Source term identification in 1-D IHCP

We propose stable numerical solutions for the simultaneous identification of temperature, temperature gradient, and general source terms in the one-dimensional inverse heat conduction problem (IHCP).

The numerical solution consists of a regularization procedure, based on the mollification method,and a marching scheme for the solution of the stabilized problem. The stability, error analysis and implementation of the algorithm are presented together with a set of numerical results.     

Solvability of the Nonlocal Inverse Parabolic Problem and Numerical Results

In this paper, we consider the unique solvability of the inverse problem of determining the right-hand side of a parabolic equation whose leading coefficient depends on time variable under nonlocal integral overdetermination condition. We obtain sufficient conditions for the unique solvability of the inverse problem. The existence and uniqueness of the solution of the inverse parabolic problem upon the data are established using the fixed point theorem. This inverse problem appears extensively in the modelling of various phenomena in engineering and physics. For example, seismology, medicine, fusion welding, continuous casting, metallurgy, aircraft, oil and gas production during drilling and operation of wells. In addition, the numerical solution of the inverse problem is studied by using the Crank-Nicolson finite difference method together with the Tikhonov regularization to find a stable and accurate approximate solution of finite differences. The resulting nonlinear system of parabolic equation is solved computationally using the MATLAB subroutine lsqnonlin. Both analytical and numerically simulated noisy input data are inverted. The root mean square error values for various noise levels for both continuous and discontinuous time-dependent heat source term are compared. Numerical results presented for two examples show the efficiency of the computational method and the accuracy and stability of the numerical solution even in the presence of noise in the input data. Furthermore, the choice of the regularization parameter is also discussed based on the trial and error technique.     

A meshless method to the numerical solution of an inverse reaction–diffusion–convection problem

In this paper, the determination of the source term in a reaction–diffusion convection problem is investigated. First with suitable transformations, the problem is reduced, then a new meshless method based on the use of the heat polynomials as basis functions is proposed to solve the inverse problem. Due to the ill-posed inverse problem, the Tikhonov regularization method with a generalized cross-validation criterion is employed to obtain a numerical stable solution. Finally, some numerical examples are presented to show the accuracy and effectiveness of the algorithm.     

Adaptive filtering of a random signal in Gaussian white noise

We consider the problem of estimating an infinite-dimensional vector θ observed in Gaussian white noise. Under the condition that components of the vector have a Gaussian prior distribution that depends on an unknown parameter β, we construct an adaptive estimator with respect to β. The proposed method of estimation is based on the empirical Bayes approach.     

求解热传导系数反问题的量子行为粒子群算法

量子行为粒子群优化算法（QPSO）和Tikhonov正则化方法用来求解热传导反问题,近似估计平板随时间变化的热传导系数。由于热传导系数的函数形式是未知的,所以问题可以归结为函数估计问题。求解过程是基于最小二乘模型的,采用的是嵌在平板中的传感器所测量得到的温度,优化过程由QPSO算法来求解。给出了由L曲线方法选择正则参数的详细过程。提出算法的有效性经过了数值实验的验证。传感器的位置和数量对结果的影响也做了研究。给出了与共轭梯度法的比较。     

GeD spline estimation of multivariate Archimedean copulas

A new multivariate Archimedean copula estimation method is proposed in a non-parametric setting. The method uses the so-called Geometrically Designed splines (GeD splines) to represent the cdf of a random variable W_θ, obtained through the probability integral transform of an Archimedean copula with parameter θ. Sufficient conditions for the GeD spline estimator to possess the properties of the underlying theoretical cdf, K(θ,t), of W_θ, are given. The latter conditions allow for defining a three-step estimation procedure for solving the resulting non-linear regression problem with linear inequality constraints. In the proposed procedure, finding the number and location of the knots and the coefficients of the unconstrained GeD spline estimator and solving the constraint least-squares optimisation problem are separated. Thus, the resulting spline estimator is used to recover the generator and the related Archimedean copula by solving an ordinary differential equation. The proposed method is truly multivariate, it brings about numerical efficiency and as a result can be applied with large volumes of data and for dimensions d≥2, as illustrated by the numerical examples presented.     

Numerical approximation of solution of nonhomogeneous backward heat conduction problem in bounded region

In this paper we consider a numerical approximation of solution of nonhomogeneous backward heat conduction problem (BHCP) in bounded region based on Tikhonov regularization method. Error estimate at t=0$t=0$ for this method is provided. According to the error estimate, a selection of regularization parameter is given. Meanwhile, a numerical implementation is described and the numerical results show that our algorithm is effective.     

