Methods For Nonlinear Operator Equations Research Articles

Numerical stability for nonlinear evolution equations

The paper deals with discretisation methods for nonlinear operator equations written as abstract nonlinear evolution equations. Brezis and Pazy showed that the solution of such problems is given by nonlinear semigroups whose theory was founded by Crandall and Liggett. By using the approximation theorem of Brezis and Pazy, we show the N-stability of the abstract nonlinear discrete problem for the implicit Euler method. Motivated by the rational approximation methods for linear semigroups, we propose a more general time discretisation method and prove its N-stability as well.

Nonlinear Stochastic Galerkin and Collocation Methods: Application to a Ferromagnetic Cylinder Rotating at High Speed

The objective of this paper is to reviewrecent developments in numerical re- construction methods for inverse transport problems in imaging applications, mainly optical tomography, fluorescence tomography and bioluminescence tomography. In those inverse problems, one aims at reconstructing physical parameters,suchas the ab- sorption coefficient, the scattering coefficient and the fluorescence light source, inside heterogeneous media, frompartialknowledge oftransport solutions onthe boundaries of the media. The physical parameters recovered can be used for diagnostic purpose. Numerical reconstruction techniques for those inverse transport problems can be roughly classified into two categories: linear reconstruction methods and nonlinear re- construction methods. In the first type of methods, the inverse problems are linearized around some known background to obtain linear inverse problems. Classical regular- ization techniques are then applied to solve those inverse problems. The second type of methods are either based on regularized nonlinear least-square techniques or based on gradient-driven iterative methods for nonlinear operator equations. In either case, the unknown parameters are iteratively updated until the solutions of the transport equations with the those parameters match the measurements to a certain extent. We reviewlinear and nonlinear reconstruction methods for inverse transport problems in medical imaging with stationary, frequency-domain and time-dependent data. The materials presented include both existing and new results. Meanwhile, we attempt to present similar algorithms for different problems in the same framework to make it more straightforward to generalize those algorithms to other inverse (transport) prob- lems.

Semilocal convergence for Halley’s method under weak Lipschitz condition

The semilocal convergence properties of Halley’s method for nonlinear operator equations are studied under the hypothesis that the second derivative satisfies some weak Lipschitz condition. The method employed in the present paper is based on a family of recurrence relations which will be satisfied by the involved operator. An application to a nonlinear Hammerstein integral equation of the second kind is provided.

Two-Sided Approximations For Nonlinear Operator Equations

AbstractWe propose an analytical-numerical method for nonlinear operator equations which converges exponentially and provides two-sided approximations. A numerical example confirms the theoretical results

A generalized conditional gradient method for nonlinear operator equations with sparsity constraints

The intention of this paper is to show the applicability of a generalized conditional gradient method for the minimization of Tikhonov-type functionals, which occur in the regularization of nonlinear inverse problems with sparsity constraints. We consider functionals of Tikhonov type where the usual quadratic penalty term is replaced by the pth power of a weighted ℓp-norm. First of all, we analyze the convergence properties of a generalized gradient algorithm. Further, we show that the considered minimization method coincides under certain conditions with a surrogate method. Numerical results are then illustrated by means of the nonlinear single photon emission computed tomography problem.

Continuous modified Newton’s-type method for nonlinear operator equations

A nonlinear operator equation $F(x)=0$, $F:H\to H,$ in a Hilbert space is considered. Continuous Newton's-type procedures based on a construction of a dynamical system with the trajectory starting at some initial point $x_0$ and becoming asymptotically close to a solution of $F(x)=0$ as $t\to +\infty$ are discussed. Well-posed and ill-posed problems are investigated.

