Tikhonov Type Research Articles

Convergence guarantees for coefficient reconstruction in PDEs from boundary measurements by variational and Newton-type methods via range invariance

Abstract A key observation underlying this paper is the fact that the range invariance condition for convergence of regularization methods for nonlinear ill-posed operator equations—such as coefficient identification in partial differential equations (PDEs) from boundary observations—can often be achieved by extending the searched for parameter in the sense of allowing it to depend on additional variables. This clearly counteracts unique identifiability of the parameter, though. The second key idea of this paper is now to restore the original restricted dependency of the parameter by penalization. This is shown to lead to convergence of variational (Tikhonov type) and iterative (Newton-type) regularization methods. We concretize the abstract convergence analysis in a framework typical of parameter identification in PDEs in a reduced and an all-at-once setting. This is further illustrated by three examples of coefficient identification from boundary observations in elliptic and time-dependent PDEs.

Identifying a response parameter in a model of brain tumour evolution under therapy

Abstract A nonlinear conjugate gradient method is derived for the inverse problem of identifying a treatment parameter in a nonlinear model of reaction–diffusion type corresponding to the evolution of brain tumours under therapy. The treatment parameter is reconstructed from additional information about the tumour taken at a fixed instance of time. Well-posedness of the direct problems used in the iterative method is outlined as well as uniqueness of a solution to the inverse problem. Moreover, the parameter identification is recasted as the minimization of a Tikhonov type functional and the existence of a minimizer to this functional is shown. Finite-difference discretization of the space and time derivatives are employed for the numerical implementation. Numerical simulations on full 3D brain data are included showing that information about a spacewise-dependent treatment parameter can be recovered in a stable way.

A regularization method based on level-sets for the problem of crack detection from electrical measurements

We investigate regularization methods for solving the problem of crack detection in bounded planar domains from electrical measurements on the boundary. Based on the multiple level-set approach introduced in Álvarez et al (2009 J. Comput. Phys. 228 5710–21) and on the regularization strategy devised in De Cezaro et al (2009 Inverse Problems 25 035004), we propose a Tikhonov type method for stabilizing the inverse problem. Convergence and stability results for this Tikhonov method are proven. An iterative method of (multiple) level-set type is derived from the optimality conditions for the Tikhonov functional, and a relation between this method and the iterated Tikhonov method is established. The proposed level-set method is tested on the same benchmark problem considered in Álvarez et al (2009 J. Comput. Phys. 228 5710–21). The numerical experiments demonstrate its ability to identify cracks in different scenarios with high accuracy even in the presence of noise.

A mixed-finite element approach for full-field residual stress estimation

In this paper, a full-field residual stress identification technique has been presented via a mixed finite element approach from limited measured detail. Being an inverse problem, complete identification of residual stresses from incomplete measurement is always suffered by inherent ill-posedness. To control this issue, the identification problem has been formulated as an optimization problem via a Tikhonov type cost functional. The first part of this cost functional is associated with an energy norm of mismatch between predicted and measured residual stress. The second part represents the regularization term. A Lagrangian is then considered to satisfy the equilibrium constraint over the region of interest. The incorporation of Lagrangian with aforementioned cost functional provides a form which is very similar to the standard Hellinger-Reissner (HR) functional except the discrepancy term between the data. The minimization of this formulation leads to the solution of two-field mixed variational form associated with residual stress and Lagrangian variable. To show the efficacy of the proposed procedure a two-dimensional welded specimen is considered where the weld bead is assumed to be spatially varying in all direction for residual stress identification. Line scan details are also presented to examine the deviation between the actual and predicted residual stress in the heat affected zone and at the region away from the weld bead. The results indicate that the mixed finite element formulation has the capability to provide the complete information of residual stress despite of having very few information of measurement.