Determination of the time-dependent reaction coefficient and the heat flux in a nonlinear inverse heat conduction problem

ABSTRACT

Diffusion processes with reaction generated by a nonlinear source are commonly encountered in practical applications related to ignition, pyrolysis and polymerization. In such processes, determining the intensity of reaction in time is of crucial importance for control and monitoring purposes. Therefore, this paper is devoted to such an identification problem of determining the time-dependent coefficient of a nonlinear heat source together with the unknown heat flux at an inaccessible boundary of a one-dimensional slab from temperature measurements at two sensor locations in the context of nonlinear transient heat conduction. Local existence and uniqueness results for the inverse coefficient problem are proved when the first three derivatives of the nonlinear source term are Lipschitz continuous functions. Furthermore, the conjugate gradient method (CGM) for separately reconstructing the reaction coefficient and the heat flux is developed. The ill-posedness is overcome by using the discrepancy principle to stop the iteration procedure of CGM when the input data is contaminated with noise. Numerical results show that the inverse solutions are accurate and stable.     

A method of fundamental solutions for radially symmetric and axisymmetric backward heat conduction problems

We propose and investigate an application of the method of fundamental solutions (MFS) to the radially symmetric and axisymmetric backward heat conduction problem (BHCP) in a solid or hollow cylinder. In the BHCP, the initial temperature is to be determined from the temperature measurements at a later time. This is an inverse and ill-posed problem, and we employ and generalize the MFS regularization approach [B.T. Johansson and D. Lesnic, A method of fundamental solutions for transient heat conduction, Eng. Anal. Boundary Elements 32 (2008), pp. 697–703] for the time-dependent heat equation to obtain a stable and accurate numerical approximation with small computational cost.     

High-order scheme for determination of a control parameter in an inverse problem from the over-specified data

The problem of finding the solution of partial differential equations with source control parameter has appeared increasingly in physical phenomena, for example, in the study of heat conduction process, thermo-elasticity, chemical diffusion and control theory. In this paper we present a high order scheme for determining unknown control parameter and unknown solution of parabolic inverse problem with both integral overspecialization and overspecialization at a point in the spatial domain. In these equations, we first approximate the spatial derivative with a fourth order compact scheme and reduce the problem to a system of ordinary differential equations (ODEs). Then we apply a fourth order boundary value method for the solution of resulting system of ODEs. So the proposed method has fourth order accuracy in both space and time components and is unconditionally stable due to the favorable stability property of boundary value methods. Several numerical examples and also some comparisons with other methods in the literature will be investigated to confirm the efficiency of the new procedure.     

Two regularization methods and the order optimal error estimates for a sideways parabolic equation

In this paper, we consider an inverse heat conduction problem which appears in some applied subjects. This problem is ill-posed 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. A numerical example shows that the computational effect of these methods are all satisfactory.     

应用QPSO算法求解二维热传导反问题

热传导反问题在国内研究起步较晚,研究方法有很多,但通常方法很难较好地接近全局最优。在经典的微粒群优化算法(PSO)的基础上,通过研究基于量子行为的微粒群优化算法(QPSO)提出了应用基于量子行为的微粒群优化算法进行二维热传导参数优化,具体介绍依据目标函数如何利用上述的算法去寻找最优参数组合。在具体应用中为了提高算法的收敛性和稳定性对算法进行了改进,并进行了大量实验,结果显示在解决热传导反问题优化问题中,基于QPSO算法的性能优越,证明QPSO在热传导领域具有很大的实际应用价值。     

A fast and adaptive method for complex-valued SAR image denoising based on <Emphasis Type="Italic">l</Emphasis><Subscript><Emphasis Type="Italic">k</Emphasis></Subscript> norm regularization

This paper developed a fast and adaptive method for SAR complex image denoising based on l _k norm regularization, as viewed from parameters estimation. We firstly establish the relationship between denoising model and ill-posed inverse problem via convex half-quadratic regularization, and compare the difference between the estimator variance obtained from the iterative formula and biased Cramer-Rao bound, which proves the theoretic flaw of the existent methods of parameter selection. Then, the analytic expression of the model solution as the function with respect to the regularization parameter is obtained. On this basis, we study the method for selecting the regularization parameter through minimizing mean-square error of estimators and obtain the final analytic expression, which resulted in the direct calculation, high processing speed, and adaptability. Finally, the effect of regularization parameter selection on the resolution of point targets is analyzed. The experiment results of simulation and real complex-valued SAR images illustrate the validity of the proposed method. Supported by the National Natural Science Foundation of China (Grant No. 60572136), the Fundamental Research Fund of NUDT (Grant No. JC0702005)     

