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Computing steering control for an approximately controllable linear system for a given target state is an ill-posed problem. We use a weighted Tikhonov regularization method and compute the regularized control. It is proved that the target state corresponding to the regularized control is close to the actual state to be attained. We also obtained error estimates and convergence rates involved in the regularization procedure using both the a priori and a posteriori parameter choice rule. Theory is substantiated with numerical experiments.

Numerical reconstruction of the missing boundary data in 2D and 3D linear elasticity.Noisy over-prescribed data are available on the remaining accessible boundary.The inverse problem is solved by a meshless method, i.e. the method of fundamental solutions (MFS).Stabilization is achieved via several SVD-based regularization methods.The regularization parameter is selected via the criteria used for inverse problems. The numerical reconstruction of the missing Dirichlet and Neumann data on an inaccessible part of the boundary in the case of two- and three-dimensional linear isotropic elastic materials from the knowledge of over-prescribed noisy measurements taken on the remaining accessible boundary part is investigated. This inverse problem is solved using the method of fundamental solutions (MFS), whilst its stabilization is achieved through several singular value decomposition (SVD)-based regularization methods, such as the Tikhonov regularization methodź48, the damped SVD and the truncated SVDź18. The regularization parameter is selected according to the discrepancy principleź40, generalized cross-validation criterionź14 and Hansen's L-curve methodź20.

In this paper, we study a non-characteristic Cauchy problem for a radially symmetric inverse heat conduction equation in a two-layer domain. This is a severely ill-posed problem in the sense that the solution (if it exists) does not depend continuously on the data. It is well-known that the classical Tikhonov regularization solutions are too smooth and the approximate solutions may lack details that might be contained in the exact solutions. Combining Fourier transform technique with a modified version of the classical Tikhonov regularization, we obtain a regularized solution which is stably convergent to the exact solution with a sharp error estimate.

Based on the smoothing Newton method and the Tikhonov regularization method, we construct a regularization Newton method for the second-order cone complementarity problem. The method uses a non-monotone line search scheme which contains the usual monotone line search as a special case. By using the theory of Euclidean Jordan algebras, we prove that the proposed method is globally and locally quadratically convergent under suitable assumptions. Some numerical results are reported which indicate the effectiveness of the method.

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.

A crucial problem concerning Tikhonov regularization is the proper choice of the regularization parameter. This paper deals with a generalization of a parameter choice rule due to Reginska (1996) [31] , analyzed and algorithmically realized through a fast fixed-point method in Bazan (2008) [3] , which results in a fixed-point method for multi-parameter Tikhonov regularization called MFP. Like the single-parameter case, the algorithm does not require any information on the noise level. Further, combining projection over the Krylov subspace generated by the Golub–Kahan bidiagonalization (GKB) algorithm and the MFP method at each iteration, we derive a new algorithm for large-scale multi-parameter Tikhonov regularization problems. The performance of MFP when applied to well known discrete ill-posed problems is evaluated and compared with results obtained by the discrepancy principle. The results indicate that MFP is efficient and competitive. The efficiency of the new algorithm on a super-resolution problem is also illustrated.

We analyze Tikhonov regularization where the forward operator is defined on a direct sum U of several Banach spaces U i , i = 1 , … , m . The regularization term on U is a sum of different regularizations on each U i . The theoretical framework, known for the case m = 1 , can be easily reformulated to the case m > 1 . Under certain assumptions on weak topologies and on the forward operator it is possible to extend the results on the well-posedness of the minimization process, on its stability with respect to the data, and on the corresponding existence and convergence results, to the general case m ⩾ 1 . We consider two particular regularizations, the total or the so-called bounded variation (BV) regularization and the smooth ( H 1 ) regularization. Assume that the domain Ω is decomposed into two non-overlapping subdomains Ω 1 , Ω 2 . The direct sum of Banach spaces for m = 2 will be U = H 1 ( Ω 1 ) ⊕ BV ( Ω 2 ) and the underlying regularization will be mixed H 1 –BV regularization. We define proper weak topology on U and prove that the assumptions for the above mentioned framework are fulfilled. In such a way we get the theoretical results for a broad range of H 1 –BV regularizations including the image denoising for the identity forward operator, the deconvolution problems for the convolution forward operator, or any other inverse problem for the continuous nonlinear forward operators. We demonstrate how the mixed regularization can be applied in practice. We suggest a denoising algorithm and compare its behavior with the BV regularization and the H 1 regularization. We discuss the results of the denoising on several real world images justifying the usefulness of the mixed H 1 –BV regularization.

