Simultaneous estimation of mass and aerodynamic rotor imbalances for wind turbines
 Jenny Niebsch^{1}Email author and
 Ronny Ramlau^{1, 2}
https://doi.org/10.1186/21905983412
© Niebsch and Ramlau; licensee Springer. 2014
Received: 10 January 2014
Accepted: 12 July 2014
Published: 9 September 2014
Abstract
The safe operation of wind turbines requires a wellbalanced rotor. The balancing of the rotor requires a method to determine its imbalances. We propose an algorithm for the reconstruction of two types of imbalances, i.e., mass and aerodynamic imbalances from pitch angle deviation. The algorithm is based on the inversion of the (nonlinear) operator equation that links the imbalance distribution of the rotor to its vibrations during operation of the wind turbine. The algorithm requires a simple finite element model of the wind turbine as well as the minimization of a Tikhonov functional with a nonlinear operator. We propose the use of a gradientbased minimization routine. The approach is validated for artificial vibration data from a model of a Nordwind NTK 500 wind turbine.
Keywords
1 Introduction
In the growing field of wind energy extraction, the topic of rotor imbalances of wind turbines (WT) is of vital importance for the operation, safety, and durability of the turbines. Approximately 20%50% of WT have significant rotor imbalances. This leads to damages of important components, high repair expenses, and reduced output [1]. Rotor imbalances can be categorized into mass and aerodynamic imbalances. Mass imbalances arise from inhomogeneous mass distributions of the rotor including blades and hub. They induce centrifugal forces that depend on the square of the rotational frequency ${\mathrm{\Omega}}_{\mathrm{rot}}$. The consequences are harmonic vibrations with ${\mathrm{\Omega}}_{\mathrm{rot}}$ in the rotor plane (lateral). The vibration amplitude depends on the ratio between ${\mathrm{\Omega}}_{\mathrm{rot}}$ and the resonance frequency (first eigenfrequency) ${\mathrm{\Omega}}_{0}$ of the turbine. Mass imbalances can be eliminated or reduced using balancing counterweights. Aerodynamic imbalances evolve from deviations in the aerodynamic properties of the blades. These lead to different thrust and tangential forces and moments, and they depend not only on the rotational frequency but also on the wind speed. Aside from lateral vibrations, they induce axial and torsional vibrations of WT. The main reason for aerodynamic imbalance is a relative deviation of the pitch angles of the blades. This can only be eliminated by a correction of those angles. Additional masses are not helpful here [1]. In practice, both types of imbalances are frequently observed both separately and in combination.
Usually, the detection of imbalances is based on spectral analysis and order analysis methods [2]. Another technique is to monitor the power characteristic [3], whereby power mean values are observed, and deviations from faultless conditions are used for the calculation of alarm limits. Unfortunately, this requires a learning phase under faultless conditions. This also holds for other signal processing methods, cf. [2]. Another disadvantage is the fact that the amount of the imbalance, e.g. absolute value and location of a mass imbalance or the deviation of one or more pitch angles, is not computable.
The state of the art in rotor imbalance determination still requires an expert team onsite. Initially, aerodynamic imbalances must be detected using optical methods. After the elimination of the aerodynamic imbalances, the mass imbalance is determined using vibration measurements with and without reference weights.
In the last few years, the authors have developed methods which allow for imbalance determination using data that can be collected by a condition monitoring system (CMS). As a consequence, the amount and location of rotor imbalance can be computed without the expensive measurements using reference weights onsite. The methods are modelbased, but they use a very simple model of the WT that can be obtained easily for each turbine type from elementary geometrical and physical parameters. The general basics of our approach are summarized in Section 2. Essentially, the method consists of the following steps: 1. modeling of the WT; 2. description of the loads (forces, moments) arising from rotor imbalance; 3. solution of the forward problem, i.e. the computation of the resulting vibrations in every node of the WT model for a given imbalance situation and 4. solution of the inverse problem where the unknown imbalance is determined from vibration data obtained in the nacelle. In our new method we began by determining the mass imbalance [4]. Later aerodynamic imbalance from pitch angle deviations was included [5].
In case of pure mass imbalances, only lateral vibrations need to be considered. Accordingly, in the model of the WT only degrees of freedom (DOF) in the lateral direction needed to be taken into account. Let us call this restricted approach the mass imbalance approach (MIA). The presence of aerodynamic imbalances in addition to mass imbalances would lead to additional lateral forces and therefore to incorrect reconstructions using MIA since the assumption in this model is that aerodynamic imbalances are not present. The model and the description of the forces were expanded in [5] in order to include the effect of pitch angle deviations as a main reason for aerodynamic imbalances. Nevertheless, MIA can serve to provide an initial guess for the solution of the problem for combined mass and aerodynamic imbalances. In the combined imbalance case, the inverse problem is nonlinear. As the problem is illposed, Tikhonov regularization is used. This includes the minimization of the Tikhonov functional which was done in [5] using a direct search method. Although we could obtain first test results, the routine was far to slow and not stable enough for extensive test computations. The acceleration of the minimization routines requires the computation of the Frechet derivative of the forward problem operator A. As most of the theory for Tikhonov regularization is formulated for real valued vector spaces, the forward operator A has to be formulated in a real setting.