Tikhonov type regularization for unbounded operators

Tikhonov type regularization for unbounded operators

A backward problem for distributed order diffusion equation: uniqueness and numerical solution

In this paper we consider the identification of the initial condition for a distributed order diffusion equation. We first prove the unique existence and the regularity properties of the strong solution on the bounded temporal-spacial domain based on the eigenfunction expansions. The ill-posedness of the backward problem is interpreted by the compactness of the observation operator. Next the Laplace transformation technique and analytic continuation method are adopted to prove the uniqueness of the backward problem. Then for stabilizing the ill-posed problem, the backward problem is formulated as a Tikhonov type optimization problems, and the conjugate gradient method is adopted to solve the optimization problem with the help of the variational adjoint technique. Finally four numerical examples are given to show the efficiency and stability of the proposed method.

On the choice of Lagrange multipliers in the iterated Tikhonov method for linear ill-posed equations in Banach spaces

ABSTRACTThis article is devoted to the study of nonstationary Iterated Tikhonov (nIT) type methods (Hanke M, Groetsch CW. Nonstationary iterated Tikhonov regularization. J Optim Theory Appl. 1998;98(1):37–53; Engl HW, Hanke M, Neubauer A. Regularization of inverse problems. Vol. 375, Mathematics and its Applications. Dordrecht: Kluwer Academic Publishers Group; 1996. MR 1408680) for obtaining stable approximations to linear ill-posed problems modelled by operators mapping between Banach spaces. Here we propose and analyse an a posteriori strategy for choosing the sequence of regularization parameters for the nIT method, aiming to obtain a pre-defined decay rate of the residual. Convergence analysis of the proposed nIT type method is provided (convergence, stability and semi-convergence results). Moreover, in order to test the method's efficiency, numerical experiments for three distinct applications are conducted: (i) a 1D convolution problem (smooth Tikhonov functional and Banach parameter-space); (ii) a 2D deblurring problem (nonsmooth Tikhonov functional and Hilbert parameter-space); (iii) a 2D elliptic inverse potential problem.

Tikhonov type regularization methods for inverse mixed variational inequalities

In this paper, we focus on a Tikhonov type regularization method for inverse mixed variational inequalities. Existence of solutions of the inverse mixed variational inequalities and its regularized problem are established. We also present the relationships among three coercivity conditions for inverse mixed variational inequalities. Finally, the nonemptiness and boundedness for the solutions set of the regularized problem are derived under some mild conditions.

An optimization approach for dynamical Tucker tensor approximation

An optimization-based approach for Tucker tensor approximation of parameter-dependent data tensors and solutions of tensor differential equations with low Tucker rank is presented. The problem of updating the tensor decomposition is reformulated as a fitting problem subject to the tangent space without relying on an orthogonality gauge condition. A discrete Euler scheme is established in an alternating least squares framework, where the quadratic subproblems are reduced to trace optimization problems, that are shown to be explicitly solvable and accessible using SVD of small size. In the presence of small singular values, instability for larger ranks is reduced, since the method does not need the (pseudo) inverse of matricizations of the core tensor. Regularization of Tikhonov type can be used to compensate for the lack of uniqueness in the tangent space. The method is validated numerically and shown to be stable also for larger ranks in the case of small singular values of the core unfoldings. Higher order explicit integrators of Runge-Kutta type can be composed.

Simultaneous inversion of the fractional order and the space-dependent source term for the time-fractional diffusion equation

In this paper, a simultaneous identification problem of the spacewise source term and the fractional order for a time-fractional diffusion equation is considered. Firstly, under some assumption and with two different kinds of observation data for one-dimensional and two-dimensional time-fractional diffusion equation, the unique results of the inverse problem are proven by the Laplace transformation method and analytic continuation technique. Then the inverse problems are transformed into Tikhonov type optimization problems, the existence of optimal solutions to the Tikhonov functional is proven. Finally, we adopt an alternating minimization algorithm to solve the optimization problems. The efficiency and stability of the inversion algorithm are tested by several one- and two-dimensional examples.

