynamic stall in aircrafts due to in-flight ice accretion [46], etc. The goal is to find optimal choice of the density of the sources which minimizes the mismatch of the temperature distribution at the final moment with the desired temperature profile.…”

Section: Introduction 1optimal Control Problemmentioning

confidence: 99%

“…The mathematical trick allowed to handle situations with erroneous information on the phase transition temperature, and opened a way to develop numerical methods with reduced computational cost due to the fact that the state vector is a solution of the PDE problem in a fixed region rather than free boundary problem. Frechet differentiability and optimality condition in the new optimal control framework was proved in [3,4], and iterative gradient method for the numerical solution was implemented in [5,6]. The approach introduced in [1, 2] is specifically designed for one phase Stefan-type free boundary problems, and is not applicable to multiphase free boundary problems.…”

Section: Introduction 1optimal Control Problemmentioning

confidence: 99%

Optimal Control of Singular Parabolic PDEs Modeling Multiphase Stefan-type Free Boundary Problems

Abdulla,

Cosgrove

2020

Preprint

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Optimal control of the singular nonlinear parabolic PDE which is a distributional formulation of multidimensional and multiphase Stefan-type free boundary problem is analyzed. Approximating sequence of finite-dimensional optimal control problems is introduced via finite differences. Existence of the optimal control and the convergence of the sequence of discrete optimal control problems both with respect to functional and control is proved. In particular, convergence of the method of finite differences, and existence, uniqueness and stability estimations are established for the singular PDE problem under minimal regularity assumptions on the coefficients.

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“…In [3,4], Frechet differentiability in Sobolev-Besov spaces was proved and the formula for the Frechet gradient and optimality condition are derived. In [6,8] gradient method was implemented in Hilbert-Besov spaces framework for the numerical solution of the ISP.…”

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confidence: 99%

Optimal Control of Multiphase Free Boundary Problems for Nonlinear Parabolic Equations

Abdulla

Cosgrove

2020

Appl Math Optim

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We consider the optimal control of singular nonlinear partial differential equation which is the distributional formulation of the multiphase Stefan type free boundary problem for the general second order parabolic equation. Boundary heat flux is the control parameter, and the optimality criteria consist of the minimization of the L 2 -norm declination of the trace of the solution to the PDE problem at the final moment from the given measurement. Sequence of finite-dimensional optimal control problems is introduced through finite differences. We establish existence of the optimal control and prove the convergence of the sequence of discrete optimal control problems to the original problem both with respect to functional and control. Proofs rely on establishing a uniform L∞ bound, and W 1,1 2 -energy estimate for the discrete nonlinear PDE problem with discontinuous coefficient.2010 Mathematics Subject Classification. 35R30, 35R35, 35K20, 35Q93, 49J20, 65M06, 65M12, 65M32, 65N21.Key words and phrases. Inverse multidimensional multiphase Stefan problem, Quasilinear parabolic PDE with discontinuous coefficients, optimal control, Sobolev spaces, method of finite differences, discrete optimal control problem, energy estimate, embedding theorems, weak compactness, convergence in functional, convergence in control, maximal monotone graph.• the following inequalities are satisfied: lim sup ε→0 J * (ε) ≥ J * , lim inf ε→0 J * (−ε) ≤ J * , (3.16) where J * (±ε) = inf GR±ε J (g).Lemma 3.3. [5] The mappings P n , Q n satisfy the conditions of Lemma 3.2.

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Optimal control of coefficients in parabolic free boundary problems modeling laser ablation

Cited by 8 publications

References 31 publications

Determination of the time-dependent convection coefficient in two-dimensional free boundary problems

Determination of the time-dependent convection coefficient in two-dimensional free boundary problems

Optimal Control of Singular Parabolic PDEs Modeling Multiphase Stefan-type Free Boundary Problems

Optimal Control of Multiphase Free Boundary Problems for Nonlinear Parabolic Equations

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