This paper focuses on a stable reconstruction method of mass imbalance and pitch angle deviation based on the minimization of the Tikhonov functional. To this end, the nonlinear forward operator A is derived in detail and its Frechet derivative is computed which is new compared to [5]. Now the minimization of the Tikhonov functional is performed using the steepest decent method. The resulting reconstruction algorithm is stable and reasonable fast. Additionally, a suitable metric is defined to measure the reconstruction quality. We present test results obtained with the gradientbased algorithm for artificial data. The paper is organized as follows: In Section 3 we derive the forward problem operator A. We will be very brief if there are no changes to the preceding paper [5] and more detailed in the description of the new parts. In particular, the Frechet derivative of A is computed. Section 4 is concerned with the inverse problem. Regularization and minimization techniques using gradientbased methods are presented as well as the final algorithm that reconstructs the imbalances from measurement data. In Section 5 we present numerical test computations for the reconstruction of pitch angle deviations only and a combination of mass imbalances and pitch angle deviations as well as results for the reconstruction quality in the presented metric.
2 General approach
The matrices M and S need to be derived for each different type of WT separately. This is the first step in our procedure. The second step pertains to the description of the load $\mathbf{p}(\mathbf{x})$ as a function of the imbalances x. In the third step, the now fully described ODE system (1) is solved. The solution operator $\tilde{\mathbf{A}}$ maps the imbalances x to the displacement u at each node of the FE model: $\tilde{\mathbf{A}}\mathbf{x}=\mathbf{u}$. Now we restrict the displacement vector u that contains the displacement of each model DOF to the vector y of the DOF that are accessible for measurements. The operator $\tilde{\mathbf{A}}$ is accordingly restricted to A. The resulting operator equation is $\mathbf{A}\mathbf{x}=\mathbf{y}$ where the operator A maps the imbalance x of the rotor in terms of mass imbalance and pitch angle deviation to the displacement y in the nodes that can be observed by measurements. This operator equation represents the forward problem. Hence step 13 are connected with the solution of the forward problem. Now in a fourth step we solve the inverse problem, i.e., we determine the imbalance x from measurements y. Since the inverse problem is illposed, we use regularization techniques, [6]. This approach has been applied successfully in the more simple case of mass imbalances only [4]. First steps in the direction of reconstructing mass imbalances and pitch angle deviation simultaneously were done in [5].
3 Forward problem
3.1 Mass and stiffness matrices
The derivation of M and S for the simple model of MIA was taken from [7] and presented in [4]. In the situation of mass and aerodynamic imbalances the model needs to be expanded to displacements in axial direction and torsion around the tower central axis. Still, the derivation of M and S follows exactly the Ritz method described in [[7], p.155160]. Once the model matrices are derived for a certain WT type they need to be optimized for each individual turbine of this type with respect to the first eigenfrequency ${\mathrm{\Omega}}_{0}$. The general optimization procedure is described in [4].
Thus far we have modeled six different types of WT in this way: Vestas V80, V82, V90, Südwind S77, NTK500/41 and GE1.5. For the test computations presented in this paper, we have used the model of a Nordtank turbine NTK500/41, see Figure 1 (bottom), with 35 m hub height because we are provided with aerodynamic data for the blades of this WT.
3.2 Forces and moments from imbalances – the load vector p
Here, the ${F}_{1}$, ${F}_{2}$, ${F}_{3}$ are the thrust forces at each blade. If there is no deviation in the pitch angle of the blades all three forces are equal. ${T}_{z}$ is the projection of the sum of the tangential forces at each blade $T={T}_{1}+{T}_{2}+{T}_{3}$ onto the zaxis. It is zero if there is no deviation in the pitch angles. ${F}_{c}$ denotes the centrifugal force induced by the mass imbalance. Its projection onto the zaxis ${({F}_{c})}_{z}$ adds to ${F}_{z}$. The moments ${M}_{x}^{(m)}$, ${M}_{z}^{(m)}$ are induced by ${F}_{c}$ of the mass imbalance, ${M}_{x}^{(F)}$, ${M}_{z}^{(F)}$, ${M}_{x}^{(T)}$, ${M}_{z}^{(T)}$ correspond to the thrust force and tangential force. Whereas the mass imbalance force can be directly computed, the aerodynamic imbalance forces need to be determined via, e.g., the BladeElementMomentum (BEM) method, a nonlinear procedure. In the BEM method, the blades are divided into a finite number of elements which are annulus segments with the center at the root of the blades. The cross section of each element is called an ‘airfoil’. If the airfoil data, the angle of attack of the wind, and the relative wind velocity as well as a lift and drag coefficient table are given, the normal or thrust forces F and the tangential forces T can be calculated according to the BEM method, see, e.g., [8]. The BEM method is used in a lot of programs for the simulation of aerodynamic effects, cf. [9], which were tested on real WT.
For the presentation of the forces and moments, we will use the following notations and abbreviations (cf. Figures 2 and 3):

