Tikhonov Functionals Research Articles

Smoothing and differentiation of data by Tikhonov and fractional derivative tools, applied to surface‐enhanced Raman scattering (SERS) spectra of crystal violet dye

AbstractAll signals obtained as instrumental response of analytical apparatus are affected by noise, as in Raman spectroscopy. Whereas Raman scattering is an inherently weak process, the noise background may lead to misinterpretations. Although surface amplification of the Raman signal using metallic nanoparticles has been a strategy employed to partially solve the signal‐to‐noise problem, the preprocessing of Raman spectral data through the use of mathematical filters has become an integral part of Raman spectroscopy analysis. In this paper, a Tikhonov modified method to remove random noise in experimental data is presented. In order to refine and improve the Tikhonov method as a filter, the proposed method includes Euclidean norm of the fractional‐order derivative of the solution as an additional criterion in Tikhonov function. In the strategy used here, the solution depends on the regularization parameter, , and on the fractional derivative order, . As will be demonstrated, with the algorithm presented here, it is possible to obtain a noise‐free spectrum without affecting the fidelity of the molecular signal. In this alternative, the fractional derivative works as a fine control parameter for the usual Tikhonov method. The proposed method was applied to simulated data and to surface‐enhanced Raman scattering (SERS) spectra of crystal violet dye in Ag nanoparticles colloidal dispersion.

Convergence analysis of critical point regularization with non-convex regularizers

One of the key assumptions in the stability and convergence analysis of variational regularization is the ability of finding global minimizers. However, such an assumption is often not feasible when the regularizer is a black box or non-convex making the search for global minimizers of the involved Tikhonov functional a challenging task. This is in particular the case for the emerging class of learned regularizers defined by neural networks. Instead, standard minimization schemes are applied which typically only guarantee that a critical point is found. To address this issue, in this paper we study stability and convergence properties of critical points of Tikhonov functionals with a possible non-convex regularizer. To this end, we introduce the concept of relative sub-differentiability and study its basic properties. Based on this concept, we develop a convergence analysis assuming relative sub-differentiability of the regularizer. The rationale behind the proposed concept is that critical points of the Tikhonov functional are also relative critical points and that for the latter a convergence theory can be developed. For the case where the noise level tends to zero, we derive a limiting problem representing first-order optimality conditions of a related restricted optimization problem. Besides this, we also give a comparison with classical methods and show that the class of ReLU-networks are appropriate choices for the regularization functional. Finally, we provide numerical simulations that support our theoretical findings and the need for the sort of analysis that we provide in this paper.

Minimum Principle for the Tikhonov Functional in the Problem of Stable Continuation of a Potential Field from a Surface

Minimum Principle for the Tikhonov Functional in the Problem of Stable Continuation of a Potential Field from a Surface

Inverse problems of identifying the unknown transverse shear force in the Euler–Bernoulli beam with Kelvin–Voigt damping

