Approximate global convergence and quasireversibility for a coefficient inverse problem with backscattering data

A numerical method possessing the approximate global convergence property is developed for a 3-D coefficient inverse problem for hyperbolic partial differential equations with backscattering data resulting from a single measurement. An important part of this technique is the quasireversibility method. An approximate global convergence theorem is proved. Results of two numerical experiments are presented. Bibliography: 46 titles. Illustrations: 2 figures.     

Numerical methods for A-optimal designs with a sparsity constraint for ill-posed inverse problems

We consider the problem of experimental design for linear ill-posed inverse problems. The minimization of the objective function in the classic A-optimal design is generalized to a Bayes risk minimization with a sparsity constraint. We present efficient algorithms for applications of such designs to large-scale problems. This is done by employing Krylov subspace methods for the solution of a subproblem required to obtain the experiment weights. The performance of the designs and algorithms is illustrated with a one-dimensional magnetotelluric example and an application to two-dimensional super-resolution reconstruction with MRI data.     

V-cycle convergence of some multigrid methods for ill-posed problems

For ill-posed linear operator equations we consider some V-cycle multigrid approaches, that, in the framework of Bramble, Pasciak, Wang, and Xu (1991), we prove to yield level independent contraction factor estimates. Consequently, we can incorporate these multigrid operators in a full multigrid method, that, together with a discrepancy principle, is shown to act as an iterative regularization method for the underlying infinite-dimensional ill-posed problem. Numerical experiments illustrate the theoretical results.

     

On the total-variation convergence of regularizing algorithms for ill-posed problems

It is well known that ill-posed problems in the space V[a, b] of functions of bounded variation cannot generally be regularized and the approximate solutions do not converge to the exact one with respect to the variation. However, this convergence can be achieved on separable subspaces of V[a, b]. It is shown that the Sobolev spaces W ₁ ^m [a, b], m ∈ ? can be used as such subspaces. The classes of regularizing functionals are indicated that guarantee that the approximate solutions produced by the Tikhonov variational scheme for ill-posed problems converge with respect to the norm of W ₁ ^m [a, b]. In turn, this ensures the convergence of the approximate solutions with respect to the variation and the higher order total variations.     

Maximum entropy solution to ill-posed inverse problems with approximately known operator

We consider the linear inverse problem of reconstructing an unknown finite measure μ from a noisy observation of a generalized moment of μ defined as the integral of a continuous and bounded operator Φ with respect to μ. Motivated by various applications, we focus on the case where the operator Φ is unknown; instead, only an approximation Φm to it is available. An approximate maximum entropy solution to the inverse problem is introduced in the form of a minimizer of a convex functional subject to a sequence of convex constraints. Under several assumptions on the convex functional, the convergence of the approximate solution is established.     

A convergence analysis of the Landweber iteration for nonlinear ill-posed problems

Summary. In this paper we prove that the Landweber iteration is a stable method for solving nonlinear ill-posed problems. For perturbed data with noise level we propose a stopping rule that yields the convergence rate ) under appropriate conditions. We illustrate these conditions for a few examples. Received February 15, 1993 / Revised version received August 2, 1994     

An inverse coefficient problem for a nonlinear parabolic variational inequality

This study is related to inverse coefficient problems for a nonlinear parabolic variational inequality with an unknown leading coefficient in the equation for the gradient of the solution. An inverse method, involving minimization of a least-squares cost functional, is developed to identify the unknown coefficient. It is proved that the solution of the corresponding direct problem depends continuously on the coefficient. On the basis of this, the existence of a quasisolution of the inverse problem is obtained in the appropriate class of admissible coefficients.     

Globally strongly convex cost functional for a coefficient inverse problem

A Carleman Weight Function (CWF) is used to construct a new cost functional for a Coefficient Inverse Problems for a hyperbolic PDE. Given a bounded set of an arbitrary size in a certain Sobolev space, one can choose the parameter of the CWF in such a way that the constructed cost functional will be strongly convex on that set. Next, convergence of the gradient method, which starts from an arbitrary point of that set, is established. Since restrictions on the size of that set are not imposed, then this is the global convergence.     

An inverse coefficient problem with nonlinear parabolic equation

The problem related to controlled potential experiments in electrochemistry is studied. Ion transport is regarded as the superposition of diffusion and migration. Modelling of the experiment leads to a problem for a nonlinear parabolic equation with additional condition. Driven by the needs of theoretical analysis, from the point of view a inverse coefficient problem, we analyze the monotonicity of input-output mappings in inverse coefficient and source problems for this parabolic equation. Additionally, we extend the nonlinear parabolic equation to a more general case. Under some proper conditions, we investigate the existence of quasisolution of the generalized nonlinear parabolic equation.     

