Source: http://www.wias-berlin.de/research/rts/inverse/?lang=1
Timestamp: 2019-04-22 12:23:29+00:00

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In many applications the quantities of interest can be observed only indirectly, or they must be derived from other measurements. Often the measurements are noisy and the reconstruction of the quantities of interest from noisy measurements is unstable. In this case the problem is an ill-posed (inverse) problem. Such problems are met in tomography and other image generating technologies, but the range of potential applications is much wider. A specific area of application is model calibration, i.e. when a model depends on additional parameters which need to be fitted to given data.
The reconstruction of the quantities of interest from such noisy and indirect obervations gives rise to statistical inverse problems, if the nature of the noise is statistical. Then we meet (nonparametric) statistical estimation problems which may be solved in a traditional sense by providing an estimator for the quantities of interest, but which may also be solved, when performing a Bayes analysis, which then provides us with additional information and which thus allows for the quantification of the uncertainty inerent in the reconstruction.
The studies in this topical area range from fundamental mathematical questions to application specific investigations. They involve expertise in regularization theory, non-parametric statistics, and within the specific application.
In recent years important contributions were made to the analysis of linear and nonlinear statistical inverse problems. One objective was the design of efficient reconstruction methods and the understanding of a proper parameter choice (model selection) for the used regularization scheme. Bayesian methods were also studied, both theoretically and in practical applications (MCMC methods). One important application was within the industrial project BOP.
P. Deuflhard, M. Grötschel, D. Hömberg, U. Horst, J. Kramer, V. Mehrmann, K. Polthier, F. Schmidt, Ch. Schütte, M. Skutella, J. Sprekels, eds., MATHEON -- Mathematics for Key Technologies, 1 of EMS Series in Industrial and Applied Mathematics, European Mathematical Society Publishing House, Zurich, 2014, 453 pages, (Collection Published).
M. Eigel, M. Marschall, R. Schneider, Sampling-free Bayesian inversion with adaptive hierarchical tensor representations, Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data, 34 (2018), pp. 035010/1--035010/29, DOI 10.1088/1361-6420/aaa998 .
The statistical Bayesian approach is a natural setting to resolve the ill-posedness of inverse problems by assigning probability densities to the considered calibration parameters. Based on a parametric deterministic representation of the forward model, a sampling-free approach to Bayesian inversion with an explicit representation of the parameter densities is developed. The approximation of the involved randomness inevitably leads to several high dimensional expressions, which are often tackled with classical sampling methods such as MCMC. To speed up these methods, the use of a surrogate model is beneficial since it allows for faster evaluation with respect to calibration parameters. However, the inherently slow convergence can not be remedied by this. As an alternative, a complete functional treatment of the inverse problem is feasible as demonstrated in this work, with functional representations of the parametric forward solution as well as the probability densities of the calibration parameters, determined by Bayesian inversion. The proposed sampling-free approach is discussed in the context of hierarchical tensor representations, which are employed for the adaptive evaluation of a random PDE (the forward problem) in generalized chaos polynomials and the subsequent high-dimensional quadrature of the log-likelihood. This modern compression technique alleviates the curse of dimensionality by hierarchical subspace approximations of the involved low rank (solution) manifolds. All required computations can be carried out efficiently in the low-rank format. A priori convergence is examined, considering all approximations that occur in the method. Numerical experiments demonstrate the performance and verify the theoretical results.
A. Anikin, A. Gasnikov, P. Dvurechensky, A. Turin, A. Chernov, Dual approaches to the minimization of strongly convex functionals with a simple structure under affine constraints, Computational Mathematics and Mathematical Physics, 57 (2017), pp. 1262--1276.
D. Belomestny, H. Mai, J.G.M. Schoenmakers, Generalized Post--Widder inversion formula with application to statistics, Journal of Mathematical Analysis and Applications, 455 (2017), pp. 89--104.
