Nonlinear inverse problem for the estimation of time-and-space dependent heat transfer coefficients

Osman, A. M.; Beck, James V.

doi:10.2514/6.1987-150

Cited by 6 publications

(5 citation statements)

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“…The inverse problem is to reconstruct the unknown u from noisy data y. It arises in corrosion detection [19,24] and analysis of quenching process [33]. For the inversion, the flux g is set to 1, and the true coefficient u is given by 1 + sin(πx 1 ).…”

Section: Stationary Robin Inverse Problemmentioning

confidence: 99%

A variational Bayesian method to inverse problems with impulsive noise

Jin

2012

Journal of Computational Physics

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We propose a novel numerical method for solving inverse problems subject to impulsive noises which possibly contain a large number of outliers. The approach is of Bayesian type, and it exploits a heavy-tailed t distribution for data noise to achieve robustness with respect to outliers. A hierarchical model with all hyper-parameters automatically determined from the given data is described.An algorithm of variational type by minimizing the Kullback-Leibler divergence between the true posteriori distribution and a separable approximation is developed. The numerical method is illustrated on several one-and two-dimensional linear and nonlinear inverse problems arising from heat conduction, including estimating boundary temperature, heat flux and heat transfer coefficient. The results show its robustness to outliers and the fast and steady convergence of the algorithm. key words: impulsive noise, robust Bayesian, variational method, inverse problems IntroductionWe are interested in Bayesian approaches for inverse problems subject to impulsive noises. Bayesian inference provides a principled framework for solving diverse inverse problems, and has demonstrated distinct features over deterministic techniques, e.g., Tikhonov regularization. Firstly, it can yield an ensemble of plausible solutions consistent with the given data. This enables quantifying the uncertainty of a specific solution, e.g., with credible intervals. In contrast, deterministic techniques generally content with singling out one solution from the ensemble. Secondly, it provides a flexible regularization since hierarchical modeling can partially resolve the nontrivial issue of choosing an appropriate regularization parameter.It is known that the underlying mechanism is balancing principle [22]. Thirdly, it allows seamlessly integrating structural/multiscale features of the problem through careful prior modeling. Therefore, it has attracted attention in a wide variety of applied disciplines, e.g., geophysics [37,34], medical imaging * [18, 27, 1] and heat conduction [13,39, 40,14], see also [32,28,30,31] for other applications. For an overview of methodological developments, we refer to the monographs [37,26].Amongst existing studies on Bayesian inference for inverse problems, the Gaussian noise model has played a predominant role. This is often justified by appealing to central limit theorem. The theorem asserts that the normal distribution is a suitable model for data that are formed as the sum of a large number of independent components. Even in the absence of such justifications, this model is still preferred due to its computational/analytical conveniences, i.e., it allows direct computation of the posterior mean and variance and easy exploration of the posterior state space (for linear models with Gaussian priors). A well acknowledged limitation of the Gaussian model is its lack of robustness against the outliers, i.e., data points that lie far away from the bulk of the data, in the observations: A single aberrant data point can significantly influ...

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Section: Stationary Robin Inverse Problemmentioning

confidence: 99%

A variational Bayesian method to inverse problems with impulsive noise

Jin

2012

Journal of Computational Physics

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“…This is a highly nonlinear and ill-posed inverse problem and arises in many applications of practical importance. The Robin coefficient may characterize the thermal properties of conductive materials on the interface or certain physical processes near the boundary, e.g., it represents the corrosion damage profile in corrosion detection [5] [7], and indicates the thermal property in quenching processes [15].…”

Section: Introductionmentioning

confidence: 99%

Quadratic convergence of Levenberg-Marquardt method for elliptic and parabolic inverse robin problems