Approximation methods for nonlinear operator equations

Let E be a real normed linear space and A : E → E be a uniformly quasi-accretive map. For arbitrary x 1 E E define the sequence x n E E by x n+1 := x n - α n Ax n , n > 1, where {an} is a positve real sequence satisfying the following conditions: (i) Σα n = ∞; (ii) lim α n = 0. For x* E N(A):= {x E E: Ax = 0}, assume that σ:= inf n ∈ N0 φ(∥x n+1 -x*∥)/∥x n+1 - x*∥ > 0 and that ∥Ax n+1 - Ax n ∥ → 0, where No:= {n ∈ N (the set of all positive integers): x n+1 ¬= x*} and ψ: [0,∞) → [0,∞) is a strictly increasing function with ψ(0) = 0. It is proved that a Mann-type iteration process converges strongly to x*. Furthermore if, in addition, A is a uniformly continuous map, it is proved, without the condition on σ, that the Mann-type iteration process converges strongly to x*. As a consequence, corresponding convergence theorems for fixed points of hemi-contractive maps are proved.

Generalization of ionospheric tomography on diverse data sources: Reconstruction of the three‐dimensional ionosphere from simultaneous vertical ionograms, backscatter ionograms, and total electron content data

An ionospheric reconstruction method is developed that is capable of simultaneously utilizing information from all possible sources of ionospheric measurement. We describe the method and demonstrate its performance on ionospheric data collected by an over‐the‐horizon radar (vertical and backscatter ionograms) and total electron content data collected by receivers operated in the Caribbean by the Applied Research Laboratories of the University of Texas at Austin. For each data set the method produces a smooth three‐dimensional ionospheric model that is consistent with all the measurements. The method is based on the Newton‐Kontorovich method for nonlinear operator equations and Tikhonov's regularization technique for ill‐posed problems. We demonstrate here the ability of our technique to reconstruct ionospheric spatial variations from global‐scale to medium‐scale irregularities or traveling ionospheric disturbances. Overall results of this study demonstrate that the technique suggested makes it possible to unite data from various ionospheric instruments to provide monitoring of the ionosphere over a large geographical area in a wide range of spatial scales.

Remarks on the approximation methods for nonlinear operator equations of the m-accretive type

Remarks on the approximation methods for nonlinear operator equations of the m-accretive type

Reconstruction of a three‐dimensional ionosphere from backscatter and vertical ionograms measured by over‐the‐horizon radar

A method for inversion of backscatter ionograms is described. Source data for the method consist of leading edge of backscatter ionograms from all azimuthal sectors of the backscatter sounder and the quasi vertical incidence (QVI) ionogram. Such data are routinely collected at over‐the‐horizon radar (OTHR) installations operated by the U.S. Navy. The output is a three‐dimensional model of the ionosphere for the coverage area of the sounder. The model produced is consistent with all the input data. The method is based on the Newton‐Kontorovich method for nonlinear operator equations and Tikhonov's regularization technique. Theory of the method and principles of its numerical realization are described. The method is applied to data acquired by the backscatter sounder operating at the Chesapeake, Virginia, OTHR facility. Maps of plasma frequency at a constant altitude over a geographic region that covers a 64°–80° azimuthal sector out to distances of more than 2000 km are presented. The backscatter ionogram inversion technique suggested makes it possible for OTHR backscatter sounders to provide real‐time monitoring of the ionosphere over a large geographical area.

Iterative methods for nonlinear operator equations

A nonlinear conjugate gradient method has been introduced and analyzed by J.W. Daniel. This method applies to nonlinear operators with symmetric Jacobians. The conjugate gradient method applied to the normal equations can be used to approximate the solution of general nonsymmetric linear systems of equations if the condition of the coefficient matrix is small. In this article, we obtain nonlinear generalizations of this method which apply directly to nonlinear operator equations. Under conditions on the Hessian and the Jacobian of the operators, we prove that these methods converge to a unique solution. Error bounds and local convergence results are also obtained.

An interval version of Chebyshev's method for nonlinear operator equations

An interval version of Chebyshev's method for nonlinear operator equations

On the β-convergence of the projection method for non-linear operator equations

On the β-convergence of the projection method for non-linear operator equations

What We Must Learn

The paper deals with discretisation methods for nonlinear operator equations written as abstract nonlinear evolution equations. Brezis and Pazy showed that the solution of such problems is given by nonlinear semigroups whose theory was founded by Crandall and Liggett. By using the approximation theorem of Brezis and Pazy, we show the N-stability of the abstract nonlinear discrete problem for the implicit Euler method. Motivated by the rational approximation methods for linear semigroups, we propose a more general time discretisation method and prove its N-stability as well.