The density function reconstruction of surface sources from a single Cauchy measurement

The inverse problem of reconstructing sources is explored when a single boundary Cauchy data is postulated on the potential. We are particularly involved in sources supported by (hyper-)surfaces. Mild assumptions are required on the location of these supports and the calculation of the charge density function is then aimed. We consider a variational formulation, based on a duplication artifice of the potential and we check the symmetry and the positive definiteness of the weak problem. Because of the severe ill-posedness, the use of a regularization is mandatory for a safe approximation of the solution. Lavrentiev’s method is therefore recommended in the context owing to the symmetry and the positivity. We check why that regularization turns out to be a Tikhonov method for some underlying shadow equation that is not needed in computations and is therefore never explicitly constructed. Results stated in a wide literature for the Tikhonov regularization applies as well to our variational problem. An important consequence is that the Morozov Discrepancy Principle, we use for the selection of the regularization parameter yields a convergent strategy. Now, that the Discrepancy Principle requires the residual of that inaccessible ‘shadow equation’, we explain how the Kohn–Vogelius function allows for the computation of that residual.     

电磁逆散射成像的一种混合正则化方法

电磁逆散射成像问题数值求解中,非线性逆散射方程及其对应的离散方程组具有明显的不适定性,针对求解通常所用Tikhonov正则化方法的参数选择在先验选取时缺乏有效的误差信息,而后验选取时需要更多计算量求解有关参数的方程的困难,本文中将小参数Tikhonov正则化方法与共轭梯度法结合,提出了不适定方程组的混合正则化方法。数据仿真表明,该方法既可保证正则化效果,也减少了计算量。     

Modified particle swarm optimization and fuzzy regularization for pseudo de-convolution of spatially variant blurs

We propose a modified particle swarm optimization (MPSO) based method for Pseudo De-convolution of the ill-posed inverse problem namely, the space-variant image degradation (SVD). In this paper, SVD is simulated by the pseudo convolution of different sub-regions of the image with different known blurring kernels and additive random noise with unknown variance. Two heuristic modifications are proposed in PSO: 1) Initialization of the swarm and 2) Mutation of the global best. Fuzzy logic is applied for the computation of regularization parameter (RP) to cater for the sensitivity of the problem. The computation of RP is crucial due to the additive noise in the SVD image. Thus mathematical morphology (MM) is applied for better extraction of spatial activity from the distorted image. The performance of the proposed method is evaluated with different test images and noise powers. Comparative analysis demonstrates the superiority of proposed restoration, in terms of quantitative measures, over well-known existing and state-of-the-art SVD approaches.     

A boundary element method for a multi-dimensional inverse heat conduction problem

In this paper, we investigate a variational method for a multi-dimensional inverse heat conduction problem in Lipschitz domains. We regularize the problem by using the boundary element method coupled with the conjugate gradient method. We prove the convergence of this scheme with and without Tikhonov regularization. Numerical examples are given to show the efficiency of the scheme.     

An optimal regularization method for space-fractional backward diffusion problem

In this paper, a space-fractional backward diffusion problem (SFBDP) in a strip is considered. By the Fourier transform, we proposed an optimal modified method to solve this problem in the presence of noisy data. The convergence estimates for the approximate solutions with the regularization parameter selected by an a priori and an a posteriori strategy are provided, respectively. Numerical experiments show that the proposed methods are effective and stable.     

Source term identification for an axisymmetric inverse heat conduction problem

We consider an inverse heat source problem of determining the heat source term from the final temperature history of a cylinder. This problem is ill-posed. A simplified Tikhonov regularization method is applied to formulate regularized solution, which is stably convergent to the exact one with a logarithmic type error estimate.     

Increasing efficiency of alternating triangular method based on improved spectral estimates

A modified alternating triangular method is constructed for three-dimensional difference elliptic equations with the linear source function. Under the special restriction on the source function, the method requires $n_0 (\varepsilon ) \cong O\left( {{1 \mathord{\left/ {\vphantom {1 {\sqrt[4]{{\left\| h \right\|}}}}} \right. \kern-0em} {\sqrt[4]{{\left\| h \right\|}}}}} \right)$ iterations. The improved estimate of the parameter is obtained for the alternating triangular method after the diagonal component of the matrix of the problem is considered separately, which helped reduce the number of iterations twice asymptotically. The improved spectral estimates and results of numerical experiments for the Dirichlet problem of the Poisson equation with the linear source function of the form q(x)u(x) and nonstationary heat conduction equation are given.