mr, ${\varphi}_{m}$ – absolute value and angle w.r.t. blade A of the mass imbalance,

– deviation of the pitch angle of i th blade to an adjusted pitch angle θ,${\theta}_{i}$

– angular frequency (note: WT rotates clockwise seen from the wind),$\omega =2\pi \mathrm{\Omega}$

– angle between blades,$\phi =2/3\pi $

ϕ – initial location of blade A w.r.t. a zero mark (equals 0 if the mark can be chosen at blade A),

L – distance from hub to nacelle midpoint (xaxis),

– thrust force at blade i with pitch angle deviation ${\theta}_{i}$, $i=1,2,3$,${F}_{i}:=F({x}_{i})$

– tangential force at blade i with pitch angle deviation ${\theta}_{i}$, $i=1,2,3$,${T}_{i}:=T({x}_{i})$

– corresponding length (distance of resulting force vector from nacelle) for pitch angle ${\theta}_{i}$, $i=1,2,3$.${l}_{i}:=l({x}_{i})$
3.2.1 Forces and moments from mass imbalance
3.2.2 Forces and moments from pitch angle deviation
3.2.3 Combination and rewriting of forces and moments – the final load vector
where ${\mathbf{p}}_{1}^{T}:=(0,\dots ,0,{F}_{y},0,0,0,0)$, and the ${\omega}_{i}$ are the roots of the eigenvalues of ${\mathbf{M}}^{1}\mathbf{S}$. For WT the only eigenfrequency close to the operating frequency ω is ${\omega}_{0}$. For safety reasons, the operation of the WT with $\omega ={\omega}_{0}$ is avoided. As only vibrations with rotating frequency ω will be extracted from the measurement, the influence of ${F}_{y}$ can be neglected. Therefore we can omit ${F}_{y}$ by setting it to zero.
3.3 Derivation of the forward operator A
3.3.1 The imbalance vector x and the data vector y
 1.
the vibration ${c}_{y}(\mathbf{x})cos(\omega t{\gamma}_{y}(\mathbf{x}))$ at DOF $(N4)$ (displacement in ydirection),
 2.
the vibration ${c}_{z}(\mathbf{x})cos(\omega t{\gamma}_{z}(\mathbf{x}))$ at DOF $(N3)$ (displacement in zdirection),
 3.
the vibration ${c}_{{\beta}_{x}}(\mathbf{x})cos(\omega t{\gamma}_{{\beta}_{x}}(\mathbf{x}))$ at DOF $(N2)$ (torsion around the xaxis).
for all three sensors.
3.3.2 Solution of the ODE system
3.4 Derivative
can be computed.
4 Regularization
The inverse problem of (14) is to find the imbalance vector x for given data y that, in case of real measurements, are contaminated by noise. We assume that the noisy data ${\mathbf{y}}^{\delta}$ fulfill the inequality $\parallel \mathbf{y}{\mathbf{y}}^{\delta}\parallel \le \delta $, i.e., a bound on the noise is known. The operator A is nonlinear and illposed. To obtain a stable solution, one has to use regularization methods, [6, 10]. Theoretical results for the regularization of nonlinear problems were derived in the last decade, cf. [11–16]. We have used the most prominent Tikhonov regularization.
4.1 Tikhonov functional
where $\mathbf{c}.\ast \mathbf{x}={({c}_{1}{x}_{1},\dots ,{c}_{5}{x}_{5})}^{T}$ denotes the componentbycomponent multiplication. We remark that ${\u3008\mathbf{x},\mathbf{z}\u3009}_{c}=\u3008\mathbf{c}.\ast \mathbf{x},\mathbf{z}\u3009$.
4.2 The gradient of Tikhonov functional
4.3 Minimization with steepest decent and algorithm
5 Numerical tests
5.1 Settings
5.1.1 Model data
We have tested the imbalance reconstruction algorithm for a wind turbine of the type NTK 500 with constant rotational frequency, since the aerodynamic data for this WT can be found in the literature. Namely, we use the following system parameters:

