An inverse problem for an integro‐differential operator on a star‐shaped graph

A partial inverse problem for an integro‐differential Sturm‐Liouville operator on a star‐shaped graph is studied. We suppose that the convolution kernels are known on all the edges of the graph except one and recover the kernel on the remaining edge from a part of the spectrum. We prove the uniqueness theorem for this problem and develop a constructive algorithm for its solution, based on the reduction of the inverse problem on the graph to the inverse problem on the interval by using the Riesz basis property of the special system of functions.     

An Iterative Algorithm for the Coefficient Inverse Problem of Differential Equations

In this paper, an iterative algorithm for solving a coefficient inverse problem is submitted. The key of the method is to project an unknown coefficient function on a finite dimensional function space. Thus, the inverse problem can be changed into a nonlinear algebraic system of equations.     

On the uniqueness and reconstruction for an inverse problem of the fractional diffusion process

Consider an inverse problem for the time-fractional diffusion equation in one dimensional spatial space. The aim is to determine the initial status and heat flux on the boundary simultaneously from heat measurement data given on the other boundary. Using the Laplace transform and the unique extension technique, the uniqueness for this inverse problem is proven. Then we construct a regularizing scheme for the reconstruction of boundary flux for known initial status. The convergence rate of the regularizing solution is established under some a priori information about the exact solution. Moreover, the initial distribution can also be recovered approximately from our regularizing scheme. Finally we present some numerical examples, which show the validity of the proposed reconstruction scheme.     

An analytical method for solving elastic system in inhomogeneous orthotropic media

In this paper, the three‐dimensional initial value problem for elastic system in inhomogeneous orthotropic media is considered and an analytical method is studied to solve this problem. The system is written in terms of Fourier images of displacements with respect to lateral variables. The resulting problem is reduced to integral equations of the Volterra type, whose solution is obtained by the method of successive approximations. Finally, using the real Paley‐Wiener theorem, it is shown that the solution of the initial value problem can be found by the inverse Fourier transform.     

The Inverse Problem Method on the Half-Line and a Nested System of Riccati Equations

A mixed problem for the nonlinear Bogoyavlenskii system on the half-line is studied by the inverse problem method. The solution of the mixed problem is reduced to the solution of the inverse spectral problem of recovering a forth-order differential operator on the half-line from the Weyl matrix. We derive evolution equations for the elements of the Weyl matrix and give an algorithm for the solution of the mixed problem. Evolution equations of the elements of the Weyl matrix are nonlinear. It is shown that they can be reduced to a nested system of three successively solvable matrix Riccati equations.     

On inverse traveling salesman problems

The inverse traveling salesman problem belongs to the class of ??inverse combinatorial optimization?? problems. In an inverse combinatorial optimization problem, we are given a feasible solution for an instance of a particular combinatorial optimization problem, and the task is to adjust the instance parameters as little as possible so that the given solution becomes optimal in the new instance. In this paper, we consider a variant of the inverse traveling salesman problem, denoted by ITSP _W,A , by taking into account a set W of admissible weight systems and a specific algorithm. We are given an edge-weighted complete graph (an instance of TSP), a Hamiltonian tour (a feasible solution of TSP) and a specific algorithm solving TSP. Then, ITSP _W,A , is the problem to find a new weight system in W which minimizes the difference from the original weight system so that the given tour can be selected by the algorithm as a solution. We consider the cases ${W \in \{\mathbb{R}^{+m}, \{1, 2\}^m , \Delta\}}$ where ?? denotes the set of edge weight systems satisfying the triangular inequality and m is the number of edges. As for algorithms, we consider a local search algorithm 2-opt, a greedy algorithm closest neighbor and any optimal algorithm. We devise both complexity and approximation results. We also deal with the inverse traveling salesman problem on a line for which we modify the positions of vertices instead of edge weights. We handle the cases ${W \in \{\mathbb{R}^{+n}, \mathbb{N}^n\}}$ where n is the number of vertices.     

Noniterative Reconstruction Method for an Inverse Potential Problem Modeled by a Modified Helmholtz Equation

This paper deals with an inverse potential problem posed in two dimensional space whose forward problem is governed by a modified Helmholtz equation. The inverse problem consists in the reconstruction of a set of anomalies embedded into a geometrical domain from partial measurements of the associated potential. Since the inverse problem, we are dealing with, is written in the form of an ill-posed boundary value problem, the idea is to rewrite it as a topology optimization problem. In particular, a shape functional is defined to measure the misfit of the solution obtained from the model and the data taken from the partial measurements. This shape functional is minimized with respect to a set of ball-shaped anomalies using the concept of topological derivatives. It means that the shape functional is expanded asymptotically and then truncated up to the desired order term. The resulting expression is trivially minimized with respect to the parameters under consideration which leads to a noniterative second-order reconstruction algorithm. As a result, the reconstruction process becomes very robust with respect to noisy data and independent of any initial guess. Finally, some numerical experiments are presented to show the effectiveness of the proposed reconstruction algorithm.     

