A procedure for determining a spacewise dependent heat source and the initial temperature

The inverse problem of determining a spacewise dependent heat source, together with the initial temperature for the parabolic heat equation, using the usual conditions of the direct problem and information from two supplementary temperature measurements at different instants of time is studied. These spacewise dependent temperature measurements ensure that this inverse problem has a unique solution, despite the solution being unstable, hence the problem is ill-posed. We propose an iterative algorithm for the stable reconstruction of both the initial data and the source based on a sequence of well-posed direct problems for the parabolic heat equation, which are solved at each iteration step using the boundary element method. The instability is overcome by stopping the iterations at the first iteration for which the discrepancy principle is satisfied. Numerical results are presented for a typical benchmark test example, which has the input measured data perturbed by increasing amounts of random noise. The numerical results show that the proposed procedure gives accurate numerical approximations in relatively few iterations.     

A variational method for identifying a spacewise-dependent heat source

The inverse problem of determining a spacewise-dependent heatsource for the parabolic heat equation using the usual conditionsof the direct problem and information from one supplementarytemperature measurement at a given instant of time is studied.This spacewise-dependent temperature measurement ensures thatthis inverse problem has a unique solution, but the solutionis unstable and hence the problem is ill-posed. We propose avariational conjugate gradient-type iterative algorithm forthe stable reconstruction of the heat source based on a sequenceof well-posed direct problems for the parabolic heat equationwhich are solved at each iteration step using the boundary elementmethod. The instability is overcome by stopping the iterativeprocedure at the first iteration for which the discrepancy principleis satisfied. Numerical results are presented which have theinput measured data perturbed by increasing amounts of randomnoise. The numerical results show that the proposed procedureyields stable and accurate numerical approximations after onlya few iterations.     

A simplified Tikhonov regularization method for determining the heat source

This paper is to discuss the inverse problem of determining a spacewise dependent heat source in one-dimensional heat equation in a bounded domain where data is given at some fixed time. This problem is ill-posed, i.e., the solution (if it exists) does not depend continuously on the data. The regularization solution is given by a simplified Tikhonov regularization. For this regularization solution, the Hölder type stability estimate between the regularization solution and the exact solution is obtained. Numerical examples show that the regularization method is effective and stable.     

Simultaneous identification of diffusion coefficient,spacewise dependent source and initial value for one‐dimensional heat equation

This paper deals with an inverse problem of determining the diffusion coefficient, spacewise dependent source term, and the initial value simultaneously for a one‐dimensional heat equation based on the boundary control, boundary measurement, and temperature distribution at a given single instant in time. By a Dirichlet series representation for the boundary observation, the identification of the diffusion coefficient and initial value can be transformed into a spectral estimation problem of an exponential series with measurement error, which is solved by the matrix pencil method. For the identification of the source term, a finite difference approximation method in conjunction with the truncated singular value decomposition is adopted, where the regularization parameter is determined by the generalized cross‐validation criterion. Numerical simulations are performed to verify the result of the proposed algorithm. Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical identification of source terms for a two dimensional heat conduction problem in polar coordinate system

This work investigates the inverse problem of reconstructing a spacewise dependent heat source in a two-dimensional heat conduction equation using a final temperature measurement. Problems of this type have important applications in several fields of applied science. Under certain assumptions, this problem can be transformed into a one-dimensional problem where the heat source only depends on the variable r  . However, being different from other one-dimensional inverse heat source problems, there exists singularity on the coefficient of our model, which may make the analysis more difficult, regardless of theoretical or numerical. The inverse problem is reduced to an operator equation of the first kind and the corresponding adjoint operator is deuced. For the two dimensional case, i.e., f=f(r,θ)$f=f(r\text{,}\theta )$, theoretical analysis can be done by similar derivation. Based on the landweber regularization framework, an iterative algorithm is proposed to obtain the numerical solution. Some typical numerical examples are presented to show the validity of the inversion method.     

Uniqueness for an inverse problem for a nonlinear parabolic system with an integral term by one-point Dirichlet data

We consider an inverse problem arising in laser-induced thermotherapy, a minimally invasive method for cancer treatment, in which cancer tissues is destroyed by coagulation. For the dosage planning quantitatively reliable numerical simulation are indispensable. To this end the identification of the thermal growth kinetics of the coagulated zone is of crucial importance. Mathematically, this problem is a nonlinear and nonlocal parabolic inverse heat source problem. We show in this paper that the temperature dependent thermal growth parameter can be identified uniquely from a one-point measurement.     