Rotational speed ${\mathrm{\Omega}}_{\mathrm{rot}}=0.45\text{Hz}=27\text{rpm}$.

Aerodynamic data: NACA63421 (lift and drag coefficients: http://airfoiltools.com/airfoil/details?airfoil=naca634421il), NKT50041 (profile data: http://www.readbag.com/13022617201extrawebdocsnordtankwtdescription), blade length = 20.5 m, wind speed = 8 m/s.

Model data:

Masses: ${m}_{r}=9\text{,}030\text{kg}$ (rotor), ${m}_{n}=15\text{,}400\text{kg}$ (nacelle), ${m}_{t}=22\text{,}500\text{kg}$ (tower).

Geometry: Tower height: $h=33.8\text{m}$, outer diameter at tower top: ${d}_{\mathrm{out}}^{\mathrm{top}}=1.69\text{m}$, outer diameter at tower bottom: ${d}_{\mathrm{out}}^{\mathrm{bot}}=2.4\text{m}$, distance from rotor to nacelle midpoint $L=1.5\text{m}$.

Eigenfrequency: ${\mathrm{\Omega}}_{0}=0.925\text{Hz}$.
We have set the zero mark at blade A. Thus blade B correspond to an angle of 240^{∘} and blade C corresponds to 120^{∘}, resp. For example, assuming a pitch angle deviation of blade B of 2^{∘} compared to blade A and C and a mass imbalance of 50 kgm at blade A, the corresponding imbalance vector is $\mathbf{x}={(0,2,0,50,0)}^{T}$; if the imbalance is at blade C, we have $\mathbf{x}={(0,2,0,50cos(2/3\pi ),50sin(2/3\pi ))}^{T}$. We obtain radial, axial and torsion vibration data y by applying the forward operator to x. To simulate real measurements, the data y is contaminated with Gaussian noise. The disturbed data ${\mathbf{y}}^{\delta}$ as well as the noise $\delta =\parallel \mathbf{y}{\mathbf{y}}^{\delta}\parallel $ are used as input data for the reconstruction process.
5.1.2 Reconstruction quality
In practice the original pitch angle deviation is not known. Here we chose the number s in a way that at least one of the ${x}_{i}$, $i=1,2,3$, is zero. The residual vibrations are the same for all representatives of the class.
provided the denominator is not zero.
5.2 Aerodynamic imbalance caused by pitch angle deviation
The first test was concerned with the reconstruction of pitch angle deviation imbalances in the absence of mass imbalances. Here we present two examples.
Reconstruction results for imbalance $\mathbf{x}\mathbf{=}{\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{)}}^{\mathit{T}}$ .
${\mathit{\delta}}_{\mathbf{rel}}$  s  ${\mathbf{x}}_{\mathit{\alpha}}^{\mathit{\delta}}\mathbf{.}\mathbf{+}\mathit{s}$  ${\mathit{\epsilon}}_{\mathbf{pitch}}^{\mathbf{rel}}\mathbf{(}{\mathbf{x}}_{\mathit{\alpha}}^{\mathit{\delta}}\mathbf{,}\phantom{\rule{0.2em}{0ex}}\mathbf{x}\mathbf{)}$  ${\mathit{\epsilon}}_{\mathbf{mass}}\mathbf{(}{\mathbf{x}}_{\mathit{\alpha}}^{\mathit{\delta}}\mathbf{,}\phantom{\rule{0.2em}{0ex}}\mathbf{x}\mathbf{)}$ 