Instead of the Tikhonov regularization method which with a scalar being the regularization parameter, Liu et al. [1] have proposed a novel regularization method with a vector as being the regularization parameter. As a continuation we further propose an optimally scaled vector regularization method (OSVRM) to solve the ill-posed linear problems, which is better than the Tikhonov regularization method. The presently proposed vector regularization method is shown to be well conditioning to overcome the ill-posedness of the linear equations system. The OSVRM causes a significant improvement of stability and accuracy in the numerical solution of ill-posed linear problem, and its convergence speed is as fast as by solving the well-posed linear problem. Some tests of the linear inverse problems confirm the efficiency and accuracy of the OSVRM.

In this paper, we propose a new method for solving large-scale ill-posed problems. This method is based on the Karush–Kuhn–Tucker conditions, Fisher–Burmeister function and the discrepancy principle. The main difference from the majority of existing methods for solving ill-posed problems is that, we do not need to choose a regularization parameter in advance. Experimental results show that the proposed method is effective and promising for many practical problems.

This paper deals with the problem of determining an unknown source which depends only on one variable in two-dimensional Poisson equation, with the aid of an extra measurement at an internal point. The problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. We obtain the regularization solution by the truncation method. For the regularization solution, the Holder type stability estimate between the regularization solution and the exact solution is given. Numerical results are presented to illustrate the accuracy and efficiency of this method.

This paper discusses the consistent regularization property of the generalized @a method when applied as an integrator to an initial value high index and singular differential-algebraic equation model of a multibody system. The regularization comes from within the discretization itself and the discretization remains consistent over the range of values the regularization parameter may take. The regularization involves increase of the smallest singular values of the ill-conditioned Jacobian of the discretization and is different from Baumgarte and similar techniques which tend to be inconsistent for poor choice of regularization parameter. This regularization also helps where pre-conditioning the Jacobian by scaling is of limited effect, for example, when the scleronomic constraints contain multiple closed loops or singular configuration or when high index path constraints are present. The feed-forward control in Kane's equation models is additionally considered in the numerical examples to illustrate the effect of regularization. The discretization presented in this work is adopted to the first order DAE system (unlike the original method which is intended for second order systems) for its A-stability and same order of accuracy for positions and velocities.

In this paper, a new gradient regularization algorithm is introduced and applied to solve an inverse problem of determining source terms in one-dimensional advection–dispersion equation with final observations. By functional approximations, the algorithm is reduced to find an optimal perturbation for a given source parameter involving computations of gradient vectors. In the case of using accurate data, the optimal perturbation can be directly worked out by the least square method; in the case of with random noisy data, regularization strategy may be useful to the realization of the algorithm. As compared with ordinary gradient regularization algorithm, the new gradient regularization algorithm gives a possible approach to the precise choices of optimal regularization parameters. Several numerical simulations under different conditions are carried out showing that the algorithm is feasible and efficient. Furthermore, the algorithm is successfully applied to solve a real life example of determining an average magnitude of groundwater pollution sources.

Although the \emph{residual method}, or \emph{constrained regularization}, is frequently used in applications, a detailed study of its properties is still missing. This sharply contrasts the progress of the theory of Tikhonov regularization, where a series of new results for regularization in Banach spaces has been published in the recent years. The present paper intends to bridge the gap between the existing theories as far as possible. We develop a stability and convergence theory for the residual method in general topological spaces. In addition, we prove convergence rates in terms of (generalized) Bregman distances, which can also be applied to non-convex regularization functionals. We provide three examples that show the applicability of our theory. The first example is the regularized solution of linear operator equations on $L^p$-spaces, where we show that the results of Tikhonov regularization generalize unchanged to the residual method. As a second example, we consider the problem of density estimation from a finite number of sampling points, using the Wasserstein distance as a fidelity term and an entropy measure as regularization term. It is shown that the densities obtained in this way depend continuously on the location of the sampled points and that the underlying density can be recovered as the number of sampling points tends to infinity. Finally, we apply our theory to compressed sensing. Here, we show the well-posedness of the method and derive convergence rates both for convex and non-convex regularization under rather weak conditions., 29 pages, one figure