One-dimensional Inverse Problem Research Articles

Estimation of heat flux in one-dimensional inverse heat conduction problem

Estimation of heat flux in one-dimensional inverse heat conduction problem

A numerical solution of the one-dimensional inverse parabolic problem using Chebyshev T polynomials

A numerical solution of the one-dimensional inverse parabolic problem using Chebyshev T polynomials

The Inverse Analysis for Hydrodynamic Lubrication Using FEM and CIP Method

This paper presents two numerical methods for an inverse problem in hydrodynamic lubrication. In the inverse problem, the film thickness profile is estimated from a given pressure distribution. In this study, the finite element method and CIP method (Cubic-Interpolated Profile method) are used to solve the inverse problems. Calculated results are presented for unsteady one-dimensional plane inclined bearing and seal in reciprocating motion. In the plane inclined bearing, it is shown that the steady-state film thickness profile is obtained as an asymptotic solution in unsteady-state. And it is shown that the periodic film thickness profiles calculated for seal in reciprocating motion are in good agreement with results given by Hirano and Kaneta. The calculated results show that the numerical methods presented in this study can be used to one-dimensional inverse problems.

On estimating thermal diffusivity using analytical inverse solution for unsteady one-dimensional heat conduction

A modified procedure for calculating the thermal diffusivity of solids based on temperature measurements at two points and the semi-infinite boundary condition is presented. The method makes use of a solution to the unsteady one-dimensional inverse heat conduction problem for the semi-infinite solid. The procedure gives accurate results based on temperature changes produced by an arbitrary fluctuating heat flux source at the boundary.

On the stability and convergence of a difference scheme for an one-dimensional parabolic inverse problem

The parabolic equation with the control parameter is a class of parabolic inverse problems and is nonlinear. While determining the solution of the problems, we shall determinate some unknown control parameter. These problems play a very important role in many branches of science and engineering. This article is devoted to the following parabolic initial and boundary problem with the control parameter: ∂ u/∂ t = ∂ 2 u/∂ x 2 + p( t) u + ϕ( x, t), 0 < x < 1, 0 < t ⩽ T satisfying u( x, 0) = f( x), 0 < x < 1; u(0, t) = g 0( t), u(1, t) = g 1( t), u( x ∗, t) = E( t),0 ⩽ t ⩽ T where ϕ( x, t), f( x), g 0( t), g 1( t) and E( t) are known functions, u( x, t) and p( t) are unknown functions. With the help of a function transformation, the nonlinear problem given is transformed into a linear problem and then the back-ward Euler scheme is constructed for the latter. The unconditional stability and convergence of the difference scheme is proved with the maximum principle. The convergence orders of the approximations of both u and p are of O( τ + h 2), which improve the result obtained by Cannon et al. in 1994. Numerical example shows validity of our analysis. The method in this article is also applicable to the two-dimensional inverse problem.

Quasi-one-dimensional method for the two-dimensional magnetotelluric sounding inverse problem

The two-dimensional magnetotelluric sounding inverse problem is solved by a quasi-one-dimensional method. A linearization method is proposed for one-dimensional inverse problems. The appearance of false conducting structures in the solution of the inverse problem is considered and some counter-measures are suggested. The quasi-one-dimensional method is applicable to many-dimensional magnetotelluric sounding inverse problems if the electrical conductivity is a priori known to vary slowly along the Earth’s surface.

On the range of the forward map for the viscoelastic seismic inverse problem

This paper examines the range of the forward map for the one-dimensional viscoelastic seismic inverse problem. It is shown that the viscoelastic problem inherits properties that characterize the range of the lossless system, in both the continuum and discrete cases. Furthermore, in the discrete case it is shown that the range of the forward map for a viscoelastic system is actually a proper subset of the range for a lossless system.

A contraction theorem for the one-dimensional seismic inverse problem on a viscoelastic medium

This paper concerns itself with an analytic proof of the solvability of the one-dimensional seismic inverse problem on a viscoelastic medium. This problem uses techniques similar to that of the attenuating system, but encounters additional difficulties due to the presence of the memory variable. The proof is based on showing that an integral equation for the impedance is a contraction mapping using energy methods.

A Tau Method for the One-Dimensional Parabolic Inverse Problem Subject to Temperature Overspecification

In this work, an approximational technique based on shifted Legendre-tau ideas is presented for the one-dimensional parabolic inverse problem with a control parameter. The method consists of expanding the required approximate solution as the elements of a shifted Legendre polynomial. Using the operational matrices we reduce the problem to a set of algebraic equations. A numerical example is included to demonstrate the validity and applicability of the technique and a comparison is made with existing results. The method is easy to implement and produces very accurate results.

