On possibility of obtaining linear accuracy evaluation of approximate solutions to inverse problems

We consider a concept of linear a priori estimate of the accuracy for approximate solutions to inverse problems with perturbed data. We establish that if the linear estimate is valid for a method of solving the inverse problem, then the inverse problem is well-posed according to Tikhonov. We also find conditions, which ensure the converse for the method of solving the inverse problem independent on the error levels of data. This method is well-known method of quasi-solutions by V. K. Ivanov. It provides for well-posed (according to Tikhonov) inverse problems the existence of linear estimates. If the error levels of data are known, a method of solving well-posed according to Tikhonov inverse problems is proposed. This method called the residual method on the correctness set (RMCS) ensures linear estimates for approximate solutions. We give an algorithm for finding linear estimates in the RMCS.     

强迫对流管道内壁结垢层几何形状的逆运算估计

利用逆运算法中的共轭梯度法与差异原理,通过测量管壁内的温度,来估算一管流系统内壁结垢层厚度的几何形状．过程中未预先设定结垢层厚度的函数形式．因此,可将这类逆运算问题归类为“函数预测”．逆运算过程的管壁温度测量值,可由直接解法所求得的温度数值来仿真实际的测量温度．并用测量误差来检验逆运算分析的正确性．数值实验结果显示,管内壁结垢层厚度的几何形状可获得极佳的估算值．所提出的技术可用作管路维修的预警系统,当管壁结垢层厚度超出某预先设定值时可适时发出维修警示．     

The inverse solution of the atomic mixing equations by an operator-splitting method

The quantification problem of recovering the original material distribution from secondary ion mass spectrometry (SIMS) data is considered in this paper. It is an inverse problem, is ill-posed and hence it requires a special technique for its solution. The quantification problem is essentially an inverse diffusion or (classically) a backward heat conduction problem. In this paper an operator-splitting method (that is proposed in a previous paper by the first author for the solution of inverse diffusion problems) is developed for the solution of the problem of recovering the original structure from the SIMS data. A detailed development of the quantification method is given and it is applied to typical data to demonstrate its effectiveness.     

Identification of a time-dependent source term for a time fractional diffusion problem

In this paper, we study an inverse problem of identifying a time-dependent term of an unknown source for a time fractional diffusion equation using nonlocal measurement data. Firstly, we establish the conditional stability for this inverse problem. Then two regularization methods are proposed to for reconstructing the time-dependent source term from noisy measurements. The first method is an integral equation method which formulates the inverse source problem into an integral equation of the second kind; and a prior convergence rate of regularized solutions is derived with a suitable choice strategy of regularization parameters. The second method is a standard Tikhonov regularization method and formulates the inverse source problem as a minimizing problem of the Tikhonov functional. Based on the superposition principle and the technique of finite-element interpolation, a numerical scheme is proposed to implement the second regularization method. One- and two-dimensional examples are carried out to verify efficiency and stability of the second regularization method.     

Reconstruction of an Impedance in Two-dimensions from Spectral Data

The two-dimensional spectral inverse problem involves the reconstruction of an unknown coefficient in an elliptic partial differential equation from spectral data, such as eigenvalues. Projection of the boundary value problem and the unknown coefficient onto appropriate vector spaces leads to a matrix inverse problem. Unique solutions of this matrix inverse problem exist provided that the eigenvalue data is close to the eigenvalues associated with the analogous constant coefficient boundary value problem. We discuss here the application of such a technique to the reconstruction of an impedance p in the boundary value problem $$ \eqalign{ -\nabla (\,p \nabla u) = \lambda p u \hbox {\quad in R} \cr u = 0 \hbox {\quad on R}}$$ where R is a rectangular domain. The matrix inverse problem, although nonstandard, is solved by a fixed-point iterative method and an impedance function p * is constructed which has the same m lowest eigenvalues as the unknown p . Numerical evidence of the success of the method will be presented.     

