Extreme Methods for Solving Ill-Posed Problems with Applications to Inverse Heat Transfer Problems

Alifanov, O. M.; Artioukhine, E.; Rumyantsev, S. A.

doi:10.1615/978-1-56700-038-2.0

Cited by 159 publications

(6 citation statements)

References 94 publications

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“…u∈ (1,2]. The Hölder exponents predicted by the theory (see theorem 2) are also observed numerically, cf the numerical exponents η i,j , listed in the left part of table 1.…”

Section: The Test Problemmentioning

confidence: 54%

“…Note that the compatibility conditions are not satisfied, in particular u / ∈ C 4+θ,2+θ/2 (Q). The iteration is started with q 0 (u) = sin(π u), u ∈ [0, 1] 0, u∈ (1,2].…”

Section: The Test Problemmentioning

confidence: 99%

“…Nonlinear diffusion processes appear, e.g., in the modelling of cooling processes for steel or glass in liquids or solids (cf, e.g., [16]), in the modelling of phase transitions, for instance in crystallization processes of polymers (cf [7]) and in reaction kinetics of chemical systems (see, e.g., [4]). In all these areas, the inverse problem of determining the nonlinearity from measurements is of interest, e.g., it was shown in [2] that one-dimensional problems like the one we consider have important applications in heat transfer at high temperatures. We treat the inverse problem of determining q(•) in (1.1) as model for such parameter identification problems, and point out possible extensions of our results to more general equations and higher dimensions later in this section (cf remark 4).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Global uniqueness and Hölder stability for recovering a nonlinear source term in a parabolic equation

2004

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Consider the semilinear parabolic equation −u t (x, t) + u xx + q(u) = f (x, t), with the initial condition u(x, 0) = u 0 (x), Dirichlet boundary conditions u(0, t) = ϕ 0 (t), u(1, t) = ϕ 1 (t) and a sufficiently regular source term q(•), which is assumed to be known a priori on the range of u 0 (x). We investigate the inverse problem of determining the function q(•) outside this range from measurements of the Neumann boundary data u x (0, t) = ψ 0 (t), u x (1, t) = ψ 1 (t). Via the method of Carleman estimates, we derive global uniqueness of a solution (u, q) to this inverse problem and Hölder stability of the functions u and q with respect to errors in the Neumann data ψ 0 , ψ 1 , the initial condition u 0 and the a priori knowledge of the function q (on the range of u 0). The results are illustrated by numerical tests. The results of this paper can be extended to more general nonlinear parabolic equations.

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“…u∈ (1,2]. The Hölder exponents predicted by the theory (see theorem 2) are also observed numerically, cf the numerical exponents η i,j , listed in the left part of table 1.…”

Section: The Test Problemmentioning

confidence: 54%

“…Note that the compatibility conditions are not satisfied, in particular u / ∈ C 4+θ,2+θ/2 (Q). The iteration is started with q 0 (u) = sin(π u), u ∈ [0, 1] 0, u∈ (1,2].…”

Section: The Test Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Global uniqueness and Hölder stability for recovering a nonlinear source term in a parabolic equation

2004

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“…Despite their ill posedness, the solution of inverse problems can be obtained through their reformulation in terms of well-posed problems, such as minimization problems together with some kind of regularization (stabilization) technique [17]. Various methods have been successfully used to derive estimates of parameters and functions in linear and nonlinear inverse heat transfer problems [18][19][20][21][22][23][24]. For example in [22], some parameters were simultaneously estimated by using only temperature measurements in moist capillary porous media in Luikov's linear dimensionless formulation.…”

Section: Introductionmentioning

confidence: 99%

Thickness determination in textile material design: dynamic modeling and numerical algorithms

Xu¹,

Ge²

2012

Inverse Problems

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Textile material design is of paramount importance in the study of functional clothing design. It is therefore important to determine the dynamic heat and moisture transfer characteristics in the human body–clothing–environment system, which directly determine the heat–moisture comfort level of the human body. Based on a model of dynamic heat and moisture transfer with condensation in porous fabric at low temperature, this paper presents a new inverse problem of textile thickness determination (IPTTD). Adopting the idea of the least-squares method, we formulate the IPTTD into a function minimization problem. By means of the finite-difference method, quasi-solution method and direct search method for one-dimensional minimization problems, we construct iterative algorithms of the approximated solution for the IPTTD. Numerical simulation results validate the formulation of the IPTTD and demonstrate the effectiveness of the proposed numerical algorithms.

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“…In many dynamic heat transfer situations, the surface temperature and/or heat flux histories of a solid must be determined from transient temperature measurements at one or more interior locations. In particular, during the past four decades many researchers were interested in the special case of estimating a surface condition from interior measurements which has come to be known as the inverse heat conduction problem (IHCP) [1,2,6,14]. Some applications of the IHCP in several engineering contexts and many industries [12] include the determination of thermal constants in some freezing and quenching processes, the estimation of surface heat transfer measurements taken within the skin of a re-entry space vehicle, the motion of a projectile over a gun barrel surface, the determination of aerodynamic heating in wind tunnels and rocket nozzles and infrared computerized tomography.…”

Section: Introductionmentioning

confidence: 99%

Differential-difference regularization for a 2D inverse heat conduction problem

Qian¹,

Zhang²

2010

Inverse Problems

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We study the differential-difference regularization for a two-dimensional inverse heat conduction problem, i.e. the heat equation is semi-discretized by a differential-difference equation, where the time derivative and a spatial second-order derivative have been replaced by the finite differences, while the other spatial second-order derivative is preserved. We analyze the properties of the discretized approximation using Fourier transform techniques. Some error estimates, which give the information about how to choose the step lengths in the discretization, show that the semi-discretized form has a 'regularized effect'. We also proved the unconditional convergence of a discretization scheme involving spatial marching.

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Extreme Methods for Solving Ill-Posed Problems with Applications to Inverse Heat Transfer Problems

Cited by 159 publications

References 94 publications

Global uniqueness and Hölder stability for recovering a nonlinear source term in a parabolic equation

Global uniqueness and Hölder stability for recovering a nonlinear source term in a parabolic equation

Thickness determination in textile material design: dynamic modeling and numerical algorithms

Differential-difference regularization for a 2D inverse heat conduction problem

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