A Numerical Method for Solving of a Nonlinear Inverse Diffusion Problem

In this paper, we will first study the existence and uniqueness of the solution for a one-dimensional nonlinear inverse diffusion problem via an auxiliary problem and the Schauder fixed-point theorem. Furthermore, a numerical algorithm based on the finite-difference method and the least-squares scheme for solving a nonlinear inverse problem is proposed. At the beginning of the numerical algorithm, Taylor's series expansion is employed to linearize nonlinear terms and then the finite-difference method is used to discretize the problem domain. The present approach is to rearrange the matrix forms of the differential governing equations and estimate the unknown diffusion coefficient. The least-squares method is adopted to find the solution. It is assumed that no prior information is available on the functional form of the unknown diffusion coefficient in the present study, and thus, it is classified as the function estimation in inverse calculation. Results show that an excellent estimation on the diffusion coefficient can be obtained within a couple of minutes of CPU time on a Pentium IV-2.4 GHz PC.

A numerical solution technique for a one-dimensional inverse nonlinear parabolic problem

This study is intended to provide a new numerical algorithm for solving a one-dimensional inverse nonlinear parabolic problem. At the beginning of the study, Taylor’s series expansion is employed to linearize nonlinear terms and then finite-difference method is used to discretize the problem domain. The present approach is to rearrange the matrix forms of the differential governing equations and estimate unknown functions. The least-squares method is adopted to find the solution. It is assumed that no prior information is available on the functional form of the unknown coefficient in the present study, thus, it is classified as the function estimation. Results show that an excellent estimation on the unknown functions of the inverse problem which can be obtained within a couple of minutes CPU time at Pentium IV-2.4 GHz PC.

Deconvolution of axisymmetric flame properties using Tikhonov regularization

We present a method based on Tikhonov regularization for solving one-dimensional inverse tomography problems that arise in combustion applications. In this technique, Tikhonov regularization transforms the ill-conditioned set of equations generated by onion-peeling deconvolution into a well-conditioned set that is less susceptible to measurement errors that arise in experimental settings. The performance of this method is compared to that of onion-peeling and Abel three-point deconvolution by solving for a known field variable distribution from projected data contaminated with an artificially generated error. The results show that Tikhonov deconvolution provides a more accurate field distribution than onion-peeling and Abel three-point deconvolution and is more stable than the other two methods as the distance between projected data points decreases.

Reflection of pulses in a nonlinear random medium with strong fluctuations

The reflection of delta pulses from a strongly fluctuating nonlinear random medium is studied. With the use of the equation of a backscattered field obtained by the invariant imbedding method, a numerical algorithm is developed for solving the one-dimensional direct and inverse problems of wave reflection in time representation. The solutions obtained can be used for investigation of the fluctuation-related features of reflections occurring in nonlinear random media and in the design of remote sensing methods. The conditions that give rise to wave stochastic resonance and dynamic effects in nonlinear random reflections are considered.

One-step GPS for the estimation of temperature-dependent thermal conductivity

In this paper we are concerned with the estimation of temperature-dependent thermal conductivity of a one-dimensional inverse heat conduction problem. First, we construct a one-step group-preserving scheme (GPS) for the semi-discretization of quasilinear heat conduction equation, and then derive a quasilinear algebraic equation to determine the unknown thermal conductivity under a given initial temperature and a measured temperature perturbed by noise at time T. The new method does not require any prior information on the functional form of thermal conductivity. Several examples are examined to show that the new approach has high accuracy and efficiency, and the number of iterations spent in solving the quasilinear algebraic equation is smaller than five even in a large temperature range.

Comparison of four different versions of the variable metric method for solving inverse heat conduction problems

In this study, four different versions of the variable metric method (VMM) are investigated in solving standard one-dimensional inverse heat conduction problems in order to evaluate their efficiency and accuracy. These versions include Davidon–Fletcher–Powell (DFP), Broydon–Fletcher–Goldfarb–Shanno (BFGS), Symmetric Rank-one (SR1), and Biggs formula of the VMM. These investigations are carried out using temperature data obtained from numerical simulations.

Local regularization method applied to estimating oxygen consumption during muscle activities

In this paper, we consider a one-dimensional inverse problem of estimating an input signal that a priori is known to be a smooth function with a few jump discontinuities. A popular method of regularizing the inverse problem with such prior information is to use a total variation (TV) penalty as a regularizing functional. In this paper, we adopt a different approach, considering the problem from the Bayesian statistics viewpoint and using a hierarchical Gaussian smoothness prior. We demonstrate that the approach allows us to construct a local regularization scheme that is computationally effective and reproduces well the jump discontinuities. The approach avoids the non-differentiability problems encountered by TV methods. Once discretized, the algorithm is completely data driven, the parameter selections requiring no user intervention. The method is applied to an inverse problem in metabolic modelling, where the objective is to estimate the mitochondrial oxygen consumption during muscle activities based on noisy observations of the time decay of the oxygen concentration on the muscle surface.