In this work we derive an inversion formula for the Laplace transform of a density observed on a curve in the complex domain, which generalizes the well known Post-Widder formula. We establish convergence of our inversion method and derive the corresponding convergence rates for the case of a Laplace transform of a smooth density. As an application we consider the problem of statistical inference for variance-mean mixture models. We construct a nonparametric estimator for the mixing density based on the generalized Post-Widder formula, derive bounds for its root mean square error and give a brief numerical example.
S. Bürger, P. Mathé, Discretized Lavrent'ev regularization for the autoconvolution equation, Applicable Analysis. An International Journal, 96 (2017), pp. 1618--1637, DOI 10.1080/00036811.2016.1212336 .
Lavrent?ev regularization for the autoconvolution equation was considered by Janno J. in Lavrent?ev regularization of ill-posed problems containing nonlinear near-to-monotone operators with application to autoconvolution equation, Inverse Prob. 2000;16:333?348. Here this study is extended by considering discretization of the Lavrent?ev scheme by splines. It is shown how to maintain the known convergence rate by an appropriate choice of spline spaces and a proper choice of the discretization level. For piece-wise constant splines the discretized equation allows for an explicit solver, in contrast to using higher order splines. This is used to design a fast implementation by means of post-smoothing, which provides results, which are indistinguishable from results obtained by direct discretization using cubic splines.
M. Hintermüller, C.N. Rautenberg, T. Wu, A. Langer, Optimal selection of the regularization function in a generalized total variation model. Part II: Algorithm, its analysis and numerical tests, Journal of Mathematical Imaging and Vision, 59 (2017), pp. 515--533.
Based on the generalized total variation model and its analysis pursued in part I (WIAS Preprint no. 2235), in this paper a continuous, i.e., infinite dimensional, projected gradient algorithm and its convergence analysis are presented. The method computes a stationary point of a regularized bilevel optimization problem for simultaneously recovering the image as well as determining a spatially distributed regularization weight. Further, its numerical realization is discussed and results obtained for image denoising and deblurring as well as Fourier and wavelet inpainting are reported on.
M. Hintermüller, C.N. Rautenberg, Optimal selection of the regularization function in a weighted total variation model. Part I: Modeling and theory, Journal of Mathematical Imaging and Vision, 59 (2017), pp. 498--514.
P. Mathé, S.V. Pereverzev, Complexity of linear ill-posed problems in Hilbert space, Journal of Complexity, 38 (2017), pp. 50--67.
D. Belomestny, J.G.M. Schoenmakers, Statistical inference for time-changed Lévy processes via Mellin transform approach, Stochastic Processes and their Applications, 126 (2016), pp. 2092--2122.
K. Lin, S. Lu, P. Mathé, Oracle-type posterior contraction rates in Bayesian inverse problems, Inverse Problems and Imaging, 9 (2015), pp. 895--915.
P. Mathé, Adaptive discretization for signal detection in statistical inverse problems, Applicable Analysis. An International Journal, 94 (2015), pp. 494--505.
S. Anzengruber, B. Hofmann, P. Mathé, Regularization properties of the sequential discrepancy principle for Tikhonov regularization in Banach spaces, Applicable Analysis. An International Journal, 93 (2014), pp. 1382--1400.
S. Lu, P. Mathé, Discrepancy based model selection in statistical inverse problems, Journal of Complexity, 30 (2014), pp. 290--308.
C. Marteau, P. Mathé, General regularization schemes for signal detection in inverse problems, Mathematical Methods of Statistics, 23 (2014), pp. 176--200.
R.I. Boţ, B. Hofmann, P. Mathé, Regularizability of ill-posed problems and the modulus of continuity, Analysis and Applications, 32 (2013), pp. 299--312.
E. Burnaev, A. Zaytsev, V. Spokoiny, Non-asymptotic properties for Gaussian field regression, Automation and Remote Control, 74 (2013), pp. 1645--1655.