A convergence analysis of nonlinear implicit iterative method for nonlinear ill-posed problems

Implicit iterative method acquires good effect in solving linear ill-posed problems. We have ever applied the idea of implicit iterative method to solve nonlinear ill-posed problems, under the restriction that α is appropriate large, we proved the monotonicity of iterative error and obtained the convergence and stability of iterative sequence, numerical results show that the implicit iterative method for nonlinear ill-posed problems is efficient. In this paper, we analyze the convergence and stability of the corresponding nonlinear implicit iterative method when α_k are determined by Hanke criterion.     

Numerical methods for solving the coefficient inverse problem

Translated from Metody Matematicheskogo Modelirovaniya i Vychislitel'noi Diagnostika, pp. 35–45, Izd. Moskovskogo Universiteta, Moscow, 1990.     

Posterior contraction rates for the Bayesian approach to linear ill-posed inverse problems

We consider a Bayesian nonparametric approach to a family of linear inverse problems in a separable Hilbert space setting with Gaussian noise. We assume Gaussian priors, which are conjugate to the model, and present a method of identifying the posterior using its precision operator. Working with the unbounded precision operator enables us to use partial differential equations (PDE) methodology to obtain rates of contraction of the posterior distribution to a Dirac measure centered on the true solution. Our methods assume a relatively weak relation between the prior covariance, noise covariance and forward operator, allowing for a wide range of applications.     

Solvability of an inverse problem for the parabolic equation with convergence

Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 33, No. 3, pp. 42–49, May–June, 1992.     

On a coefficient inverse problem for a parabolic equation in a domain with free boundary

The present paper deals with the inverse problem of determination of the coefficient of the first derivative of the unknown function with respect to a spatial variable for a one-dimensional parabolic equation in the domain whose boundary is determined by two unknown functions. The conditions of local existence and uniqueness of a solution to the inverse problem are established.     

On the second-order asymptotical regularization of linear ill-posed inverse problems

ABSTRACT

In this paper, we establish an initial theory regarding the second-order asymptotical regularization (SOAR) method for the stable approximate solution of ill-posed linear operator equations in Hilbert spaces, which are models for linear inverse problems with applications in the natural sciences, imaging and engineering. We show the regularizing properties of the new method, as well as the corresponding convergence rates. We prove that, under the appropriate source conditions and by using Morozov's conventional discrepancy principle, SOAR exhibits the same power-type convergence rate as the classical version of asymptotical regularization (Showalter's method). Moreover, we propose a new total energy discrepancy principle for choosing the terminating time of the dynamical solution from SOAR, which corresponds to the unique root of a monotonically non-increasing function and allows us to also show an order optimal convergence rate for SOAR. A damped symplectic iterative regularizing algorithm is developed for the realization of SOAR. Several numerical examples are given to show the accuracy and the acceleration effect of the proposed method. A comparison with other state-of-the-art methods are provided as well.     

Source and coefficient inverse problems for an elliptic equation in a rectangle

The inverse problems of determining the source and coefficient of an elliptic equation in a rectangle are studied. Additional information on the solution to the direct problem (overdetermination) is the trace of its solution on an interval inside the rectangle. Sufficient existence and uniqueness conditions (global) are derived for the inverse problems. The study is performed in the class of continuously differentiable functions whose derivatives satisfy a Hölder condition.     

Iterative method for solving an inverse coefficient problem for a hyperbolic equation

For a hyperbolic equation, we consider an inverse coefficient problem in which the unknown coefficient occurs in both the equation and the initial condition. The solution values on a given curve serve as additional information for determining the unknown coefficient. We suggest an iterative method for solving the inverse problem based on reduction to a nonlinear operator equation for the unknown coefficient and prove the uniform convergence of the iterations to a function that is a solution of the inverse problem.     

Parameter choice by discrepancy principles for ill-posed problems leading to optimal convergence rates

Schock (Ref. 1) considered a general a posteriori parameter choice strategy for the Tikhonov regularization of the ill-posed operator equationTx=y which provides nearly the optimal rate of convergence if the minimal-norm least-squares solution belongs to the range of the operator (T ^* T) ^v , o<v1. Recently, Nair (Ref. 2) improved the result of Schock and also provided the optimal rate ifv=1. In this note, we further improve the result and show in particular that the optimal rate can be achieved for 1/2v1.The final version of this work was written while M. T. Nair was a Visiting Fellow at the Centre for Mathematics and Its Applications, Australian National University, Canberra, Australia. The work of S. George was supported by a Senior Research Fellowship from CSIR, India.