Poisson方程热源识别反问题

探讨了半带状区域上二维Poisson方程只含有一个空间变量的热源识别反问题.这类问题是不适定的,即问题的解(如果存在的话)不连续依赖于测量数据.利用Carasso-Tikhonov正则化方法,得到了问题的一个正则近似解,并且给出了正则解和精确解之间具有Holder型误差估计.数值实验表明Carasso-Tikhonov正则化方法对于这种热源识别是非常有效的.     

Linear estimates of accuracy for approximate solutions of inverse problems

In this paper, we explore the question of which non-linear inverse problems, which are solved by a selected regularization method, may have so-called linear a priori accuracy estimates – that is, the accuracy of corresponding approximate solutions linearly depends on the error level of the data. In particular, we prove that if such a linear estimate exists, then the inverse problem under consideration is well posed, according to Tikhonov. For linear inverse problems, we find that the existence of linear estimates lead to, under some assumptions, the well-posedness (according to Tikhonov) on the whole space of solutions. Moreover, we consider a method for solving inverse problems with guaranteed linear estimates, called the residual method on the correctness set (RMCS). The linear a priori estimates of absolute and relative accuracy for the RMCS are presented, as well as analogous a posteriori estimates. A numerical illustration of obtaining linear a priori estimates for appropriate parametric sets of solutions using RMCS is given in comparison with Tikhonov regularization. The a posteriori estimates are calculated on these parametric sets as well.     

Identification of the zeroth-order coefficient and fractional order in a time-fractional reaction-diffusion-wave equation

In this paper, we investigate an inverse problem of recovering the zeroth-order coefficient and fractional order simultaneously in a time-fractional reaction-diffusion-wave equation by using boundary measurement data from both of uniqueness and numerical method. We prove the uniqueness of the considered inverse problem and the Lipschitz continuity of the forward operator. Then the inverse problem is formulated into a variational problem by the Tikhonov-type regularization. Based on the continuity of the forward operator, we prove that the minimizer of the Tikhonov-type functional exists and converges to the exact solution under an a priori choice of regularization parameter. The steepest descent method combined with Nesterov acceleration is adopted to solve the variational problem. Three numerical examples are presented to support the efficiency and rationality of our proposed method.     

Stability estimates for the solutions to inverse extremal problems for the Helmholtz equation

The inverse problems are under study for the Helmholtz equation describing acoustic scattering at a three-dimensional inclusion. Some optimization method reduces these problems to the inverse extremum problems with variable refraction index and boundary source density as controls. We prove that these problems are solvable and derive the optimality systems that describe necessary optimality conditions. Analysis of the optimality systems leads us to some sufficient conditions on the input data ensuring the uniqueness and stability of optimal solutions.     

A multi-scale image reconstruction algorithm for electrical capacitance tomography

Electrical capacitance tomography (ECT) is considered as a promising process tomography (PT) technology, and its successful applications depend mainly on the precision and speed of the image reconstruction algorithms. In this paper, based on the wavelet multi-scale analysis method, an efficient image reconstruction algorithm is presented. The original inverse problem is decomposed into a sequence of inverse problems, which are solved successively from the largest scale to the smallest scale. At different scales, the inverse problem is solved by a generalized regularized total least squares (TLS) method, which is developed using a combinational minimax estimation method and an extended stabilizing functional, until the solution of the original inverse problem is found. The homotopy algorithm is employed to solve the objective functional. The proposed algorithm is tested by the noise-free capacitance data and the noise-contaminated capacitance data, and excellent numerical performances and satisfactory results are observed. In the cases considered in this paper, the reconstruction results show remarkable improvement in the accuracy. The spatial resolution of the reconstructed images by the proposed algorithm is enhanced and the artifacts in the reconstructed images can be eliminated effectively. As a result, a promising algorithm is introduced for ECT image reconstruction.     

Parameter selection by discrete mollification and the numerical solution of the inverse heat conduction problem

A procedure for the numerical solution of the one-dimensional inverse heat conduction problem, based on the computaion of the solution associated with a suitable filtered version of the noisy data by discrete mollification is presented and a parameter choice criterion, which automatically determines the radius of mollification as a function of the amount of noise in the data, is introduced. Several numerical examples of interest are also analyzed, showing the accuracy and stability properties of the method.     