Abstract In this paper, we study the inverse problems of determining the unknown transverse shear force g ⁢ ( t ) {g(t)} in a system governed by the damped Euler–Bernoulli equation ρ ⁢ ( x ) ⁢ u t ⁢ t + μ ⁢ ( x ) ⁢ u t + ( r ⁢ ( x ) ⁢ u x ⁢ x ) x ⁢ x + ( κ ⁢ ( x ) ⁢ u x ⁢ x ⁢ t ) x ⁢ x = 0 , ( x , t ) ∈ ( 0 , ℓ ) × ( 0 , T ] , \rho(x)u_{tt}+\mu(x)u_{t}+(r(x)u_{xx})_{xx}+(\kappa(x)u_{xxt})_{xx}=0,\quad(x,% t)\in(0,\ell)\times(0,T], subject to the boundary conditions u ⁢ ( 0 , t ) = 0 , u x ⁢ ( 0 , t ) = 0 , [ r ⁢ ( x ) ⁢ u x ⁢ x + κ ⁢ ( x ) ⁢ u x ⁢ x ⁢ t ] x = ℓ = 0 , - [ ( r ⁢ ( x ) ⁢ u x ⁢ x + κ ⁢ ( x ) ⁢ u x ⁢ x ⁢ t ) x ] x = ℓ = g ⁢ ( t ) , u(0,t)=0,\quad u_{x}(0,t)=0,\quad[r(x)u_{xx}+\kappa(x)u_{xxt}]_{x=\ell}=0,% \quad-[(r(x)u_{xx}+\kappa(x)u_{xxt})_{x}]_{x=\ell}=g(t), for t ∈ [ 0 , T ] {t\in[0,T]} , from the measured deflection ν ⁢ ( t ) := u ⁢ ( ℓ , t ) {\nu(t):=u(\ell,t)} , t ∈ [ 0 , T ] {t\in[0,T]} , and from the bending moment ω ⁢ ( t ) := - ( r ⁢ ( 0 ) ⁢ u x ⁢ x ⁢ ( 0 , t ) + κ ⁢ ( 0 ) ⁢ u x ⁢ x ⁢ t ⁢ ( 0 , t ) ) , t ∈ [ 0 , T ] , \omega(t):=-(r(0)u_{xx}(0,t)+\kappa(0)u_{xxt}(0,t)),\quad t\in[0,T], where the terms ( κ ⁢ ( x ) ⁢ u x ⁢ x ⁢ t ) x ⁢ x {(\kappa(x)u_{xxt})_{xx}} and μ ⁢ ( x ) ⁢ u t {\mu(x)u_{t}} account for the Kelvin–Voigt damping and external damping, respectively. The main purpose of this study is to analyze the Kelvin–Voigt damping effect on determining the unknown transverse shear force (boundary input) through the given boundary measurements. The inverse problems are transformed into minimization problems for Tikhonov functionals, and it is shown that the regularized functionals admit unique solutions for the inverse problems. By suitable regularity on the admissible class of shear force g ⁢ ( t ) {g(t)} , we prove that these functionals are Fréchet differentiable, and the derivatives are expressed through the solutions of corresponding adjoint problems posed with measured data as boundary data associated with the direct problem. The solvability of these adjoint problems is obtained under the minimal regularity of the boundary data g ⁢ ( t ) {g(t)} , which turns out to be the regularizing effect of the Kelvin–Voigt damping in the direct problem. Furthermore, using the Fréchet derivative of the more regularized Tikhonov functionals, we obtain remarkable Lipschitz stability estimates for the transverse shear force in terms of the given measurement by a feasible condition only on the Kelvin–Voigt damping coefficient.

Solution for ill-posed EIV model regularization attending to its decreasing regularization characteristic

Abstract The errors-in-variables (EIV) model is used for data processing in the field of geodesy. However, the EIV model may be ill-posed. By analyzing the decreasing regularization (D-regularization) characteristic of solutions for EIV models, algorithms using traditional methods such as singular value decomposition or the Tikhonov function can directly determine the irrationality of a model. When an EIV model is ill-posed, solutions in which the observation errors in the coefficient matrix are simulated by variables make the ill-posed nature of the model more serious. This is because the simulated observation errors are subtracted from the coefficient matrix in subsequent computations, which reduces the singular value of the normal matrix. This point is verified using an example. To account for the D-regularization of solutions in EIV models, a modified algorithm is derived by classifying the models into two categories, and the regularization parameters are iteratively revised based on the mean squared error. Finally, some conclusions are drawn from two separate examples.

A method for determining the parameters in a rheological model for viscoelastic materials by minimizing Tikhonov functionals

ABSTRACT Parameter estimation for generalized Maxwell models for viscoelastic materials can become ill-posed when insufficient experimental data is available. In this article, we introduce a rheological model containing Maxwell elements, define the associated forward operator and the inverse problem in order to determine the number of Maxwell elements and the material parameters of the underlying viscoelastic material. We simulate a relaxation experiment by applying a strain to the material and measure the generated stress. Since the mechanical response varies with the number of Maxwell elements, the forward operator of the underlying inverse problem depends on parts of the solution. Thereby, the forward problem consists in computing stress responses for a given number of Maxwell elements, stiffness parameters and relaxation times. The inverse problem means to compute these parameters from given stress measurements, where an additional difficulty lies in the fact that the forward mapping changes with the number of Maxwell elements and, thus, with a quantity to be computed as part of the solution. Under the assumption that every relaxation time is located in one temporal decade we propose a clustering algorithm to resolve this problem. We provide the calculations that are necessary for the minimization process and conclude by investigating unperturbed as well as noisy data. Different reconstruction approaches for the stiffnesses and relaxation times based on minimizing a least squares functional are presented. We look at individual stress components to analyze different strain rates and displacement rates, respectively, and study how experimental duration affects the identified material parameters.