Numerical approximation of a Tikhonov type regularizer by a discretized frozen steepest descent method

We present a frozen regularized steepest descent method and its finite dimensional realization for obtaining an approximate solution for the nonlinear ill-posed operator equation F ( x ) = y . The proposed method is a modified form of the method considered by Argyros et al. (2014). The balancing principle considered by Pereverzev and Schock (2005) is used for choosing the regularization parameter. The error estimate is derived under a general source condition and is of optimal order. The provided numerical example proves the efficiency of the proposed method.

A sub-domain based regularization method with prior information for human thorax imaging using electrical impedance tomography

Electrical impedance tomography (EIT) is a non-invasive imaging technique that can be used as a bed-side monitoring tool for human thorax imaging. EIT has high temporal resolution characteristics but at the same time it suffers from poor spatial resolution due to ill-posedness of the inverse problem. Often regularization methods are used as a penalty term in the cost function to stabilize the sudden changes in resistivity. In human thorax monitoring, with conventional regularization methods employing Tikhonov type regularization, the reconstructed image is smoothed between the heart and the lungs, that is, it makes it difficult to distinguish the exact boundaries of the lungs and the heart. Sometimes, obtaining structural information of the object prior to this can be incorporated into the regularization method to improve the spatial resolution along with helping create clear and distinct boundaries between the objects. However, the boundary of the heart is changed rapidly due to the cardiac cycle hence there is no information concerning the exact boundary of the heart. Therefore, to improve the spatial resolution for human thorax monitoring during the cardiac cycle, in this paper, a sub-domain based regularization method is proposed assuming the lungs and part of background region is known. In the proposed method, the regularization matrix is modified anisotropically to include sub-domains as prior information, and the regularization parameter is assigned with different weights to each sub-domain. Numerical simulations and phantom experiments for 2D human thorax monitoring are performed to evaluate the performance of the proposed regularization method. The results show a better reconstruction performance with the proposed regularization method.

The Generalized Forward-Backward Splitting Method for the Minimization of the Sum of Two Functions in Banach Spaces

We propose a convergence analysis of the generalized forward-backward splitting iterative procedure for the minimization of the sum of two functions in Banach spaces. The generalized forward-backward splitting method is applied to the minimization of functionals of the Tikhonov type with semi-norm regularization. We also investigate the perturbed forward-backward splitting method and prove its stability.

Tikhonov regularization for mathematical programs with generalized complementarity constraints

In this paper, in order to solve mathematical programs with generalized nonlinear comple-mentarity constraints, we present a new regularization method of Tikhonov type, based on regularizing an optimization problem with equality constraints. The stability and convergence of the regularized solutions are considered. Numerical examples are given for illustration.

The Cauchy problem for Tikhonov’s system in the case of a multiple root of the degenerate equation

For a system of the Tikhonov type in the case of a double root of the degenerate equation, the asymptotics of the boundary-layer solution is constructed and justified. The character of the asymptotics and the algorithm for constructing it differ substantially from the classical case when the degenerate equation has a simple root.

Controllability of a backward fractional semilinear differential equation

In this paper we study the approximate controllability of a fractional semilinear differential equation involving the right fractional Caputo derivative. More precisely, we construct by means of Tikhonov type regularization method, the controllability operator. Then under certain condition on this operator, we obtain that the associate backward fractional linear system can be steered to an arbitrary small neighborhood of the state at initial time. This allows us to prove the approximate controllability of the semilinear system.