Q. Jin, P. Mathé, Oracle inequality for a statistical Raus--Gfrerer-type rule, SIAM ASA J. Uncertainty Quantification, 1 (2013), pp. 386--407.
A. Zaitsev, E. Burnaev, V. Spokoiny, Properties of the posterior distribution of a regression model based on Gaussian random fields, Automation and Remote Control, 74 (2013), pp. 1645--1655.
B. Hofmann, P. Mathé, Some note on the modulus of continuity for ill-posed problems in Hilbert space, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 18 (2012), pp. 34--41.
G. Blanchard, P. Mathé, Discrepancy principle for statistical inverse problems with application to conjugate gradient iteration, Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data, 28 (2012), pp. 115011/1--115011/23.
B. Hofmann, P. Mathé, Parameter choice in Banach space regularization under variational inequalities, Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data, 28 (2012), pp. 104006/1--104006/17.
S.M.A. Becker, Regularization of statistical inverse problems and the Bakushinskii veto, Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data, 27 (2011), pp. 115010/1--115010/22.
In the deterministic context Bakushinskii's theorem excludes the existence of purely data-driven convergent regularization for ill-posed problems. We will prove in this work that in the statistical setting we can either construct a counter example or develop an equivalent formulation depending on the considered class of probability distributions. Hence, Bakushinskii's theorem does not generalize to the statistical context, although this has often been assumed in the past. To arrive at this conclusion, we will deduce from the classic theory new concepts for a general study of statistical inverse problems and perform a systematic clarification of the key ideas of statistical regularization.
F. Bauer, P. Mathé, Parameter choice methods using minimization schemes, Journal of Complexity, 27 (2011), pp. 68--85.
In this paper we establish a generalized framework, which allows to prove convergenence and optimality of parameter choice schemes for inverse problems based on minimization in a generic way. We show that the well known quasi-optimality criterion falls in this class. Furthermore we present a new parameter choice method and prove its convergence by using this newly established tool.
J. Flemming, B. Hofmann, P. Mathé, Sharp converse results for the regularization error using distance functions, Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data, 27 (2011), pp. 025006/1--025006/18.
In the analysis of ill-posed inverse problems the impact of solution smoothness on accuracy and convergence rates plays an important role. For linear ill-posed operator equations in Hilbert spaces and with focus on the linear regularization schema we will establish relations between the different kinds of measuring solution smoothness in a point-wise or integral manner. In particular we discuss the interplay of distribution functions, profile functions that express the regularization error, index functions generating source conditions, and distance functions associated with benchmark source conditions. We show that typically the distance functions and the profile functions carry the same information as the distribution functions, and that this is not the case for general source conditions. The theoretical findings are accompanied with examples exhibiting applications and limitations of the approach.
P. Mathé, U. Tautenhahn, Enhancing linear regularization to treat large noise, Journal of Inverse and Ill-Posed Problems, 19 (2011), pp. 859--879.
P. Mathé, U. Tautenhahn, Regularization under general noise assumptions, Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data, 27 (2011), pp. 035016/1--035016/15.
The authors explain how the major results which were obtained recently in Eggermont et al (2009 Inverse Problems 25 115018) can be derived from a more general perspective of recent regularization theory. By pursuing this further, the authors provide a general view on regularization under general noise assumptions, including weakly and strongly controlled noise. The prospect is not to generalize previous work in this direction, but rather to envision the intrinsic structure present in regularization under general noise assumptions. In particular, the authors find variants of the discrepancy and the Lepski$vrm i$ principle to choose the regularization parameter, albeit within different context and under different assumptions.
B. Hoffmann, P. Mathé, H. VON Weizsäcker, Regularization in Hilbert space under unbounded operators and general source conditions, Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data, 25 (2009), pp. 115013/1--115013/15.
D. Belomestny, G.N. Milstein, V. Spokoiny, Regression methods in pricing American and Bermudan options using consumption processes, Quantitative Finance, 9 (2009), pp. 315--327.