3%  0.67  ${(0.03,1.93,0.05,0.04,0.06)}^{T}$  5.9%  0.07 
5%  0.67  ${(0.06,1.97,0.04,0.03,0.05)}^{T}$  4.6%  0.07 
8%  0.67  ${(0.10,1.94,0.04,0.04,0.05)}^{T}$  8.5%  0.06 
10%  0.67  ${(0.06,1.89,0.16,0.04,0.05)}^{T}$  13.2%  0.06 
12%  0.67  ${(0.06,1.88,0.18,0.02,0.06)}^{T}$  15%  0.07 
15%  0.67  ${(0.25,1.82,0.07,0.02,0.06)}^{T}$  21.5%  0.07 
17%  0.67  ${(0.09,1.83,0.25,0.03,0.06)}^{T}$  21.7%  0.07 
Example 2 The reconstruction result for an imbalance $\mathbf{x}={(1,0,0.5,0,0)}^{T}$ for a reconstruction with noisy data and noise level of ${\delta}_{\mathrm{rel}}=7\text{\%}$ is shown in Figure 5, bottom picture. Here, the parameter s is computed as $s=0.167$ and the relative pitch angle reconstruction error is ${\epsilon}_{\mathrm{pitch}}^{\mathrm{rel}}({\mathbf{x}}_{\alpha}^{\delta},\mathbf{x})=7\text{\%}$.
The initial approximation ${\mathbf{x}}_{0}$ for both examples was chosen as ${\mathbf{x}}_{0}={(0,0,0,0,0)}^{T}$.
5.3 Combined mass and aerodynamic imbalance
Reconstruction results for imbalance $\mathbf{x}\mathbf{=}{\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{50}\mathbf{,}\mathbf{0}\mathbf{)}}^{\mathit{T}}$ .
Start vector  t  ${\mathbf{x}}_{\mathit{\alpha}}^{\mathit{\delta}}\mathbf{.}\mathbf{+}\mathit{s}$  ${\mathit{\epsilon}}_{\mathbf{pitch}}^{\mathbf{rel}}\mathbf{(}{\mathbf{x}}_{\mathit{\alpha}}^{\mathit{\delta}}\mathbf{,}\phantom{\rule{0.2em}{0ex}}\mathbf{x}\mathbf{)}$  ${\mathit{\epsilon}}_{\mathbf{mass}}^{\mathbf{rel}}\mathbf{(}{\mathbf{x}}_{\mathit{\alpha}}^{\mathit{\delta}}\mathbf{,}\phantom{\rule{0.2em}{0ex}}\mathbf{x}\mathbf{)}$ 