О некорректных задачах, экстремалях функционала Тихонова и регуляризованных принципах Лагранжа

The problem of finding a normal solution to an operator equation of the first kind on a pair of Hilbert spaces is classical in the theory of ill-posed problems. In accordance with the theory of regularization, its solutions are approximated by the extremals of the Tikhonov functional. From the point of view of the theory of problems for constrained extremum, the problem of minimizing a functional, equal to the square of the norm of an element, with an operator equality constraint (that is, given by an operator with an infinite-dimensional image) is equivalent to the classical ill-posed problem. The paper discusses the possibility of regularizing the Lagrange principle (LP) in the specified constrained extremum problem. This regularization is a transformation of the LP that turns it into a universal tool of stable solving illposed problems in terms of generalized minimizing sequences (GMS) and preserves its “general structural arrangement” based on the constructions of the classical Lagrange function. The transformed LP “contains” the classical analogue as its limiting variant when the numbers of the GMS elements tend to infinity. Both non-iterative and iterative variants of the regularization of the LP are discussed. Each of them leads to stable generation of the GMS in the original constrained extremum problem from the extremals of the regular Lagrange functional taken at the values of the dual variable generated by the corresponding procedure for the regularization of the dual problem. In conclusion, the article discusses the relationship between the extremals of the Tikhonov and Lagrange functionals in the considered classical ill-posed problem.

On the existence and stability of minimizers for generalized Tikhonov functionals with general similarity data

<abstract><p>We study a unified form of Tikhonov regularization for ill-posed problems with a general data similarity term. We discuss sufficient conditions on this generalized Tikhonov functional that guarantee existence and stability of solutions. Furthermore, we show that some particular cases of similarity functionals and regularization techniques can be cast into a unified theoretical framework. In particular, we consider the cases of $ p $-power norm, Bregman distance and mutual information as examples of a data similarity term, and Tikhonov regularization of order one or using a Sobolev norm, total variation penalization and powers of semi-norms associated to closed operators as examples of a regularization term. Finally, we take up the case of the image registration problem.</p></abstract>

Compound Regularization of Full-Waveform Inversion for Imaging Piecewise Media

The nonlinear and ill-posed nature of full waveform inversion (FWI) requires us to use sophisticated regularization techniques to solve it. In most applications, the model parameters may be described by physical properties (e.g., wave speeds, density, attenuation, anisotropic parameters) which are piecewise functions of space. Compound regularizations are thus necessary to reconstruct properly such parameters by FWI. We consider different implementations of compound regularizations in the wavefield reconstruction inversion (WRI) method, a formulation of FWI that extends its search space and prevent the so-called cycle skipping pathology. Our hybrid regularizations rely on Tikhonov and total variation (TV) functionals, from which we build two classes of hybrid regularizers: the first class is simply obtained by a convex combination (CC) of the two functionals, while the second relies on their infimal convolution (IC). In the former class, the model of parameters is required to simultaneously satisfy different priors, while in the latter the model is broken into its basic components, each satisfying a distinct prior (e.g. smooth, piecewise constant, piecewise linear). We implement these types of compound regularizations in the WRI optimization problem using the alternating direction method of multipliers (ADMM). Then, we assess our regularized WRI in the framework of seismic imaging applications. Using a wide range of subsurface models, we conclude that compound regularizer based on IC leads to the lowest error in the parameter reconstruction compared to that obtained with the CC counterpart and the Tikhonov and TV regularizers when used independently.

Regularization by Architecture: A Deep Prior Approach for Inverse Problems

The present paper studies so-called deep image prior (DIP) techniques in the context of ill-posed inverse problems. DIP networks have been recently introduced for applications in image processing; also first experimental results for applying DIP to inverse problems have been reported. This paper aims at discussing different interpretations of DIP and to obtain analytic results for specific network designs and linear operators. The main contribution is to introduce the idea of viewing these approaches as the optimization of Tikhonov functionals rather than optimizing networks. Besides theoretical results, we present numerical verifications.

