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Can we recover a complex signal from its Fourier magnitudes? More generally, given a set of m measurements, yk=ak∗x for k=1,…,m, is it possible to recover x∈Cn (i.e., length-n complex vector)? This generalized phase retrieval (GPR) problem is a fundamental task in various disciplines and has been the subject of much recent investigation. Natural nonconvex heuristics often work remarkably well for GPR in practice, but lack clear theoretic explanations. In this paper, we take a step toward bridging this gap. We prove that when the measurement vectors ak’s are generic (i.i.d. complex Gaussian) and numerous enough (m≥Cnlog3n), with high probability, a natural least-squares formulation for GPR has the following benign geometric structure: (1) There are no spurious local minimizers, and all global minimizers are equal to the target signal x, up to a global phase, and (2) the objective function has a negative directional curvature around each saddle point. This structure allows a number of iterative optimization methods to efficiently find a global minimizer, without special initialization. To corroborate the claim, we describe and analyze a second-order trust-region algorithm. [ABSTRACT FROM AUTHOR]

This paper investigates the inverse problem of determining a heat source in the parabolic heat equation using the usual conditions. Firstly, the problem is reduced to an equivalent problem which is easy to handle using variational iteration method. Secondly, variational iteration method is used to solve the reduced problem. Using this method a rapid convergent sequence can be produced which tends to the exact solution of the problem. Furthermore, variational iteration method does not require the discretization of the problem. Two numerical examples are presented to illustrate the strength of the method. [ABSTRACT FROM AUTHOR]

The impact of this paper is twofold. First, we study convergence rates of the iteratively regularized Gauss-Newton (IRGN) algorithm with a linear penalty term under a generalized source assumption and show how the regularizing properties of new iterations depend on the solution smoothness. Secondly, we introduce an adaptive IRGN procedure, which is investigated under a relaxed smoothness condition. The introduction and analysis of a more general penalty term are of great importance since, apart from bringing stability to the numerical scheme designed for solving a large class of applied inverse problems, it allows us to incorporate various types of a priori information available on the model. Both a priori and a posteriori stopping rules are investigated. For the a priori stopping rule, optimal convergence rates are derived. A numerical example illustrating convergence rates is considered [ABSTRACT FROM AUTHOR]

This paper presents a method for identifying the general-order poles of a meromorphic function algebraically from its values on the unit circle, which has various applications in inverse source problems in potential analysis. First, we derive a system of Dth-degree equations for N distinct poles zn of order Dn, where n = 1, 2, . . . ,N and D = max1≤n≤N{Dn}. Then, we transform these equations into linear equations for the coefficients of the Nth-degree equation whose roots are zn so that the poles are obtained algebraically from data. The obtained poles can be used as an initial solution for iterative algorithms. A method for estimating the order Dn of each pole is also proposed and is numerically verified. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we will present a generalization for a minimization problem from I. Daubechies, M. Defrise, and C. DeMol (2004) . This generalization is useful for solving many practical problems in which more than one constraint are involved. In this regard, we will conclude the findings of many papers (most of which are on image processing) from this generalization. It is hoped that the approach proposed in this paper will be a suitable reference for some applied works where multi-frames, multi-wavelets, or multi-constraints are present in linear inverse problems. [Copyright &y& Elsevier]

Abstract  We develop a general convergence analysis for a class of inexact Newton-type regularizations for stably solving nonlinear ill-posed problems. Each of the methods under consideration consists of two components: the outer Newton iteration and an inner regularization scheme which, applied to the linearized system, provides the update. In this paper we give a novel and unified convergence analysis which is not confined to a specific inner regularization scheme but applies to a multitude of schemes including Landweber and steepest decent iterations, iterated Tikhonov method, and method of conjugate gradients. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we present an iterative method, inspired by the inverse iteration with shift technique of finite linear algebra, designed to find the eigenvalues and eigenfunctions of the Laplacian with homogeneous Dirichlet boundary condition for arbitrary bounded domains . This method, which has a direct functional analysis approach, does not approximate the eigenvalues of the Laplacian as those of a finite linear operator. It is based on the uniform convergence away from nodal surfaces and can produce a simple and fast algorithm for computing the eigenvalues with minimal computational requirements, instead of using the ubiquitous Rayleigh quotient of finite linear algebra. Also, an alternative expression for the Rayleigh quotient in the associated infinite dimensional Sobolev space which avoids the integration of gradients is introduced and shown to be more efficient. The method can also be used in order to produce the spectral decomposition of any given function . [Copyright &y& Elsevier]