一类双曲反问题的逼近算法及收敛性

该文考虑地球物理勘探中出现的间断特性阻抗的反演问题．利用样条插值理论，把无穷维空间上的反问题用有限维空间上的反问题来近似．利用半群理论，证明了近似反问题之解收敛于原反问题之解．据此可得到求解反问题的一种稳定的近似算法．     

Performance comparisons of greedy algorithms in compressed sensing

Compressed sensing has motivated the development of numerous sparse approximation algorithms designed to return a solution to an underdetermined system of linear equations where the solution has the fewest number of nonzeros possible, referred to as the sparsest solution. In the compressed sensing setting, greedy sparse approximation algorithms have been observed to be both able to recover the sparsest solution for similar problem sizes as other algorithms and to be computationally efficient; however, little theory is known for their average case behavior. We conduct a large‐scale empirical investigation into the behavior of three of the state of the art greedy algorithms: Normalized Iterative Hard Thresholding (NIHT), Hard Thresholding Pursuit (HTP), and CSMPSP. The investigation considers a variety of random classes of linear systems. The regions of the problem size in which each algorithm is able to reliably recover the sparsest solution is accurately determined, and throughout this region, additional performance characteristics are presented. Contrasting the recovery regions and the average computational time for each algorithm, we present algorithm selection maps, which indicate, for each problem size, which algorithm is able to reliably recover the sparsest vector in the least amount of time. Although no algorithm is observed to be uniformly superior, NIHT is observed to have an advantageous balance of large recovery region, absolute recovery time, and robustness of these properties to additive noise across a variety of problem classes. A principle difference between the NIHT and the more sophisticated HTP and CSMPSP is the balance of asymptotic convergence rate against computational cost prior to potential support set updates. The data suggest that NIHT is typically faster than HTP and CSMPSP because of greater flexibility in updating the support that limits unnecessary computation on incorrect support sets. The algorithm selection maps presented here are the first of their kind for compressed sensing. Copyright © 2014 John Wiley & Sons, Ltd.     

求解一类逆鲁棒优化问题的牛顿型扰动方法

逆优化问题是指通过调整目标函数和约束中的某些参数使得已知的一个解成为参数调整后的优化问题的最优解.本文考虑求解一类逆鲁棒优化问题.首先,我们将该问题转化为带有一个线性等式约束,一个二阶锥互补约束和一个线性互补约束的极小化问题;其次,通过一类扰动方法来对转化后的极小化问题进行求解,然后利用带Armijo线搜索的非精确牛顿法求解每一个扰动问题.最后,通过数值实验验证该方法行之有效.     

Diffeomorphisms as time one maps

Summary. We investigate the inverse ODE problem of finding a vector field such that the time one map associated to its flow coincides with a given diffeomorphism. Using a constructive approach we solve this problem for a class of diffeomorphisms having a globally attracting fixed point. Furthermore we consider how the solution fields depend on the diffeomorphism. As an example we show that for certain parameters, the Hénon map is the time one map of a two dimensional flow.     

A dynamical system approach to Heisenberg uniqueness pairs

A canonical system is a kind of first-order system of ordinary differential equations on an interval of the real line parametrized by complex numbers. It is known that any solution of a canonical system generates an entire function of the Hermite-Biehler class. In this paper, we deal with the inverse problem to recover a canonical system from a given entire function of the Hermite-Biehler class satisfying appropriate conditions. This inverse problem was solved by de Branges in 1960s. However his results are often not enough to investigate a Hamiltonian of recovered canonical system. In this paper, we present an explicit way to recover a Hamiltonian from a given exponential polynomial belonging to the Hermite-Biehler class. After that, we apply it to study distributions of roots of self-reciprocal polynomials.     

