Determination of space‐dependent coefficients from temperature measurements using the conjugate gradient method

Cao, Kai; Lesnic, D.

doi:10.1002/num.22262

Cited by 14 publications

(9 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The Sobolev gradient that arises from the Sobolev space setting is smoother than the L 2 -gradient and was firstly used to generate preconditioners for the steepest descent method in [26]. The preconditioned CGM using Sobolev gradient has been extensively applied to the solution of inverse problems, e.g., in electrical impedance tomography [20], for the Robin inverse problem [18,19] and in the parameter identification for the bio-heat equation [4]. In this paper, a new preconditioner generated from the inner product is introduced.…”

Section: Preconditioningmentioning

confidence: 99%

“…Another major novelty of our study is that apart from the reconstruction of the TCC, we consider the numerical reconstruction of the SBC governing nonlinear fourth-order radiative contact. Within the numerical innovation, an important contribution of this paper is the development in section 3 of a preconditioned CGM in a Hilbert space setting with inner product for the noise removal in the reconstructed solutions and overcoming the vanishing of the gradient at final time, encountered in the conventional CGM [4,16,17]. Moreover, note that in all the works mentioned above, the coefficient identification problems concerned homogeneous materials only.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Reconstruction of the heat transfer coefficient at the interface of a bi-material

Zhuo

Lesnic

Meng

2019

Inverse Problems in Science and Engineering

Self Cite

View full text Add to dashboard Cite

The knowledge of heat transfer behaviour of composite thermal systems requires the characterization of the heat transfer coefficient at the contact interfaces between the constituent materials. The present work is devoted to investigating an inverse problem with generalized interface condition containing an unknown space-and time-varying interface coefficient from non-invasive temperature measurements on an accessible boundary. The uniqueness of the solution holds, but the problem does not depend continuously on the input measured temperature data. A new preconditioned conjugate gradient method (CGM) is utilized to address the ill-posedness of the inverse problem. In comparison with the standard CGM with no preconditioning, this method has the merit that the gradient of the objective functional does not vanish at the final time, which restores accuracy and stability when the input data is contaminated with noise and when the initial guess is not close to the true solution. Several numerical examples corresponding to linear thermal contact and nonlinear Stefan-Boltzmann radiation condition are tested for determining thermal contact conductance and Stefan-Boltzmann coefficient, respectively.The numerical results in both one-and two-dimensions illustrate that the reconstructions are robust and stable.

show abstract

Section: Preconditioningmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Reconstruction of the heat transfer coefficient at the interface of a bi-material

Zhuo

Lesnic

Meng

2019

Inverse Problems in Science and Engineering

Self Cite

View full text Add to dashboard Cite

show abstract

“…Assume that the function ( , ), ( , ) ∈ Ω , can be well-approximated by the expression in (8) with the function ( ), ∈ {1, 2, … , }, given in (7). Then the Fourier coefficients satisfies the overdetermined system of partial differential equations (11)- (12).…”

Section: For Allmentioning

confidence: 99%

“…Coefficient inverse problems for parabolic equations were studied intensively. Up to the knowledge of the author, the widely used method to solve this problem is the optimal control approach, see e.g., [11,12,1,13,14] and references therein. The authors of [11] applied the optimal control method involving a preconditioner to numerically compute the heat conductivity with high quality.…”

Section: Introductionmentioning

confidence: 99%

A new algorithm to determine the creation or depletion term of parabolic equations from boundary measurements

Nguyen¹

2019

Preprint

View full text Add to dashboard Cite

We propose a robust numerical method to find the coefficient of the creation or depletion term of parabolic equations from the measurement of the lateral Cauchy information of their solutions. Most papers in the field study this nonlinear and severely ill-posed problem using optimal control. The main drawback of this widely used approach is the need of some advanced knowledge of the true solution. In this paper, we propose a new method that opens a door to solve nonlinear inverse problems for parabolic equations without any initial guess of the true coefficient. This claim is confirmed numerically. The key point of the method is to derive a system of nonlinear elliptic equations for the Fourier coefficients of the solution to the governing equation with respect to a special basis of 2 . We then solve this system by a predictor-corrector process, in which our computation to obtain the first and second predictors is effective. The desired solution to the inverse problem under consideration follows. Problem 1.1 (Coefficient Inverse Problem (CIP)). Assume that ( ) ≠ 0 for all ∈ Ω. Given a time > 0 and the lateral Cauchy data ( , ) = ( , ) and ( , ) = ( , )for all ∈ Ω and ∈ [0, ], determine the coefficient ( ), ∈ Ω. Here is the outward normal vector of Ω.

show abstract

“…The reconstruction of coefficients in the parabolic heat equation, [3,10], has been the focus of attention in several fields, e.g. finance, groundwater flow, oil recovery, and heat transfer.…”

Section: Introductionmentioning

confidence: 99%

Reconstruction of an orthotropic thermal conductivity from non-local heat flux measurements

Huntul

Hussein

Lesnic

et al. 2020

IJMMNO

View full text Add to dashboard Cite

Raw materials are anisotropic and heterogeneous in nature, and recovering their conductivity is of utmost importance to the oil, aerospace and medical industries concerned with the identification of soils, reinforced fiber composites and organs. Due to the ill-posedness of the anisotropic inverse conductivity problem certain simplifications are required to make the model tracktable. Herein, we consider such a model reduction in which the conductivity tensor is orthotropic with the main diagonal components independent of one space variable. Then, the conductivity components can be taken outside the divergence operator and the inverse problem requires reconstructing one or two components of the orthotropic conductivity tensor of a two-dimensional rectangular conductor using initial and Dirichlet boundary conditions, as well as non-local heat flux over-specifications on two adjacent sides of the boundary. We prove the unique solvability of this inverse coefficient problem. Afterwards, numerical results indicate that accurate and stable solutions are obtained.

show abstract

Determination of space‐dependent coefficients from temperature measurements using the conjugate gradient method

Cited by 14 publications

References 37 publications

Reconstruction of the heat transfer coefficient at the interface of a bi-material

Reconstruction of the heat transfer coefficient at the interface of a bi-material

A new algorithm to determine the creation or depletion term of parabolic equations from boundary measurements

Reconstruction of an orthotropic thermal conductivity from non-local heat flux measurements

Contact Info

Product

Resources

About