A novel mollifier inversion scheme for the Laplace transform

In this article we present a novel inversion method for the Laplace transform for non‐equidistant scanning points applying the approximate inverse to this transform. The approximate inverse is a regularization technique for inverse problems based on evaluations of scalar products of the given data with so called reconstruction kernels. Each kernel solves a system of linear equations defined by the adjoint of the Laplace transform and dilatation invariant mollifiers, which are designed articularly for this operator. The paper includes numerical results.     

An Algorithm for Solving Inverse Geoelectrics Problems Based on the Neural Network Approximation

A neural network approximation algorithm for solving inverse geoelectrics problems in the class of grid (block) models of media is presented. The algorithm is based on using neural networks for constructing an approximate inverse operator and enables formalized construction of solutions of inverse geoelectrics problem with a total number of sought-for medium parameters of ~ n · 10³. The correctness of the problem of constructing neural network inverse operators is considered. A posteriori estimates of the degree of ambiguity of solutions of the resulting inverse problem are calculated. The operation of the algorithm is illustrated by examples of 2D and 3D inversions of synthetic and field geoelectric data obtained by the MTS method.     

Discrete-analytic schemes for solving an inverse coefficient heat conduction problem in a layered medium with gradient methods

A method for constructing numerical schemes for an inverse coefficient heat conduction problem with boundary measurement data and piecewise-constant coefficients is considered. Some numerical schemes for a gradient optimization algorithm to solve the inverse problem are presented. The method is based on locally-adjoint problems in combination with approximation methods in Hilbert spaces.     

Recovering the Initial Distribution for a Time-Fractional Diffusion Equation

We consider the inverse problem of reconstructing the initial condition of a one-dimensional time-fractional diffusion equation from measurements collected at a single interior location over a finite time-interval. The method relies on the eigenfunction expansion of the forward solution in conjunction with a Tikhonov regularization scheme to control the instability inherent in the problem. We show that the inverse problem has a unique solution provided exact data is given, and prove stability results regarding the regularized solution. Numerical realization of the method and illustrations using a finite-element discretization are given at the end of this paper.     

A modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation

In this paper, we consider an inverse source problem for a time-fractional diffusion equation with variable coefficients in a general bounded domain. That is to determine a space-dependent source term in the time-fractional diffusion equation from a noisy final data. Based on a series expression of the solution, we can transform the original inverse problem into a first kind integral equation. The uniqueness and a conditional stability for the space-dependent source term can be obtained. Further, we propose a modified quasi-boundary value regularization method to deal with the inverse source problem and obtain two kinds of convergence rates by using an a priori and an a posteriori regularization parameter choice rule, respectively. Numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed method.     

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).     

用3个向量对构造梁振动系统的刚度矩阵

针对梁的离散化模型的刚度矩阵是五对角矩阵,梁振动反问题的实质是实对称五对角矩阵的特征值反问题.该文利用向量对、Moore-Penrose广义逆给出了实对称五对角矩阵向量对反问题存在唯一解的条件,并结合矩阵分块讨论了双对称五对角矩阵向量对反问题解存在唯一的条件,进而计算了次对角线位置元素为负,其它位置元素均为正的实对称五对角矩阵特征值反问题.由于构造梁的离散模型需要的数据可由测试得到,故而其结果适合于模态分析、系统结构的分析与设计等方面应用.最后给出了数值算例,通过数值讨论说明方法的有效性.     

Some classes of inverse problems of determining the source function in convection–diffusion systems

We consider the generalized solvability of convection–diffusion systems and study the inverse problem of determining the right-hand side (the source function) of such a system from integral overdetermination data. The solution of a parabolic system is understood in the generalized sense, and distributions of certain classes are allowed as right-hand sides. Under certain conditions on the problem data, we show that the inverse problem is well-posed in the Sobolev classes.