Anisotropic mesh generation methods based on ACVT and natural metric for anisotropic elliptic equation

Anisotropic meshes are known to be well-suited for problems which exhibit anisotropic solution features.Defning an appropriate metric tensor and designing an efcient algorithm for anisotropic mesh generation are two important aspects of the anisotropic mesh methodology.In this paper,we are concerned with the natural metric tensor for use in anisotropic mesh generation for anisotropic elliptic problems.We provide an algorithm to generate anisotropic meshes under the given metric tensor.We show that the inverse of the anisotropic difusion matrix of the anisotropic elliptic problem is a natural metric tensor for the anisotropic mesh generation in three aspects:better discrete algebraic systems,more accurate fnite element solution and superconvergence on the mesh nodes.Various numerical examples demonstrating the efectiveness are presented.     

The reconstruction of a special kind of periodic Jacobi matrices

In this paper, we investigate the properties of a special kind of periodic Jacobi matrices. We show that the solution of the inverse problem for periodic Jacobi matrices is unique if and only if the matrix is of that special kind. Moreover, we present an algorithm to construct the solution of that inverse problem and give some illustrative examples. Finally, we perform a stability analysis of the algorithm.     

Shape inverse problem for the two‐dimensional unsteady Stokes flow

In this article, the shape inverse problem for the two‐dimensional unsteady Stokes flow has been presented. We employ Piola transformation to bypass the divergence free condition for the flow and prove the differentiability of the solution to the initial boundary value problem. For the approximate solution of the ill‐posed and nonlinear problem, we propose a regularized Gauss‐Newton method. The numerical examples show that our theory is useful for practical purpose and the proposed algorithm is feasible. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Explicit analytical formulation and exact inversion of decomposable fuzzy systems with singleton consequents

This paper is devoted to the inversion of fuzzy systems expressed by fuzzy rules with singleton consequents if input variables are described using strong triangular partitions. As pointed out in recent works, such fuzzy systems can be decomposed into collections of multi-linear subsystems. In this paper, an analytical formulation of the system output is explicitly developed and directly used in order to determine solutions to the inversion problem. Based on this analytical methodology, an algorithm is proposed for computing inverse solutions. As the inversion is handled analytically, the exactness of the obtained solutions is guaranteed. Furthermore, according to the decomposability of the studied fuzzy systems, all inverse solutions are found. Finally, whatever the fuzzy system under consideration, there is no need to study its invertibility beforehand since the algorithm is able to handle all possible situations (no solution, one unique solution, multiple solutions, an infinity of solutions).The proposed approach can be easily extended to other types of fuzzy systems provided that decomposability is preserved. In other words, with regard to exact inversion which often plays a key role in engineering applications such as control or diagnosis, decomposability is probably the first criterion that should be considered when choosing a specific fuzzy system structure.     

A column generation method for inverse shortest path problems

In this paper we formulate an inverse shortest path problem as a special linear programming problem. A column generation scheme is developed that is able to keep most columns of the LP model implicit and to generate necessary columns by shortest path algorithms. This method can get an optimal solution in finitely many steps. Some numerical results are reported to show that the algorithm has a good performance.The authors gratefully acknowledge the partial support of Croucher Foundation.     

Super-fast validated solution of linear systems

Validated solution of a problem means to compute error bounds for a solution in finite precision. This includes the proof of existence of a solution. The computed error bounds are to be correct including all possible effects of rounding errors. The fastest known validation algorithm for the solution of a system of linear equations requires twice the computing time of a standard (purely) numerical algorithm. In this paper we present a super-fast validation algorithm for linear systems with symmetric positive definite matrix. This means that the entire computing time for the validation algorithm including computation of an approximated solution is the same as for a standard numerical algorithm. Numerical results are presented.     

双倍维Jacobi矩阵逆问题的改进算法

本文给出了一种解决双倍维Jacobi矩阵逆问题的改进算法.该算法避免了重新构造顺序主子矩阵J_n,也避免了计算尾主子矩阵J_n+1,2n的特征多项式以及特征值,因此本文的改进算法具有更好的稳定性和精度.给出的两个数值实例说明,本文的改进算法是有效的,比现有的几种算法具有更高的精度.     

对流弥散方程的时间依赖反应系数反演及应用

探讨了一维对流弥散方程的时间依赖反应系数函数的反演问题及其在一个土柱渗流试验中的应用.借助一个积分恒等式,讨论了正问题单调解的存在条件及反问题的数据相容性.进一步考虑一个扰动土柱试验模型模拟问题,应用一种最佳摄动量正则化算法,对反应系数函数进行了数值反演模拟,并应用于实际试验数据的反分析,反演重建结果不仅与相容性分析一致,而且与实际观测数据基本吻合.