Numerical solution of forward and backward problem for 2-D heat conduction equation

For a two-dimensional heat conduction problem, we consider its initial boundary value problem and the related inverse problem of determining the initial temperature distribution from transient temperature measurements. The conditional stability for this inverse problem and the error analysis for the Tikhonov regularization are presented. An implicit inversion method, which is based on the regularization technique and the successive over-relaxation (SOR) iteration process, is established. Due to the explicit difference scheme for a direct heat problem developed in this paper, the inversion process is very efficient, while the application of SOR technique makes our inversion convergent rapidly. Numerical results illustrating our method are also given.     

A numerical method for solving an inverse thermoacoustic problem

In this paper, we consider an inverse problem of determining the initial condition of an initial boundary value problem for the wave equation with some additional information about solving a direct initial boundary value problem. The information is obtained from measurements at the boundary of the solution domain. The purpose of our paper is to construct a numerical algorithm for solving the inverse problem by an iterative method called a method of simple iteration (MSI) and to study the resolution quality of the inverse problem as a function of the number and location of measurement points. Three two-dimensional inverse problem formulations are considered. The results of our numerical calculations are presented. It is shown that the MSI decreases the objective functional at each iteration step. However, due to the ill-posedness of the inverse problem the difference between the exact and approximate solutions decreases up to some fixed number k _min, and then monotonically increases. This shows the regularizing properties of the MSI, and the iteration number can be considered a regularization parameter.     

Inverse coefficient problem for a second‐order elliptic equation with nonlocal boundary conditions

In this research article, the inverse problem of finding a time‐dependent coefficient in a second‐order elliptic equation is investigated. The existence and the uniqueness of the classical solution of the problem under consideration are established. Numerical tests using the finite‐difference scheme combined with an iteration method are presented, and the sensitivity of this scheme with respect to noisy overdetermination data is illustrated. Copyright © 2015 John Wiley & Sons, Ltd.     

Two regularization methods for a spherically symmetric inverse heat conduction problem

In this paper, we consider a spherically symmetric inverse heat conduction problem of determining the internal surface temperature of a hollow sphere from the measured data at a fixed location inside it. This is an ill-posed problem in the sense that the solution (if it exists) does not depend continuously on the data. A Tikhonov type’s regularization method and a Fourier regularization method are applied to formulate regularized solutions which are stably convergent to the exact ones with order optimal error estimates.     

Noniterative solution of inverse problems by the linear least square method

In this paper, a noniterative linear least-squares error method developed by Yang and Chen for solving the inverse problems is re-examined. For the method, condition for the existence of a unique solution and the error bound of the resulting inverse solution considering the measurement errors are derived. Though the method was shown to be able to give the unique inverse solution at only one iteration in the literature, however, it is pointed out with two examples that for some inverse problems the method is practically not applicable, once the unavoidable measurement errors are included. The reason behind this is that the so-called reverse matrix for these inverse problems has a huge number of 1-norm, thus, magnifying a small measurement error to an extent that is unacceptable for the resulting inverse solution in a practical sense. In other words, the method fails to yield a reasonable solution whenever applied to an ill-conditioned inverse problem. In such a case, two approaches are recommended for decreasing the very high condition number: (i) by increasing the number of measurements or taking measurements as close as possible to the location at which the to-be-estimated unknown condition is applied, and (ii) by using the singular value decomposition (SVD).     

Identification of a spacewise dependent heat source

The inverse problem of determining the temperature of a heat conductor together with an unknown spacewise dependent heat source from measured final data or time-average temperature observation is studied. The weak solution theory is applied for calculating the gradient of the least-squares functional that is minimized. For the general case when the heat source is the product between a known function h(x,t)$h(x\text{,}t)$ and the unknown source function f(x)$f(x)$ new explicit formulae, derived via the solution of the corresponding adjoint problem, are obtained. Numerical results obtained using the conjugate gradient method are presented and discussed.     

