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.     

An inverse problem of identifying the coefficient of first-order in a degenerate parabolic equation

This work studies an inverse problem of determining the first-order coefficient of degenerate parabolic equations using the measurement data specified at a fixed internal point. Being different from other ordinary parameter identification problems in parabolic equations, in our mathematical model there exists degeneracy on the lateral boundaries of the domain, which may cause the corresponding boundary conditions to go missing. By the contraction mapping principle, the uniqueness of the solution for the inverse problem is proved. A numerical algorithm on the basis of the predictor-corrector method is designed to obtain the numerical solution and some typical numerical experiments are also performed in the paper. The numerical results show that the proposed method is stable and the unknown function is recovered very well. The results obtained in the paper are interesting and useful, and can be extended to other more general inverse coefficient problems of degenerate PDEs.     

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.     

一类二次规划逆问题的交替方向数值方法

考虑求解一类二次规划逆问题的交替方向数值算法．首先给出矩阵变量子问题解的显示表达式,而后构造了两个求解向量变量子问题近似解的数值算法,其中一个算法基于不动点原理,另一算法则应用半光滑牛顿法．数值实验表明,所提出的算法能够快速高效地求解二次规划逆问题．     

AN ADAPTIVE INVERSE ITERATION FEM FOR THE INHOMOGENEOUS DIELECTRIC WAVEGUIDES

We introduce an adaptive finite element method for computing electromagnetic guided waves in a closed, inhomogeneous, pillared three-dimensional waveguide at a given frequency based on the inverse iteration method. The problem is formulated as a generalized eigenvalue problems. By modifying the exact inverse iteration algorithm for the eigenvalue problem, we design a new adaptive inverse iteration finite element algorithm. Adaptive finite element methods based on a posteriori error estimate are known to be successful in resolving singularities of eigenfunctions which deteriorate the finite element convergence. We construct a posteriori error estimator for the electromagnetic guided waves problem. Numerical results are reported to illustrate the quasi-optimal performance of our adaptive inverse iteration finite element 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.     

Numerical methods for some inverse problems of heart electrophysiology

We consider numerical methods for solving inverse problems that arise in heart electrophysiology. The first inverse problem is the Cauchy problem for the Laplace equation. Its solution algorithm is based on the Tikhonov regularization method and the method of boundary integral equations. The second inverse problem is the problem of finding the discontinuity surface of the coefficient of conductivity of a medium on the basis of the potential and its normal derivative given on the exterior surface. For its numerical solution, we suggest a method based on the method of boundary integral equations and the assumption on a special representation of the unknown surface.     

Solving an Inverse Problem for an Elliptic Equation by d.c. Programming

An inverse problem of determination of a coefficient in an elliptic equation is considered. This problem is ill-posed in the sense of Hadamard and Tikhonov's regularization method is used for solving it in a stable way. This method requires globally solving nonconvex optimization problems, the solution methods for which have been very little studied in the inverse problems community. It is proved that the objective function of the corresponding optimization problem for our inverse problem can be represented as the difference of two convex functions (d.c. functions), and the difference of convex functions algorithm (DCA) in combination with a branch-and-bound technique can be used to globally solve it. Numerical examples are presented which show the efficiency of the method.     

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.     

A sequential algorithm and error sensitivity analysis for the inverse heat conduction problems with multiple heat sources

This paper proposes a sequential approach to determine the unknown parameters for inverse heat conduction problems which have multiple time-dependent heat sources. There are two main aims in this study, one is to derive an inverse algorithm that can estimate the unknown conditions effectively, and the other is to bring up a theoretical sensitivity analysis to discuss what causes the growth of errors. This paper has three major achievements with regard to the literature on IHCPs, as follows: (1) proposing an efficient sequential inverse algorithm that can simultaneously determine several unknown time-dependent parameters; (2) exploring why the sequential function specification method can provide a stable but inaccurate estimation when tackling problems with larger measurement errors; and (3) discussing the sensitivity problem and analyzing what factors cause the growth in error sensitivity. Three examples are applied to demonstrate the performance of the proposed method, and the numerical results show that the accurate estimations can be obtained by alleviating the error sensitivity when the measurement error is considered.     

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.     

The Kirsch-Kress method for inverse scattering by infinite locally rough interfaces

This paper is concerned with the inverse problem of reconstructing an infinite, locally rough interface from the scattered field measured on line segments above and below the interface in two dimensions. We extend the Kirsch-Kress method originally developed for inverse obstacle scattering problems to the above inverse transmission problem with unbounded interfaces. To this end, we reformulate our inverse problem as a nonlinear optimization problem with a Tikhonov regularization term. We prove the convergence of the optimization problem when the regularization parameter tends to zero. Finally, numerical experiments are carried out to show the validity of the inversion algorithm.     