${(0,0,0,0,0)}^{T}$  403 s  ${(0.06,2.02,0.08,44.7,2.88)}^{T}$  6%  13% 
${(0,0,0,51.5,4.8)}^{T}$  120 s  ${(0.14,1.93,0.07,50.1,0.67)}^{T}$  11%  1.4% 
As mentioned in the Introduction, the problem of simultaneous reconstruction of mass imbalances and pitch angle deviation was already treated in [5]. In this first attempt, a direct search method for minimizing the nonlinear Tikhonov functional was used, which was far too slow for extensive test computations. Also convergence was not always be achieved. With our gradientbased reconstruction method the computation time was reduced significantly to a more reasonable time of about 2 minutes on 2.8 GHz Intel Core2 Duo processor. Convergence could also be observed in all tests.
6 Summary and outlook
The paper deals with the mathematical determination of two types of imbalances in a WT: aerodynamic imbalances from pitch angle deviation of the blades and mass imbalances of the rotor. Presently, both types have to be determined onsite by an expert team performing time consuming measurement procedures. The mathematical approach provides a reconstruction method that can be implemented into a condition monitoring system. Instead of the expert team, the method requires the knowledge of geometrical and physical parameters of the WT and aerodynamic airfoil data of the blades. Whereas the former usually can be obtained from inspection reports, the latter represent very sensible data and are hardly available for newer WT. For that reason, so far the method could only be tested for an older NTK500 turbine. Unfortunately, our industrial partners could not provide the necessary vibration data for this type. Thus we had to restrict all tests to artificial data. For future applicability, the WT owners and manufacturers need to be persuaded that tit in beneficial for the safe operation of the turbine to provide the sensible airfoil data in a confined way.
The problem of imbalance determination was addressed by the authors in former papers. In this paper we have presented an improved algorithm to reconstruct mass imbalance and pitch angle deviation of the rotor of a WT at the same time. Including pitch angle deviations into the reconstruction leads to a nonlinear forward operator. The regularization of the inverse problem requires the minimization of the Tikhonov functional. Straight forward direct search method proved to be to slow and not stable enough for our purpose. Hence we employed gradient based minimization methods. To that end, we have derived the Frechet derivative of the forward operator that occurs in the gradient of the Tikhonov functional. The accelerated algorithm was successfully applied to several test examples with artificial vibration data. In case of real vibration data we expect equally good reconstruction results as in the test computations, provided that the necessary aerodynamic data of the blades are available.
Declarations
Acknowledgements
The authors work supported by the Austrian Research Promotion Agency (FFG).
Authors’ Affiliations
References
 Donth A, Grunwald A, Heilmann C, Melsheimer M: Improving performance of wind turbines through blade angle optimization and rotor balancing. EWEA O/M strategies 350; 2011.Google Scholar
 Caselitz P, Giebhardt J: Rotor condition monitoring for improved operational safety of offshore wind energy converters. ASME J Sol Energy Eng 2005, 127: 253–261. 10.1115/1.1850485View ArticleGoogle Scholar
 Hameed Z, Hong YS, Cho YM, Ahn SH, Song CK: Condition monitoring and fault detection of wind turbines and related algorithms: a review. Renew Sustain Energy Rev 2009, 13: 1–39. 10.1016/j.rser.2007.05.008View ArticleGoogle Scholar
 Ramlau R, Niebsch J: Imbalance estimation without test masses for wind turbines. ASME J Sol Energy Eng 2009.,131(1): Article ID 011010 Article ID 011010Google Scholar
 Niebsch J, Ramlau R, Nguyen TT: Mass and aerodynamic imbalance estimates of wind turbines. Energies 2010, 3: 696–710. 10.3390/en3040696View ArticleGoogle Scholar
 Engl HW, Hanke M, Neubauer A: Regularization of Inverse Problems. Kluwer Academic, Dordrecht; 1996.MATHView ArticleGoogle Scholar
 Gasch R, Knothe K: Strukturdynamik 2. Springer, Berlin; 1989.MATHView ArticleGoogle Scholar
 Ingram G: Wind turbine blade analysis using the blade element momentum method. Durham University, Durham; 2005.Google Scholar
 Ahlström A: Aeroelastic simulations of wind turbine dynamics. PhD thesis. Royal Institute of Technology, Department of Mechanics, Stockholm; 2005. Ahlström A: Aeroelastic simulations of wind turbine dynamics. PhD thesis. Royal Institute of Technology, Department of Mechanics, Stockholm; 2005.Google Scholar
 Louis AK: Inverse und Schlecht Gestellte Probleme. Teubner, Stuttgart; 1989.MATHView ArticleGoogle Scholar
 Kaltenbacher B, Neubauer A, Scherzer O: Iterative Regularization Methods for Nonlinear IllPosed Problems. de Gruyter, Berlin; 2008.MATHView ArticleGoogle Scholar
 Ramlau R, Teschke G: Tikhonov replacement functionals for iteratively solving nonlinear operator equations. Inverse Probl 2005,21(5):1571–1592. 10.1088/02665611/21/5/005MATHMathSciNetView ArticleGoogle Scholar
 Ramlau R: TIGRA – an iterative algorithm for regularizing nonlinear illposed problems. Inverse Probl 2003,19(2):433–4670. 10.1088/02665611/19/2/312MATHMathSciNetView ArticleGoogle Scholar
 Ramlau R: Morozov’s discrepancy principle for Tikhonov regularization of nonlinear operators. Numer Funct Anal Optim 2002,23(1,2):147–172.MATHMathSciNetView ArticleGoogle Scholar
 Scherzer O: The use of Morozov’s discrepancy principle for Tikhonov regularization for solving nonlinear illposed problems. Computing 1993, 51: 45–60. 10.1007/BF02243828MATHMathSciNetView ArticleGoogle Scholar
 Ramlau R: On the use of fixed point iterations for the regularization of nonlinear illposed problems. J Inverse IllPosed Probl 2005,13(2):175–200.MATHMathSciNetGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.