Linear Ill-posed Problems Research Articles

The structure of iterative methods for symmetric linear discrete ill-posed problems

The iterative solution of large linear discrete ill-posed problems with an error contaminated data vector requires the use of specially designed methods in order to avoid severe error propagation. Range restricted minimal residual methods have been found to be well suited for the solution of many such problems. This paper discusses the structure of matrices that arise in a range restricted minimal residual method for the solution of large linear discrete ill-posed problems with a symmetric matrix. The exploitation of the structure results in a method that is competitive with respect to computer storage, number of iterations, and accuracy.

A double optimal descent algorithm for iteratively solving ill-posed linear inverse problems

In the iterative solution of an ill-posed linear system , how to select a fast and easily established descent direction to reduce the residual is an important issue. A mathematical procedure to find a double optimal descent direction in , without inverting , is developed in an -dimensional Krylov subspace. The novelty is that we expand in an affine Krylov subspace with undetermined coefficients, and then two optimization techniques are used to determine these coefficients in closed form, which can greatly accelerate the convergence speed in solving the ill-posed linear problems. The double optimal descent algorithm is proven to be absolutely convergent very fast, accurate and robust against noise, which is confirmed by numerical tests of several linear inverse problems, including the heat source identification problem, the backward heat conduction problem, the inverse Cauchy problem and the external force identification problem.

A control Liapunov function approach to generalized and regularized descent methods for zero finding1

This paper revisits a class of recently proposed so-called invariant manifold methods for zero finding of ill-posed problems, showing that they can be profitably viewed as homotopy methods, in which the homotopy parameter is interpreted as a learning parameter. Moreover, it is shown that the choice of this learning parameter can be made in a natural manner from a control Liapunov function approach CLF. From this viewpoint, maintaining manifold invariance is equivalent to ensuring that the CLF satisfies a certain ordinary differential equation, involving the learning parameter, that allows an estimate of rate of convergence. In order to illustrate this approach, algorithms recently proposed using the invariant manifold approach, are rederived, via CLFs, in a unified manner. Adaptive regularization parameters for solving linear algebraic ill-posed problems were also proposed. This paper also shows that the discretizations of the ODEs to solve the zero finding problem, as well as the different adaptive choices of the regularization parameter, yield iterative methods for linear systems, which are also derived using the Liapunov optimizing control LOC method.

A strongly ill-posed integrodifferential singular parabolic problem in the unit cube of $\mathbb{R}^n$

Via Carleman's estimates we prove uniqueness and continuous dependence results for a severely ill-posed linear integro-differential singular parabolic problems without initial conditions.

Enhancing magnetic signals in unexploded ordnances (UXO) detection based on edge-preserved stable downward continuation method

This paper describes an efficient edge-preserved regularization algorithm for downward continuation of magnetic data in detecting unexploded ordnance (UXO). The magnetic anomalies arising from multi-source UXO can overlap at a height over the ground surface while causative sources may not be readily separated due to low level of signal-to-noise ratio of the observed data. To effectively work the magnetic method in the cleanup stage of contaminated area with UXO, the magnetic anomalies of UXO sources should be enhanced in order to separate the locations of different sources. The stable downward continuation of magnetic data can increase the signal-to-noise ratio, which subsequently causes the separation of UXO sources by enhancing the signals. In this study the researchers formulated the downward continuation as a linear ill-posed deconvolution problem. To obtain a reasonable downward continued field, the proposed filter is stabilized in a Fourier domain to regularize the downward solution using the edge-preserved (or total-variation) algorithm. The L-curve method was used to choose the optimum value of the regularization parameter, which is a trade-off between the misfit and the solution norms in the cost function of optimization problem. A synthetic magnetic field was constructed from isolated multi-source UXO anomalies, the results of which show that the data can be stably downward continued to the ground surface. Likewise, a field data set was provided to demonstrate the capability of the applied method in UXO detection. The results of the synthetic and real case study revealed that the observed magnetic anomalies at a specific height of survey over the ground surface have low amplitude, indeed, the causative UXO sources may not be readily distinguished in detection process, especially anomalies from small UXOs. It was shown that the continued data can enhance the locations of UXOs while small ones are not distinguishable in the primary data.

Simplified GSVD computations for the solution of linear discrete ill-posed problems

The generalized singular value decomposition (GSVD) often is used to solve Tikhonov regularization problems with a regularization matrix without exploitable structure. This paper describes how the ...