Here we develop methods for efficient pricing multidimensional discrete-time American and Bermudan options by using regression based algorithms together with a new approach towards constructing upper bounds for the price of the option. Applying sample space with payoffs at the optimal stopping times, we propose sequential estimates for continuation values, values of the consumption process, and stopping times on the sample paths. The approach admits constructing both low and upper bounds for the price by Monte Carlo simulations. The methods are illustrated by pricing Bermudan swaptions and snowballs in the Libor market model.
P. Mathé, S.V. Pereverzev, The use of higher order finite difference schemes is not dangerous, Journal of Complexity, 25 (2009), pp. 3--10.
V. Spokoiny, C. Vial, Parameter tuning in pointwise adaptation using a propagation approach, The Annals of Statistics, 37 (2009), pp. 2783--2807.
This paper discusses the problem of adaptive estimating a univariate object like the value of a regression function at a given point or a linear functional in a linear inverse problem. We consider an adaptive procedure originated from Lepski (1990) which selects in a data-driven way one estimate out of a given class of estimates ordered by their variability. A serious problem with using this and similar procedures is the choice of some tuning parameters like thresholds. Numerical results show that the theoretically recommended proposals appear to be too conservative and lead to a strong oversmoothing effects. A careful choice of the parameters of the procedure is extremely important for getting the reasonable quality of estimation. The main contribution of this paper is the new approach for choosing the parameters of the procedure by providing the prescribed behavior of the resulting estimate in the simple parametric situation. We establish a non-asymptotical “oracle” bound which shows that the estimation risk is, up to a logarithmic multiplier, equal to the risk of the “oracle” estimate which is optimally selected from the given family. A numerical study demonstrates the nice performance of the resulting procedure in a number of simulated examples.
B. Hofmann, P. Mathé, M. Schieck, Modulus of continuity for conditionally stable ill-posed problems in Hilbert space, Journal of Inverse and Ill-Posed Problems, 16 (2008), pp. 567-585.
P. Mathé, B. Hofmann, Direct and inverse results in variable Hilbert scales, Journal of Approximation Theory, 154 (2008), pp. 77--89.
P. Mathé, B. Hofmann, How general are general source conditions?, Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data, 24 (2008), pp. 015009/1--015009/5.
P. Mathé, N. Schöne, Regularization by projection in variable Hilbert scales, Applicable Analysis. An International Journal, 2 (2008), pp. 201-- 219.
B. Hofmann, P. Mathé, S.V. Pereverzev, Regularization by projection: Approximation theoretic aspects and distance functions, Journal of Inverse and Ill-Posed Problems, 15 (2007), pp. 527--545.
P. Mathé, U. Tautenhahn, Error bounds for regularization methods in Hilbert scales by using operator monotonicity, Far East Journal of Mathematical Sciences (FJMS), 24 (2007), pp. 1--21.
P. Mathé, B. Hofmann, Analysis of profile functions for general linear regularization methods, SIAM Journal on Numerical Analysis, 45 (2007), pp. 1122--1141.
The stable approximate solution of ill-posed linear operator equations in Hilbert spaces requires regularization. Tight bounds for the noise-free part of the regularization error are constitutive for bounding the overall error. Norm bounds of the noise-free part which decrease to zero along with the regularization parameter are called profile functions and are subject of our analysis. The interplay between properties of the regularization and certain smoothness properties of solution sets, which we shall describe in terms of source-wise representations is crucial for the decay of associated profile functions. On the one hand, we show that a given decay rate is possible only if the underlying true solution has appropriate smoothness. On the other hand, if smoothness fits the regularization, then decay rates are easily obtained. If smoothness does not fit, then we will measure this in terms of some distance function. Tight bounds for these allow us to obtain profile functions. Finally we study the most realistic case when smoothness is measured with respect to some operator which is related to the one governing the original equation only through a link condition. In many parts the analysis is done on geometric basis, extending classical concepts of linear regularization theory in Hilbert spaces. We emphasize intrinsic features of linear ill-posed problems which are frequently hidden in the classical analysis of such problems.