Note on “Inverse minimum cost flow problems under the weighted Hamming distance”

Jiang et al. proposed an algorithm to solve the inverse minimum cost flow problems under the bottleneck-type weighted Hamming distance [Y. Jiang, L. Liu, B. Wuc, E. Yao, Inverse minimum cost flow problems under the weighted Hamming distance, European Journal of Operational Research 207 (2010) 50–54]. In this note, it is shown that their proposed algorithm does not solve correctly the inverse problem in the general case due to some incorrect results in that article. Then, a new algorithm is proposed to solve the inverse problem in strongly polynomial time. The algorithm uses the linear search technique and solves a shortest path problem in each iteration.     

On the complexity of inverse convex ordered 1-median problem on the plane and on tree networks

An ordered median function is used in location theory to generalize a class of problems, including median and center problems. In this paper we consider the complexity of inverse ordered 1-median problems on the plane and on trees, where the multipliers are sorted nondecreasingly. Based on the convexity of the objective function, we prove that the problems with variable weights or variable coordinates on the line are NP-hard. Then we can directly get the NP-hardness result for the corresponding problem on the plane. We finally develop a cubic time algorithm that solves the inverse convex ordered 1-median problem on trees with relaxation on modification bounds.     

二维椭圆型方程反问题中优化算法的比较

近年来，决定椭圆型方程系数反问题在地磁、地球物理、冶金和生物等实际问题上有着广泛的应用．本文讨论了二维的决定椭圆型方程系数反问题的数值求解方法．由误差平方和最小原则，这个反问题可化为一个变分问题，并进一步离散化为一个最优化问题，其目标函数依赖于要决定的方程系数．本文着重考察非线性共轭梯度法在此最优化问题数值计算中的表现，并与拟牛顿法作为对比．为了提高算法的效率我们适当选择加快收敛速度的预处理矩阵．同时还考察了线搜索方法的不同对优化算法的影响．数值实验的结果表明，非线性共轭梯度法在这类大规模优化问题中相对于拟牛顿法更有效．     

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     

A non-iterative linear inverse solution for the block approach in EIT

Electrical impedance tomography (EIT) is a simple, economic and healthy technique to capture images from the internal area of the body. Although EIT is cheaper and smaller than other imaging systems and requires no ionizing radiation, the resolution associated with this technique is intrinsically limited and the image reconstruction algorithms proposed up to now are not efficient enough. In addition to low resolution EIT is an ill-posed inverse problem. Block method in EIT is based on electrical properties of materials and used to enhance image resolution and also to improve the reconstruction algorithm. Recently an inverse solution for EIT based on block method has been developed, however, this method uses non-linear algorithm. The present article provides a non-iterative linear inverse solution for the block approach on EIT. Using linear equations in this new approach provides a fast algorithm and the ability to solve complicated block problems. We have assumed that the subject has a 2D rectangular shape and is made up of identical fixed size blocks and all of the particles of each block have the same electrical conductivities. It is shown by computer simulations that this linear reconstruction algorithm employing the block method results in an accurate identification.     

A complementary pivoting algorithm for linear network problems

In this paper, an algorithm is presented for the transshipment problem that is an adaption of the method used by Jones, Saigal, and Schneider for solving single-commodity, spatial-equilibrium problems. The approach uses a variable-dimension strategy in which a sequence of subproblems is formed by solving the problem one-node-at-a-time. The algorithm is tested on uncapacitated transportation problems. Although the computational results are not directly comparable to other methods (since the algorithm is implemented in C under UNIX), the results show that the method is very effective and may e competitive with the best available algorithms for linear network problems.This research was supported, in part, by grant number ECS-85 04 195 from the National Science Foundation.     

A Variable Krasnoselski–Mann Algorithm for a New Class of Fixed Point Problems

We study the convergence of a variable version of the Krasnoselski–Mann algorithm applied to a primal dual fixed point problem. The link with Spingarn's partial inverse method is made, and an application to feasibility problems and mathematical programming is also proposed.     

The determination of two moving heat sources in two-dimensional inverse heat problem

There are some restrictions in the estimation of location and strength problems in recent studies. One of the restrictions is that the problem is limited to the static sources and the single moving source. In other words, there is no method available to estimate the location and the strength of two moving sources presently. Therefore, it is necessary to develop a robust method to estimate the location and the strength of two moving heat sources. In this paper, a numerical algorithm is proposed to determine the problem sequentially. Special feature about this method is that no preselect functional form for the unknown sources is necessary and no sensitivity analysis is needed in the algorithm. Two examples are used to demonstrate the characteristics of the proposed method. From the results, they show that the proposed method is an accurate and efficient method to determine the location and the strength of two moving sources in the inverse heat conduction problem.