An inverse backward problem for degenerate parabolic equations

This work studies the inverse problem of reconstructing an initial value function in the degenerate parabolic equation using the final measurement data. Problems of this type have important applications in the field of financial engineering. Being different from other inverse backward parabolic problems, the mathematical model in our article may be allowed to degenerate at some part of boundaries, which may lead to the corresponding boundary conditions missing. The conditional stability of the solution is obtained using the logarithmic convexity method. A finite difference scheme is constructed to solve the direct problem and the corresponding stability and convergence are proved. The Landweber iteration algorithm is applied to the inverse problem and some typical numerical experiments are also performed in the paper. The numerical results show that the proposed method is stable and the unknown initial value is recovered very well.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1900–1923, 2017     

一类抛物型方程系数反问题的分裂算法

考虑利用终端时刻的温度u(x,T)=Z_T(x)反演热传导方程u_t-a~2u_(xx) q(x)u=0,x∈(0,1)中的未知系数q(x)的反问题.通过引进变换v(x,t)=(u_t(x,t)/u(x,t))将此非线性不适定问题的求解分解为两步.首先利用输入数据迭代求解一个非线性的正问题(该过程独立于未知系数),得到其迭代解v~(k)(x,t).其次利用q(x)与v(x,t)的关系式求出q(x)的近似解.对提出的反演方法,证明了采用的变换的可行性,得到了原反问题与由变换后的非线性正问题反演q(x)的等价性并且证明了迭代解的收敛性,给出了收敛速度.数值结果表明了该方法的有效性.     

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

The article presents a quasi-one-dimensional method for solving the inverse problem of electromagnetic sounding. The quasi-one-dimensional method is an iteration process that in each iteration solves a parametric one-dimensional inverse problem and a two-dimensional direct problem. The solution results of these problems are applied to update the input values for the parametric one-dimensional inverse problem in the next iteration. The method has been implemented for a two-dimensional inverse problem of magnetotelluric sounding in a quasi-layered medium.     

An inverse hyperbolic heat conduction problem in estimating surface heat flux of a living skin tissue

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to solve the inverse hyperbolic heat conduction problem in estimating the unknown time-dependent surface heat flux in a living skin tissue from the temperature measurements taken within the tissue. The inverse solutions will be justified based on the numerical experiments in which three different heat flux distributions are to be determined. The temperature data obtained from the direct problem are used to simulate the temperature measurements. The influence of measurement errors upon the precision of the estimated results is also investigated. Results show that an excellent estimation on the time-dependent surface heat flux can be obtained for the test cases considered in this study.     

Large time behaviors of hot spots for the heat equation with a potential

We consider the Cauchy problem of the heat equation with a potential which behaves like the inverse square at infinity. In this paper we study the large time behavior of hot spots of the solutions for the Cauchy problem, by using the asymptotic behavior of the potential at the space infinity.     

Spacewise coefficient identification in steady and unsteady one-dimensional diffusion problems

In this paper the determination of the spacewise dependent materialproperty coefficients and the function solution in both steadyand unsteady diffusion problems are analysed. For a one-dimensionalquasi-heterogeneous material with square-root harmonic conductivityit is shown that a single measurement of the conductivity andthe flux on the boundary is sufficient to determine uniquelythe unknown physical property and the function solution.     

Uniqueness and stability in determining the heat radiative coefficient,the initial temperature and a boundary coefficient in a parabolic equation

We consider an inverse parabolic problem. We prove that the heat radiative coefficient, the initial temperature and a boundary coefficient can be simultaneously determined from the final overdetermination, provided that the heat radiative coefficient is a priori known in a small subdomain. Moreover we establish a stability estimate for this inverse problem.     

Boundary Identification for a Blast Furnace

In this paper,the authors discuss an inverse boundary problem for the axi- symmetric steady-state heat equation,which arises in monitoring the boundary corrosion for the blast-furnace.Measure temperature at some locations are used to identify the shape of the corrosion boundary. The numerical inversion is complicated and consuming since the wear-line varies during the process and the boundary in the heat problem is not fixed.The authors suggest a method that the unknown boundary can be represented by a given curve plus a small perturbation,then the equation can be solved with fixed boundary,and a lot of computing time will be saved. A method is given to solve the inverse problem by minimizing the sum of the squared residual at the measuring locations,in which the direct problems are solved by axi- symmetric fundamental solution method. The numerical results are in good agreement with test model data as well as industrial data,even in severe corrosion case.