Comparative study of cryogen spray cooling with R-134a and R-404a: implications for laser treatment of dark human skin

Cutaneous laser treatment in dark skin patients is challenging due to significant light absorption by the melanin at the basal layer of epidermis, which can result in irreversible nonspecific thermal injury to the epidermis. Cryogen spray cooling (CSC) with R-134a (boiling point approximately -26.2 degrees C at 1 atm), which is currently used during cutaneous laser treatment, has shown poor efficacy in protecting dark human skin. We investigated the potential of CSC with R-404a (boiling point approximately -46.5 degrees C at 1 atm), which has a lower boiling point than R-134a, for improved therapeutic outcome in dark human skin at three levels: in vitro (epoxy resin skin phantom), ex vivo (normal dark human skin sample), and in vivo (skin of the rabbit external ear). The skin phantom was used to acquire the surface and internal temperature profiles in response to CSC with R-134a or R-404a at various spurt durations, based upon which CSC-induced heat removal from the skin phantom was estimated using an algorithm that solved a one-dimensional inverse heat conduction problem. CSC with R-404a increased the temperature reductions within the phantom and subsequently the amount of heat removal from the phantom in comparison to that with R-134a. Normal ex vivo Fitzpatrick types V-VI human skin samples were used to investigate the thermal response of dark human skin epidermis to CSC (R-134a or R-404a) at various spurt durations in conjunction with 595-nm pulsed dye laser irradiation at various radiant exposures. Cryogen R-404a increased the threshold radiant exposures for irreversible thermal injury to the epidermis in dark pigmentation skin. No obvious CSC-induced morphological changes to human skin was observed when sprayed with R404-a spurts using durations up to 300 ms. In vivo rabbit ear vasculature was used as a model of cutaneous anomalies to assess the influences of CSC (with R-134a or R-404a) on the photothermolysis of dermal blood vessels. CSC (R-134a or R-404a) with the spurt durations of 100 to 300 ms increased the most superficial depth of thermally damaged dermal blood vessel compared with the sites without CSC, implying possible nonspecific cooling of superficial dermal blood vessels by the cryogen spurts with the settings applied.

Casting/Mold Thermal Contact Heat Transfer during Solidification of Al-Cu-Si Alloy (LM 21) Plates in Thick and Thin Molds

Heat flow at the casting/mold interface was assessed and studied during solidification of Al-Cu-Si (LM 21) alloy in preheated cast iron molds of two different thicknesses, coated with graphite and alumina based dressings. The casting and the mold were instrumented with thermocouples connected to a computer controlled temperature data acquisition system. The thermal history at nodal locations in the mold and casting obtained during experimentation was used to estimate the heat flux by solving the one-dimensional inverse heat conduction problem. The cooling rate and solidification time were measured using the computer-aided cooling curve analysis data. The estimated heat flux transients showed a peak due to the formation of a stable solid shell, which has a higher thermal conductivity compared with the liquid metal in contact with the mold wall prior to the occurrence of the peak. The high values of heat flux transients obtained with thin molds were attributed to mold distortion due to thermal stresses. For thin molds, assumption of Newtonian heating yielded reliable interfacial heat transfer coefficients as compared with one-dimensional inverse modeling. The time of occurrence of peak heat flux increased with a decrease in the mold wall thickness and increase in the casting thickness.

Solvability of a class of integro-differential equations and connections to one-dimensional inverse problems

A new approach to one-dimensional inverse problem was recently introduced by Barry Simon. We continue the study on an intermediate object A, which satisfies a nonlinear integro-differential equation. We prove local solvability of this A-equation and find a necessary condition for global solvability. Some exact solutions are presented.

Solving inverse heat conduction problem with discrete wavelet transform

We consider as a test case the linear one-dimensional inverse heat conduction problem. Usual regularization methods are equivalent to filtering out high-frequency components, in a uniform way on the whole stretch of data. This sets a limit to the trade-off that can be obtained between good resolution and efficient noise reduction. To overcome this limit, we propose the use of wavelet decomposition, which enables us to perform time–frequency analysis of the data content. A simple thresholding algorithm is used to eliminate noise components. The method is compared with the usual singular-value decomposition. Preliminary results on simple test cases show that wavelet decomposition permits us to enhance the results of the inversion procedure, by giving smoother estimates of slow-varying parts of the solution while maintaining a sharp estimation of fast transients.