Imputing a variational inequality function or a convex objective function: A robust approach

To impute the function of a variational inequality and the objective of a convex optimization problem from observations of (nearly) optimal decisions, previous approaches constructed inverse programming methods based on solving a convex optimization problem [17], [7]. However, we show that, in addition to requiring complete observations, these approaches are not robust to measurement errors, while in many applications, the outputs of decision processes are noisy and only partially observable from, e.g., limitations in the sensing infrastructure. To deal with noisy and missing data, we formulate our inverse problem as the minimization of a weighted sum of two objectives: 1) a duality gap or Karush–Kuhn–Tucker (KKT) residual, and 2) a distance from the observations robust to measurement errors. In addition, we show that our method encompasses previous ones by generating a sequence of Pareto optimal points (with respect to the two objectives) converging to an optimal solution of previous formulations. To compare duality gaps and KKT residuals, we also derive new sub-optimality results defined by KKT residuals. Finally, an implementation framework is proposed with applications to delay function inference on the road network of Los Angeles, and consumer utility estimation in oligopolies.     

Ekeland's inverse function theorem in graded Fréchet spaces revisited for multifunctions

In this paper, we present some inverse function theorems and implicit function theorems for set-valued mappings between Fréchet spaces. The proof relies on Lebesgue's Dominated Convergence Theorem and on Ekeland's variational principle. An application to the existence of solutions of differential equations in Fréchet spaces with non-smooth data is given.     

Reasoning about relationships between quantities to reorganize inverse function meanings: The case of Arya

Researchers have argued high school students, college students, pre-service teachers, and in-service teachers do not construct productive inverse function meanings. In this report, I first summarize the literature examining students’ and teachers’ inverse function meanings. I then provide my theoretical perspective, including my use of the terms understanding and meaning and my operationalization of productive inverse function meanings. I describe a conceptual analysis of ways students may reorganize their limited inverse function meanings into productive meanings via reasoning about relationships between covarying quantities. I then present one pre-service teacher’s activity in a semester long teaching experiment to characterize how her quantitative, covariational, and bidirectional reasoning supported her in reorganizing her limited inverse function meanings into more productive meanings. I describe how this reorganization required her to reconstruct her meanings for various related mathematical ideas. I conclude with research and pedagogical implications stemming from this work and directions for future research.     

Inverse temperature-dependent perfusion coefficient reconstruction

The identification of the temperature-dependent perfusion coefficient in the one-dimensional transient bio-heat conduction equation is investigated. If Neumann boundary conditions are prescribed, then the additional measurement sufficient to render a unique solution is a temperature measurement on a part of the boundary. A numerical approach based on a Crank-Nicolson-type finite-difference scheme combined with the first-order Tikhonov regularization method is developed. Numerical results are presented and discussed.     

Nondestructive testing of objects of complex shape using infrared thermography: Determination of the spatiotemporal distribution of the irradiation heat flux

The existing inverse methods used to determine the heat flux density require that the forward problem and the problem domain (geometry) be known. In this paper, in order to determine the spatiotemporal heat flux density without knowing the real problem domain, we propose an approach based on temporal tracking of the thermal front. The proposed approach is particularly relevant when a three-dimensional formulation is adopted for nondestructive testing using infrared thermography. For such a formulation, heat flux density resulting from the external thermal stimulus is needed and must be determined to accurately characterize the defects and reconstruct the internal geometry of the inspected objects. The proposed approach uses only two inputs: the time-dependent temperature of the frontal surface recorded by an infrared camera and the 3D point cloud of the frontal surface collected by a 3D scanner. The method is evaluated numerically on an object of complex shape. We consider the case of pulsed thermal stimulus as well as the cases of unit step and modulated thermal stimuli. An experimental validation is performed on a cylindrical object submitted to a pulsed thermal stimulus and a modulated thermal stimulus. The results show the accuracy of the method which can easily be implemented as the initial step of the three-dimensional quantitative nondestructive testing of objects using infrared thermography.     

改进的傅里叶域小波域联合去模糊算法

傅里叶域与小波域的联合去模糊算法在低噪声时具有优越的恢复效果,但是这种联合去模糊算法并不适用于含噪声的模糊图像.为了解决这一问题,本文将先验约束分别引入傅里叶域的去模糊步骤和小波域的去噪步骤.在傅里叶域,用矩阵形式表示目标函数.对目标函数添加平滑约束并且通过噪声水平和模糊图像高频信息计算得到平滑约束项的滤波系数.同样方式,在小波域对小波域目标函数添加能量约束,实现小波域目标函数的正则化过程.分析傅里叶域的噪声放大程度,通过傅里叶域的滤波系数计算得到小波域能量约束的滤波系数.傅里叶域的平滑约束可以抑制滤波过程中噪声的产生,小波域的能量约束可以提高小波域滤波的鲁棒性.仿真实验表明,改进的算法相比于原始算法具有更好的鲁棒性,可以有效提高图像的恢复质量.对于噪声标准差为0.010.1的模糊图像,改进算法恢复图像峰值信噪比比原始算法恢复图像的峰值信噪比高1左右.并且改进算法对于高斯型点扩散函数误差具有鲁棒性,当点扩散函数估计方差与实际方差相差0.4时,改进算法的恢复效果仍优于原始算法.     

Inverse scattering with fixed energy and an inverse eigenvalue problem on the half-line

Recently A. G. Ramm (1999) has shown that a subset of phase shifts , , determines the potential if the indices of the known shifts satisfy the Müntz condition . We prove the necessity of this condition in some classes of potentials. The problem is reduced to an inverse eigenvalue problem for the half-line Schrödinger operators.

     

Cramer’s rule for some quaternion matrix equations

Cramer’s rules for some left, right and two-sided quaternion matrix equations are obtained within the framework of the theory of the column and row determinants.     

Computing an eigenvector of an inverse Monge matrix in max-plus algebra

The problem of computing an eigenvector of an inverse Monge matrix in max-plus algebra is addressed. For a general matrix, the problem can be solved in at most O(n³) time. This note presents an O(n²) algorithm for computing one max-plus algebraic eigenvector of an inverse Monge matrix . It is assumed that is irreducible.