A. Goldenshluger, V. Spokoiny, Recovering convex edges of image from noisy tomographic data, Institute of Electrical and Electronics Engineers. Transactions on Information Theory, 52 (2006), pp. 1322--1334.
D. Belomestny, M. Reiss, Spectral calibration of exponential Lévy models, Finance and Stochastics, 10 (2006), pp. 449--474.
P. Mathé, U. Tautenhahn, Interpolation in variable Hilbert scales with application to inverse problems, Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data, 22 (2006), pp. 2271--2297.
For solving linear ill-posed problems with noisy data regularization methods are required. In the present paper regularized approximations in Hilbert scales are obtained by a general regularization scheme. The analysis of such schemes is based on new results for interpolation in Hilbert scales. Error bounds are obtained under general smoothness conditions.
J. Polzehl, V. Spokoiny, Propagation-separation approach for local likelihood estimation, Probability Theory and Related Fields, 135 (2006), pp. 335--362.
The paper presents a unified approach to local likelihood estimation for a broad class of nonparametric models, including, e.g., regression, density, Poisson and binary response models. The method extends the adaptive weights smoothing (AWS) procedure introduced by the authors [Adaptive weights smoothing with applications to image sequentation. J. R. Stat. Soc., Ser. B 62, 335-354 (2000)] in the context of image denoising. The main idea of the method is to describe a greatest possible local neighborhood of every design point in which the local parametric assumption is justified by the data. The method is especially powerful for model functions having large homogeneous regions and sharp discontinuities. The performance of the proposed procedure is illustrated by numerical examples for density estimation and classification. We also establish some remarkable theoretical non-asymptotic results on properties of the new algorithm. This includes the “propagation” property which particularly yields the root-$n$ consistency of the resulting estimate in the homogeneous case. We also state an “oracle” result which implies rate optimality of the estimate under usual smoothness conditions and a “separation” result which explains the sensitivity of the method to structural changes.
D. Belomestny, Reconstruction of the general distribution from the distribution of some statistics, Theory of Probability and its Applications, 49 (2005), pp. 1--15.
We investigate the problem of characterizing the distribution of independent identically distributed random variables $X_1,ldots,X_m$ (general distribution) by the distribution of linear statistics and statistics of maximum with positive coefficients. Necessary and sufficient conditions are found under which such a characterization takes place.
A. Goldenshluger, V. Spokoiny, On the shape-from-moments problem and recovering edges from noisy Radon data, Probability Theory and Related Fields, 128 (2004), pp. 123--140.
D. Belomestny, Constraints on distributions imposed by properties of linear forms, ESAIM. Probability and Statistics, 7 (2003), pp. 313-328.
Let $(X_1,Y_1)...,(X_m,Y_m)$ be $m$ independent identically distributed bivariate vectors and $L_1=b_1X_1+...+b_mX_m$ ,$L_2=b_1Y_1+...+b_mY_m$ are two linear forms with positive coefficients. We study two problems: under what conditions does the equidistribution of $L_1$ and $L_2$ imply the same property for $X_1$ and $Y_1$, and under what conditions does the independence of $L_1$ and $L_2$ entail independence of $X_1$ and $Y_1$? Some analytical sufficient conditions are obtained and it is shown that in general they can not be weakened.
P. Mathé, S.V. Pereverzev, Discretization strategy for linear ill-posed problems in variable Hilbert scales, Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data, 19 (2003), pp. 1263-1277.
The authors study the regularization of projection methods for solving linear ill-posed problems with compact and injective linear operators in Hilbert spaces. Smoothness of the unknown solution is given in terms of general source conditions, such that the framework of variable Hilbert scale s is suitable. The structure of the error is analyzed in terms of the noise level, the regularization parameter and as a function of other parameters, driving the discretization. As a result, a strategy is proposed, which automatically adapts to the unknown source condition, uniformly for certain classes, and provides the optimal order of accuracy.