Approximation of solutions with singularities of various types for linear ill-posed problems

Approximation of solutions with singularities of various types for linear ill-posed problems

The Exact Condition Number of the Truncated Singular Value Solution of a Linear Ill-Posed Problem

The main result of this paper is the formulation of an explicit expression for the condition number of the truncated least squares solution of $Ax=b$. This expression is given in terms of the singular values of $A$ and the Fourier coefficients of $b$. The result is derived using the notion of the Fréchet derivative together with the product norm on the data $[A,b]$ and the 2-norm on the solution. Numerical experiments are given to confirm our results by comparing them to those obtained by means of a finite difference approach.

Inverse subspace problems with applications

SUMMARY Given a square matrix A, the inverse subspace problem is concerned with determining a closest matrix to A with a prescribed invariant subspace. When A is Hermitian, the closest matrix may be required to be Hermitian. We measure distance in the Frobenius norm and discuss applications to Krylov subspace methods for the solution of large-scale linear systems of equations and eigenvalue problems as well as to the construction of blurring matrices. Extensions that allow the matrix A to be rectangular and applications to Lanczos bidiagonalization, as well as to the recently proposed subspace-restricted SVD method for the solution of linear discrete ill-posed problems, also are considered.Copyright © 2013 John Wiley & Sons, Ltd.

Fractional regularization matrices for linear discrete ill-posed problems

The numerical solution of linear discrete ill-posed problems typically requires regularization. Two of the most popular regularization methods are due to Tikhonov and Lavrentiev. These methods require the choice of a regularization matrix. Common choices include the identity matrix and finite difference approximations of a derivative operator. It is the purpose of the present paper to explore the use of fractional powers of the matrices $$A^\mathrm{T}\!A$$ (for Tikhonov regularization) and A (for Lavrentiev regularization) as regularization matrices, where A is the matrix that defines the linear discrete ill-posed problem. Both small- and large-scale problems are considered.

On the interplay of basis smoothness and specific range conditions occurring in sparsity regularization

The convergence rates results in ℓ1-regularization when the sparsity assumption is narrowly missed, presented by Burger et al (2013 Inverse Problems 29 025013), are based on a crucial condition which requires that all basis elements belong to the range of the adjoint of the forward operator. Partly it was conjectured that such a condition is very restrictive. In this context, we study sparsity-promoting varieties of Tikhonov regularization for linear ill-posed problems with respect to an orthonormal basis in a separable Hilbert space using ℓ1 and sublinear penalty terms. In particular, we show that the corresponding range condition is always satisfied for all basis elements if the problems are well-posed in a certain weaker topology and the basis elements are chosen appropriately related to an associated Gelfand triple. The Radon transform, Symm’s integral equation and linear integral operators of Volterra type are examples for such behaviour, which allows us to apply convergence rates results for non-sparse solutions, and we further extend these results also to the case of non-convex ℓq-regularization with 0 < q < 1.

An objective rationale for the choice of regularisation parameter with application to global multiple-frequency &amp;lt;i&amp;gt;S&amp;lt;/i&amp;gt;-wave tomography

Abstract. In a linear ill-posed inverse problem, the regularisation parameter (damping) controls the balance between minimising both the residual data misfit and the model norm. Poor knowledge of data uncertainties often makes the selection of damping rather arbitrary. To go beyond that subjectivity, an objective rationale for the choice of damping is presented, which is based on the coherency of delay-time estimates in different frequency bands. Our method is tailored to the problem of global multiple-frequency tomography (MFT), using a data set of 287 078 S-wave delay times measured in five frequency bands (10, 15, 22, 34, and 51 s central periods). Whereas for each ray path the delay-time estimates should vary coherently from one period to the other, the noise most likely is not coherent. Thus, the lack of coherency of the information in different frequency bands is exploited, using an analogy with the cross-validation method, to identify models dominated by noise. In addition, a sharp change of behaviour of the model &amp;amp;ell;∞-norm, as the damping becomes lower than a threshold value, is interpreted as the signature of data noise starting to significantly pollute at least one model component. Models with damping larger than this threshold are diagnosed as being constructed with poor data exploitation. Finally, a preferred model is selected from the remaining range of permitted model solutions. This choice is quasi-objective in terms of model interpretation, as the selected model shows a high degree of similarity with almost all other permitted models (correlation superior to 98% up to spherical harmonic degree 80). The obtained tomographic model is displayed in the mid lower-mantle (660–1910 km depth), and is shown to be compatible with three other recent global shear-velocity models. A wider application of the presented rationale should permit us to converge towards more objective seismic imaging of Earth's mantle.