Inverse reconstruction of energy spectra of clinical electron beams using the generalized simulated annealing method

In radiotherapy, the electron spectrum should be considered for accurate dose calculation. There are mainly three approaches to obtain the central axis electron energy spectrum, namely: (i) radiation source modeling; (ii) direct measurement and (iii) inverse reconstruction. To determine the electron spectra, it is proposed to use the inverse reconstruction approach. Inverse reconstruction consists in deriving the electron spectra from measured depth dose distributions. Since inverse reconstruction is a poorly conditioned problem the Tikhonov regularization is implemented to avoid the nonphysical oscillations inherent to this problem. Generalized simulated annealing method is used for minimizing the Tikhonov function in order to find the electron spectra. The effectiveness of the spectral reconstruction is verified by the gamma index criterion. Gamma Index defines the degree of agreement between two depth-dose distributions based on predefined tolerances in terms of distance-to-agreement (DTA) and difference dose (DD). Results show a good agreement between provided (measured in a water phantom) and reproduced (simulated from a reconstructed spectrum) depth dose curves within the gamma index tolerance level defined by >95% passing rate with 1% difference dose/1 mm distance to agreement. In addition, it is possible to evidence an acceptable agreement between provided and reproduced dose profiles according to >95% with 2%/2 mm for every beam, excepting the lowest energy beam. Therefore, it is concluded that generalized simulated annealing is able to reconstruct central-axis electron spectra incident on water phantom surface within restrictive limits of gamma index clinical acceptance.

A Survey on Surrogate Approaches to Non-negative Matrix Factorization

Motivated by applications in hyperspectral imaging, we investigate methods for approximating a high-dimensional non-negative matrix Y by a product of two lower-dimensional, non-negative matrices K and X. This so-called non-negative matrix factorization is based on defining suitable Tikhonov functionals, which combine a discrepancy measure for Y ≈KX with penalty terms for enforcing additional properties of K and X. The minimization is based on alternating minimization with respect to K and X, where in each iteration step one replaces the original Tikhonov functional by a locally defined surrogate functional. The choice of surrogate functionals is crucial: It should allow a comparatively simple minimization and simultaneously its first-order optimality condition should lead to multiplicative update rules, which automatically preserve non-negativity of the iterates. We review the most standard construction principles for surrogate functionals for Frobenius-norm and Kullback–Leibler discrepancy measures. We extend the known surrogate constructions by a general framework, which allows to add a large variety of penalty terms. The paper finishes by deriving the corresponding alternating minimization schemes explicitly and by applying these methods to MALDI imaging data.

Inverting Prediction Models in Micro Production for Process Design

Databased prediction models are used to estimate a possible outcome for previously unknown production parameters. These forward models enable to test new production designs and parameters virtually before applying them in the real world. Cause-effect networks are one way to generate such a prediction model. Multiple inputs and stages are being connected to one large prediction model. The functional behaviour and correlation of inputs as well as outputs is obtained through data based learning. In general, these models are non-linear and not invertible, especially for micro cold forming processes. While already being useful in process design, such models have their highest impact if inverted to find process parameters for a given output. Combining methods from the mathematical field of inverse problems as well as machine learning, a generalized inverse can be approximated. This allows finding process parameters for a given output without inverting the model directly but still using inherit information of the forward model. In this work, Tikhonov functionals are used to perform a parameter identification. The classical approach is altered by changing the discrepancy term to incorporate tolerances. Thereby, small deviations of a certain pattern are being neglected and the parameter finding process is being stabilized. In addition, different types of regularization are taken into consideration. Besides theoretical aspects of this method, examples are provided to demonstrate advantages and boundaries of an application for the process design in micro cold forming processes.

Identifying Elastic and Viscoelastic Material Parameters by Means of a Tikhonov Regularization

For studying the interaction of displacements, stresses, and acting forces for elastic and viscoelastic materials, it is of utmost importance to have a decent mathematical model available. Usually such a model consists of a coupled set of nonlinear differential equations together with appropriate boundary conditions. However, since the different material classes vary significantly with respect to their physical and mechanical behavior, the parameters which appear in these equations are unknown and therefore have to be determined before the equations can be used for further investigations or simulations. It is this very step which is addressed in this article where we consider elastic as well as viscoelastic material behavior. The idea is to compute the parameters as solutions of a minimization problem for Tikhonov functionals. Tikhonov regularization is a well-established solution technique for tackling inverse problems. On the one hand, it assures a computation that is stable with respect to noisy input data, and on the other hand, it involves desired a priori information on the solution. In this article we develop problem adapted Tikhonov functionals and prove that a Tikhonov regularization improves the accuracy especially when the underlying system is ill-conditioned.

