Volume 24, Issue 4

The windowed ray transform is the generalization of the “Analytic-Signal Transform” which was developed to extend arbitrary functions from ℝn${\mathbb{R}^{n}}$ to ℂn${\mathbb{C}^{n}}$ in a natural way. The X-ray transform is also a special case of this transform. Similar to the X-ray transform, the problem of inverting this transform is overdetermined. Hence it is possible to prove the existence of several inversion formulas. Kaiser obtained one such inversion formula in 1993. We present several inversion formulas here. One of them is similar to that of Kaiser, but it requires a weaker condition. The others are new.

According to Pareto’s theory of consumer demand a rational representative consumer should choose their consumption bundle as the solution of a mathematical programming problem of maximization of the utility function under their budget constraint. The inverse problem of demand analysis is to recover the utility function from the demand functions. The answer to the question of solvability of this problem is based on the revealed preference theory. If the problem is unsolvable, one should apply regularization procedure by introducing irrationality indices. When recovering the utility function one puts a priori requirements on it. In this paper, we suggest including positively-homogeneity property into these requirements. We compare the setups with and without this requirement both theoretically and empirically and provide evidence in favor of requiring this property when computing economic indices for consumer representing consumption behavior of large number of various households for large time intervals.

In this paper we derive convergence rates results for Landweber iteration in Hilbert scales. The assumptions that are necessary to prove these results are less restrictive than the ones given in an earlier paper. The relaxed conditions enlarge the range of applicability to a much wider class of nonlinear problems. The theory is applied to nonlinear Hammerstein operators.

In this paper a boundary value problem is considered generated by the Sturm–Liouville equation and non-separated boundary conditions, one of which contains a spectral parameter. We give a uniqueness theorem, develop an algorithm for solving the inverse problem of reconstruction of boundary value problems with spectral data. We use the spectra of two boundary value problems and some sequence of signs as a spectral data.

Normalized differences of several adjacent observations, referred to as pseudo-measurement errors in this paper, are used in so-called difference-based estimation methods as building blocks for the variance estimate of measurement errors. Numerical results demonstrate that pseudo-measurement errors can be used to serve the role of measurement errors. Based on this information, we propose the use of pseudo-measurement errors to determine an appropriate statistical model and then to subsequently investigate whether there is a mathematical model misspecification or error. We also propose to use the information provided by pseudo-measurement errors to quantify uncertainty in parameter estimation by bootstrapping methods. A number of numerical examples are given to illustrate the effectiveness of these proposed methods.

In this paper we present a method for the regularized solution of nonlinear inverse problems, based on Ivanov regularization (also called method of quasi solutions or constrained least squares regularization). This leads to the minimization of a nonconvex cost function under a norm constraint, where nonconvexity is caused by nonlinearity of the inverse problem. Minimization is done by iterative approximation, using (nonconvex) quadratic Taylor expansions of the cost function. This leads to repeated solution of quadratic trust region subproblems with possibly indefinite Hessian. Thus, the key step of the method consists in application of an efficient method for solving such quadratic subproblems, developed by Rendl and Wolkowicz [10]. We here present a convergence analysis of the overall method as well as numerical experiments.

We investigate a class of iterative regularization methods for solving nonlinear irregular operator equations in Hilbert spaces. The operator of an equation is supposed to have a normally solvable derivative at the desired solution. The operators and right parts of equations can be given with errors. A priori and a posteriori stopping rules for the iterations are analyzed. We prove that the accuracy of delivered approximations is proportional to the total error level in the operator and the right part of an equation. The obtained results improve known accuracy estimates for the class of iterative regularization methods, as applied to general irregular operator equations. The results also extend previous similar estimates related to regularization methods for linear ill-posed equations with normally solvable operators.

A series of recent numerical experiments for parameter estimation inverse problems in epidemiology [7, 6] have indicated that applicability of the discrepancy principle (DP) does not depend on the structure of a particular regularizing operator. This observation confirmed the original theoretical analysis by Bakushinsky [1, 2] on the construction of fairly general stabilizing algorithms in Banach and Hilbert spaces. In [7, 6], a unified approach to the implementation of the DP for linear ill-posed problems, the Abstract Discrepancy Principle, has been proposed and justified. The current paper investigates the convergence rates of the ADP. Special cases of sectorial and self-adjoint operators are studied.

In this article, we study an inverse problem of reconstructing a space dependent coefficient in a generalized Korteweg–de Vries (KdV) equation arising in physical systems with variable topography from final time overdetermination data. First the identification problem is transformed into an optimization problem by using optimal control framework and existence of a minimizer for the cost functional is established. Then we prove a stability estimate for retrieving the unknown coefficient in KdV equation with the upper bound of given measurements. The local uniqueness of the coefficient is also discussed.

We concerned with the asymptotic analysis for the dynamical probe method which is a reconstruction scheme to identify an anomaly inside a heat conductor from the Neumann-to-Dirichlet map. In this paper an inclusion was considered as an anomaly and we succeeded giving an improved asymptotic behavior of the indicator function defined in terms of the Neumann-to-Dirichlet map to identify not only the location of the inclusion but also some of its physical properties simultaneously. The two major improvements made for analyzing the asymptotic behavior of the indicator function are as follows. Firstly, we can know the distance to the boundary of unknown inclusion as we probe it from its outside. This improvement can avoid overshooting the boundary points as much as possible if we probe it from outside the inclusion numerically. Secondly, we can know the value of heat conductivity of inclusion as we probe close to the inclusion even without touching it.