A primal–dual hybrid gradient method for nonlinear operators with applications to MRI

We study the solution of minimax problems min xmax yG(x) + 〈K(x), y〉 − F*(y) in finite-dimensional Hilbert spaces. The functionals G and F* we assume to be convex, but the operator K we allow to be nonlinear. We formulate a natural extension of the modified primal–dual hybrid gradient method, originally for linear K, due to Chambolle and Pock. We prove the local convergence of the method, provided various technical conditions are satisfied. These include in particular the Aubin property of the inverse of a monotone operator at the solution. Of particular interest to us is the case arising from Tikhonov type regularization of inverse problems with nonlinear forward operators. Mainly we are interested in total variation and second-order total generalized variation priors. For such problems, we show that our general local convergence result holds when the noise level of the data f is low, and the regularization parameter α is correspondingly small. We verify the numerical performance of the method by applying it to problems from magnetic resonance imaging (MRI) in chemical engineering and medicine. The specific applications are in diffusion tensor imaging and MR velocity imaging. These numerical studies show very promising performance.

MCMC-Based Image Reconstruction with Uncertainty Quantification

The connection between Bayesian statistics and the technique of regularization for inverse problems has been given significant attention in recent years. For example, Bayes' law is frequently used as motivation for variational regularization methods of Tikhonov type. In this setting, the regularization function corresponds to the negative-log of the prior probability density; the fit-to-data function corresponds to the negative-log of the likelihood; and the regularized solution corresponds to the maximizer of the posterior density function, known as the maximum a posteriori (MAP) estimator of the unknown, which in our case is an image. Much of the work in this direction has focused on the development of techniques for efficient computation of MAP estimators (or regularized solutions). Less explored in the inverse problems community, and of interest to us in this paper, is the problem of sampling from the posterior density. To do this, we use a Markov chain Monte Carlo (MCMC) method, which has previously appeared in the Bayesian statistics literature, is straightforward to implement, and provides a means of both estimation and uncertainty quantification for the unknown. Additionally, we show how to use the preconditioned conjugate gradient method to compute image samples in cases where direct methods are not feasible. And finally, the MCMC method provides samples of the noise and prior precision (inverse-variance) parameters, which makes regularization parameter selection unnecessary. We focus on linear models with independent and identically distributed Gaussian noise and define the prior using a Gaussian Markov random field. For our numerical experiments, we consider test cases from both image deconvolution and computed tomography, and our results show that the approach is effective and surprisingly computationally efficient, even in large-scale cases.

A new choice rule for regularization parameters in Tikhonov regularization

This article proposes and analyses a novel heuristic rule for choosing regularization parameters in Tikhonov regularization for inverse problems. The existence of solutions to the regularization parameter equation is shown, and a variational characterization of the inverse solution is provided. Some a posteriori error estimates of the approximate solutions to the inverse problem are also established. An iterative algorithm is suggested for the efficient numerical realization of the new choice rule, which is shown to have a practically desirable monotone convergence. Numerical experiments for both mildly and severely ill-posed benchmark inverse problems with various regularizing functionals of Tikhonov type, e.g. L 2–L 2 with constraints, L 2–ℓ1 and L 1–TV, are presented to demonstrate its effectiveness, and compared with the quasi-optimality criterion.

Preface: numerical and applied linear algebra

This special issue of Advances in Computational Mathematics is devoted to Numerical and Applied Linear Algebra, which is an important area of mathematics and a fundamental part of engineering and computational science problems. One of the main tools of this area is matrices, properties of which are studied and computations of them are performed, with different methods, in a variety of applications. The special issue contains 12 papers reflecting some specialized themes on numerical and applied linear algebra and at the same time gives an idea of the broadness of this area, which has grown spectacularly in the last decades. The work entitled “Structured linear algebra problems in adaptive optics imaging”, by Johnathan M. Bardsley, Sarah Knepper, and James Nagyy, discusses an adaptive optics image problem to remove the effects of the phase error. For solving the Kronecker product structured, rank-deficient least-squares problem on the Fried geometry, two types of regularization are considered: a truncated singular value decomposition type and a Tikhonov type. The paper “On the numerical solution of large-scale sparse discrete-time Riccati equations”, by Peter Benner and Heike Fasbender, deals with the numerical solution of discrete algebraic Riccati equations. Since the systems considered are large-scale, the solution process for the computation of a