Tikhonov Functionals Incorporating Tolerances

AbstractTikhonov functionals are a well known method for solving inverse problems. They consist of a discrepancy and a penalty term. The first term evaluates the deviation of simulated data from measured data. We alternate this term by incorporating tolerances, which neglects small deviations from the data within a prescribed tolerance. This approach adapts ideas from support vector regression, which utilizes such a tolerance for identity operators and semi discrete problems. Furthermore, the application for inverse problems is motivated by applications where such tolerances naturally occur, e.g. application with multiple measurements. In this case instead of one measurement a confidence interval for the measurement can be used.In this work we provide an overview on the necessary analysis and alternation of Tikhonov functionals incorporating tolerances. In addition, an example of applications are shown and discussed. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A global minimization algorithm for Tikhonov functionals with convex () penalty terms in Banach spaces

We extend the globally convergent TIGRA method in Ramlau (2003 Inverse Prob. 19 433–65) for the computation of a minimizer of the Tikhonov-type functional with the convex () penalty terms Θ for nonlinear forward operators in Banach spaces. The Θ are allowed to be non-smooth to include or TV (total variation) functionals, which are significant in reconstructing special features of solutions such as sparsity and discontinuities. The proposed TIGRA-Θ method uses a dual gradient descent method in the inner iteration and linearly decreases the regularization parameter in the outer iteration. We present the global convergence analysis for the algorithm under suitable parameter selections, and the convergence rate results are provided under both a priori and a posteriori stopping rules. Two numerical examples—an auto-convolution problem and a parameter identification problem—are presented to illustrate the theoretic analysis and verify the effectiveness of the method.

Solutions of Tikhonov Functional Equations and Applications to Multiplication Operators on Szegö Spaces

We consider a natural representation of solutions for Tikhonov functional equations. This will be done by applying the theory of reproducing kernels to the approximate solutions of general bounded linear operator equations (when defined from reproducing kernel Hilbert spaces into general Hilbert spaces), by using the Hilbert–Schmidt property and tensor product of Hilbert spaces. As a concrete case, we shall consider generalized fractional functions formed by the quotient of Bergman functions by Szego functions considered from the multiplication operators on the Szego spaces.

Carleman estimates for the regularization of ill-posed Cauchy problems

This work is a survey of results for ill-posed Cauchy problems for PDEs of the author with co-authors starting from 1991. A universal method of the regularization of these problems is presented here. Even though the idea of this method was previously discussed for specific problems, a universal approach of this paper was not discussed, at least in detail. This approach consists in constructing of such Tikhonov functionals which are generated by unbounded linear operators of those PDEs. The approach is quite general one, since it is applicable to all PDE operators for which Carleman estimates are valid. Three main types of operators of the second order are among them: elliptic, parabolic and hyperbolic ones. The key idea is that convergence rates of minimizers are established using Carleman estimates. Generalizations to nonlinear inverse problems, such as problems of reconstructions of obstacles and coefficient inverse problems are also feasible.

Global convergence of damped semismooth Newton methods for ℓ1 Tikhonov regularization

We are concerned with Tikhonov regularization of linear ill-posed problems with ℓ1 coefficient penalties. Griesse and Lorenz (2008 Inverse Problems 24 035007) proposed a semismooth Newton method for the efficient minimization of the corresponding Tikhonov functionals. In the class of high-precision solvers for such problems, semismooth Newton methods are particularly competitive due to their superlinear convergence properties and their ability to solve piecewise affine equations exactly within finitely many iterations. However, the convergence of semismooth Newton schemes is only local in general. In this work, we discuss the efficient globalization of B(ouligand)-semismooth Newton methods for ℓ1 Tikhonov regularization by means of damping strategies and suitable descent with respect to an associated merit functional. Numerical examples are provided which show that our method compares well with existing iterative, globally convergent approaches.

A global minimization algorithm for Tikhonov functionals with sparsity constraints

In this paper, we present a globally convergent algorithm for the computation of a minimizer of the Tikhonov functional with sparsity promoting penalty term for nonlinear forward operators in Banach space. The dual TIGRA method uses a gradient descent iteration in the dual space at decreasing values of the regularization parameter , where the approximation obtained with serves as the starting value for the dual iteration with parameter . With the discrepancy principle as a global stopping rule, the method further yields an automatic parameter choice. We prove convergence of the algorithm under suitable step-size selection and stopping rules and illustrate our theoretic results with numerical experiments for the nonlinear autoconvolution problem.