A GCV based Arnoldi-Tikhonov regularization method

For the solution of linear discrete ill-posed problems, in this paper we consider the Arnoldi-Tikhonov method coupled with the Generalized Cross Validation for the computation of the regularization parameter at each iteration. We study the convergence behavior of the Arnoldi method and its properties for the approximation of the (generalized) singular values, under the hypothesis that Picard condition is satisfied. Numerical experiments on classical test problems and on image restoration are presented.

FGMRES for linear discrete ill-posed problems

GMRES is one of the most popular iterative methods for the solution of large linear systems of equations. However, GMRES does not always perform well when applied to the solution of linear systems of equations that arise from the discretization of linear ill-posed problems with error-contaminated data represented by the right-hand side. Such linear systems are commonly referred to as linear discrete ill-posed problems. The FGMRES method, proposed by Saad, is a generalization of GMRES that allows larger flexibility in the choice of solution subspace than GMRES. This paper explores application of FGMRES to the solution of linear discrete ill-posed problems. Numerical examples illustrate that FGMRES with a suitably chosen solution subspace may determine approximate solutions of higher quality than commonly applied iterative methods.

Square smoothing regularization matrices with accurate boundary conditions

This paper is concerned with the solution of large-scale linear discrete ill-posed problems. The determination of a meaningful approximate solution of these problems requires regularization. We discuss regularization by the Tikhonov method and by truncated iteration. The choice of regularization matrix in Tikhonov regularization may significantly affect the quality of the computed approximate solution. The present paper describes the construction of square regularization matrices from finite difference equations with a focus on the boundary conditions. The regularization matrices considered have a structure that makes them easy to apply in iterative methods, including methods based on the Arnoldi process. Numerical examples illustrate the properties and effectiveness of the regularization matrices described.

Fast nonstationary preconditioned iterative methods for ill-posed problems, with application to image deblurring

We introduce a new iterative scheme for solving linear ill-posed problems, similar to nonstationary iterated Tikhonov regularization, but with an approximation of the underlying operator to be used for the Tikhonov equations. For image deblurring problems, such an approximation can be a discrete deconvolution that operates entirely in the Fourier domain. We provide a theoretical analysis of the new scheme, using regularization parameters that are chosen by a certain adaptive strategy. The numerical performance of this method turns out to be superior to state-of-the-art iterative methods, including the conjugate gradient iteration for the normal equation, with and without additional preconditioning.

Automatic parameter setting for Arnoldi–Tikhonov methods

In the framework of iterative regularization techniques for large-scale linear ill-posed problems, this paper introduces a novel algorithm for the choice of the regularization parameter when performing the Arnoldi–Tikhonov method. Assuming that we can apply the discrepancy principle, this new strategy can work without restrictions on the choice of the regularization matrix. Moreover, this method is also employed as a procedure to detect the noise level whenever it is just overestimated. Numerical experiments arising from the discretization of integral equations and image restoration are presented.

A predictor–corrector iterated Tikhonov regularization for linear ill-posed inverse problems

Many questions in science and engineering often give rise to linear ill-posed inverse problems. To enable meaningful approximate solutions of the inverse problems, various regularization methods have subsequently been proposed to make these problems less sensitive to perturbations. One of the most popular regularization techniques is iterated Tikhonov regularization, which has attracted considerable attention due to its interesting applications in practical. However, this regularization often suffers from inaccurate results and low computational efficiency in some situations. In this paper, we proposed an accelerated predictor–corrector iterated Tikhonov regularization. This method combined the classical iterated Tikhonov regularization with modified Euler method, which could improve computational efficiency without sacrificing numerical accuracy. The convergence rate of the accelerated regularization was investigated theoretically when the right-hand side was corrupted by noise. In particular, we derived an error estimate of optimal order for the convergence of the accelerated regularization. Both one-and two-dimensional numerical experiments were implemented to illustrate the accuracy and efficiency of the accelerated version of iterated Tikhonov regularization.

Two-parameter discrepancy principle for combined projection and Tikhonov regularization of ill-posed problems

Abstract. For solving a linear ill-posed problem, a combination of Tikhonov regularization and a finite dimensional projection method is considered. In the present paper we treat the dimension of the projection as the second parameter of regularization. The method using the discrepancy set of all regularization parameter pairs which satisfy a certain discrepancy principle is investigated. For the case of truncated SVD and LSQ projection methods, under the standard source conditions, an order of convergence is derived. This order of convergence is optimal under the same conditions as those for Tikhonov regularization with the discrepancy principle and without additional discretization.

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.