2018

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We study the Levenberg-Marquardt (L-M) method for solving the highly nonlinear and ill-posed inverse problem of identifying the Robin coefficients in elliptic and parabolic systems. The L-M method transforms the Tikhonov regularized nonlinear non-convex minimizations into convex minimizations. And the quadratic convergence of the L-M method is rigorously established for the nonlinear elliptic and parabolic inverse problems for the first time, under a simple novel adaptive strategy for selecting regularization parameters during the L-M iteration. Then the surrogate functional approach is adopted to solve the strongly ill-conditioned convex minimizations, resulting in an explicit solution of the minimisation at each L-M iteration for both the elliptic and parabolic cases. Numerical experiments are provided to demonstrate the accuracy and efficiency of the methods.The coefficients a(x) and c(x) are the heat conductivity and radiation coefficient, satisfying that a ≤ a(x) ≤ā and c ≤ c(x) ≤c in Ω, where a,ā and c,c are positive constants. Functions f , g and h are the source strength, ambient temperature and heat flux respectively. Both coefficients γ(x) in (1.1) and (1.2) represent the Robin coefficients, which will be the focus of our interest and is assumed to stay in the following feasible constraint set:where γ 1 and γ 2 are two positive constants. For convenience, we often write the solutions of the systems (1.1) and (1.2) as u(γ) to emphasize their dependence on the Robin coefficient γ.We are now ready to formulate the inverse problems of our interest in this work. Elliptic Inverse Robin Problem: recover the Robin coefficient γ(x) in (1.1) on the inaccessible part Γ i from the measurable data z of u on the accessible part Γ a .Parabolic inverse Robin problem: recover the Robin coefficient γ(x) in (1.2) on the inaccessible part Γ i from the measurable data z of u on the accessible part Γ a over the whole time range [0, T ].The inverse Robin problems have been widely studied in literature; see [5] [2] [10] [11][12] and the references therein. The Gauss-Newton method was applied in [5] to solve the least-squares formulation of the elliptic inverse Robin problem, but with no consideration of regularizations. An L 1 -tracking functional approach was suggested for the elliptic inverse Robin problem in [2]. Effectiveness and justifications of least-squares formulations with regularizations were analysed in [10] [11][12] for the Robin inverse problems, and some iterative methods were applied to solve the resulting nonlinear least-squares minimizations. However, we may observe a common feature of these existing methods, which solve directly the nonlinear optimizations resulting from least-squares formulations with regularisations, but these optimisation problems are highly non-convex as the forward solution u(γ) is nonlinear with respect to γ, and strongly unstable at discrete level with fine mesh sizes and time step sizes due to the severe ill-posedness of the inverse problems and the fact that noise is always present ...

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“…On the boundary c the Neumann boundary condition is specified with the Neumann data q(x). The Robin coefficient (x) is of practical interest in thermal problems [31] and non-destructive evaluation [32][33][34]. Here, we consider the inverse problem of estimating scalar Robin coefficients 241 from temperature measurements on the boundary c .…”

Section: Setup Of Numerical Experimentsmentioning

confidence: 99%

Fast Bayesian approach for parameter estimation

Jin

2008

Numerical Meth Engineering

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SUMMARYThis paper presents two techniques, i.e. the proper orthogonal decomposition (POD) and the stochastic collocation method (SCM), for constructing surrogate models to accelerate the Bayesian inference approach for parameter estimation problems associated with partial differential equations. POD is a model reduction technique that derives reduced-order models using an optimal problem-adapted basis to effect significant reduction of the problem size and hence computational cost. SCM is an uncertainty propagation technique that approximates the parameterized solution and reduces further forward solves to function evaluations. The utility of the techniques is assessed on the non-linear inverse problem of probabilistically calibrating scalar Robin coefficients from boundary measurements arising in the quenching process and non-destructive evaluation. A hierarchical Bayesian model that handles flexibly the regularization parameter and the noise level is employed, and the posterior state space is explored by the Markov chain Monte Carlo. The numerical results indicate that significant computational gains can be realized without sacrificing the accuracy.

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Nonlinear inverse problem for the estimation of time-and-space dependent heat transfer coefficients

Cited by 6 publications

References 5 publications

A variational Bayesian method to inverse problems with impulsive noise

A variational Bayesian method to inverse problems with impulsive noise

Quadratic convergence of Levenberg-Marquardt method for elliptic and parabolic inverse robin problems

Fast Bayesian approach for parameter estimation

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