Abstract: It is known that the Schur complements of doubly diagonally dominant matrices are doubly diagonally dominant. In this paper, we obtain an estimate for the doubly diagonally dominant degree on the Schur complement of strictly doubly diagonally dominant matrices. Then, as an application we obtain that the eigenvalues of the Schur complements are located in the Brauer Ovals of Cassini of the original matrices under certain conditions. As another application, we obtain an upper bound for the infinity norm on the inverse on the Schur complement of strictly doubly diagonally dominant matrices. Further, based on the derived results, we give a kind of iteration called the Schur-based iteration, which can solve large scale linear systems though reducing the order by the Schur complement and can compute out the results faster. [Copyright &y& Elsevier]

In this paper we consider a class of regularized Gauss-Newton methods for solving nonlinear inverse problems for which an a posteriori stopping rule is proposed to terminate the iteration. Such methods have the frozen feature that they require only the computation of the Fréchet derivative at the initial approximation. Thus the computational work is considerably reduced. Under certain mild conditions, we give the convergence analysis and derive various estimates, including the order optimality, on these methods. [ABSTRACT FROM AUTHOR]

Abstract: This paper is devoted to the class of inverse problems for coefficient identification with an adjoint problem approach related to nonlinear parabolic partial differential equations. The unknown coefficient depends on the gradient of the solution and belongs to a set of admissible coefficients. First, we present a coefficient-to-data mapping. Then, based on maximum principle and the adjoint problem of the direct problem, the integral identities will be obtained. Using these identities, we can show that the coefficient-to-data mapping is continuous and strictly monotone. Furthermore, an approximate solution to the inverse problem is constructed, and the error is analyzed. Finally, the applicability of the method is demonstrated in numerical examples. [Copyright &y& Elsevier]

We discuss informally two approaches to solving convex and nonconvex feasibility problems – via entropy optimization and via algebraic iterative methods. We shall highlight the advantages and disadvantages of each and give various related applications and limiting examples. While some of the results are very classical, they are not as well-known to practitioners as they should be. A key role is played by the Fenchel conjugate. [ABSTRACT FROM PUBLISHER]

We consider the methods xδ=xnδ-gαn for solving nonlinear ill-posed inverse problems F(x) = y using the only available noise data yδsatisfying ||Yδ - y|| ≤ δ with a given small noise level δ > 0. We terminate the iteration by the discrepancy principle Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed., 0 ≤ n < nδ, with a given number τ > 1. Under certain conditions on {αn} and F, we prove for a large class of spectral filter functions {gα} the convergence of Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. to a true solution as δ → 0. Moreover, we derive the order optimal rates of convergence when certain Hölder source conditions hold. Numerical examples are given to test the theoretical results. [ABSTRACT FROM AUTHOR]

This article combines techniques from two fields of applied mathematics: optimization theory and inverse problems. We investigate a generalized conditional gradient method and its connection to an iterative shrinkage method, which has been recently proposed for solving inverse problems. The iterative shrinkage method aims at the solution of non-quadratic minimization problems where the solution is expected to have a sparse representation in a known basis. We show that it can be interpreted as a generalized conditional gradient method. We prove the convergence of this generalized method for general class of functionals, which includes non-convex functionals. This also gives a deeper understanding of the iterative shrinkage method. [ABSTRACT FROM AUTHOR]

The article discusses a study which explores the strong convergence of an iterative method for inverse strongly accretive operators in the framework of Banach spaces. The study found that the variational inequality for an inverse strongly monotone operator in the framework of Banach spaces is equivalent to some fixed-point problem. The study also assessed that the strong convergence for an iterative method for inverse strongly accretive operator is based on that of the inverse strongly monotone operator.

We study an inverse problem of the mathematical tide theory: a tidal potential problem for a semi-discrete model. The closure equation is constructed using observation data on the sea-level function on a part of the boundary and in the interior of the domain. Problems of existence and uniqueness of the solution are considered. An iteration algorithm for solving the problem is formulated. [ABSTRACT FROM AUTHOR]