P. Mathé, S.V. Pereverzev, Direct estimation of linear functionals from indirect noisy observations, Journal of Complexity, 18 (2002), pp. 500--516.
The authors study the efficiency of the linear functional strategy, as introduced by Anderssen (1986), for inverse problems with observations blurred by Gaussian white noise with known intensity $delta$. The optimal accuracy is presented and it is shown, how this can be achieved by a linear--functional strategy based on the noisy observations. This optimal linear--functional strategy is obtained from Tikhonov regularization of some dual problem. Next, the situation is treated, when only a finite number of noisy observations, given beforehand is available. Under appropriate smoothness assumptions best possible accuracy still can be attained, if the number of observations corresponds to the noise intensity in a proper way. It is also shown, that, at least asymptotically this number of observations cannot be reduced.
P. Mathé, Stable summation of orthogonal series with noisy coefficients, Journal of Approximation Theory, 117 (2002), pp. 66--80.
We study the recovery of continuous functions from Fourier coefficients with respect to certain given orthonormal systems, blurred by noise. For deterministic noise this is a classical ill--posed problem. Emphasis is laid on a priori smoothness assumptions on the solution, which allows to apply regularization to reach the best possible accuracy. Results are obtained for systems obeying norm growth conditions. In the white noise setting mild additional assumptions have to be made to have accurate bounds. We finish our study with the recovery of functions from noisy coefficients with respect to the Haar system.
P. Mathé, S.V. Pereverzev, Optimal discretization of inverse problems in Hilbert scales. Regularization and self-regularization of projection methods, SIAM Journal on Numerical Analysis, 38 (2001), pp. 1999--2021.
We study the efficiency of the approximate solution of ill--posed problems, based on discretized observations, which we assume to be given afore--hand. We restrict ourselves to problems which can be formulated in Hilbert scales. Within this framework we shall quantify the degree of ill--posedness, provide general conditions on projection schemes to achieve the best possible order of accuracy. We pay particular attention on the problem of self--regularization vs. Tikhonov regularization. Moreover, we study the information complexity. Asymptotically, any method, which achieves the best possible order of accuracy must use at least such amount of noisy observations. We accomplish our study with two specific problems, Abel's integral equation and the recovery of continuous functions from noisy coefficients with respect to a given orthonormal system, both classical ill--posed problems.
L. Bogulubsky, P. Dvurechensky, A. Gasnikov, G. Gusev, Y. Nesterov, A. Raigorodskii, A. Tikhonov, M. Zhukovskii, Learning supervised PageRank with gradient-based and gradient-free optimization methods, in: Advances in Neural Information Processing Systems 29, D.D. Lee, M. Sugiyama, U.V. Luxburg, I. Guyon, R. Garnett, eds., Curran Associates, Inc., 2016, pp. 4907--4915.
A. Chernov, P. Dvurechensky, A. Gasnikov, Fast primal-dual gradient method for strongly convex minimization problems with linear constraints, in: Discrete Optimization and Operations Research -- 9th International Conference, DOOR 2016, Vladivostok, Russia, September 19--23, 2016, Proceedings, Y. Kochetov, M. Khachay, V. Beresnev, E. Nurminski, P. Pardalos, eds., 9869 of Theoretical Computer Science and General Issues, Springer International Publishing Switzerland, Cham, 2016, pp. 391--403.
M. Arias Chao, D.S. Lilley, P. Mathé, V. Schlosshauer, Calibration and uncertainty quantification of gas turbines performance models, in: ASME Turbo Expo 2015: Turbine Technical Conference and Exposition, Volume 7A: Structures and Dynamics, ASME and Alstom Technologie AG, 2015, pp. V07AT29A001--V07AT29A012.
G. Blanchard, N. Krämer, Kernel partial least squares is universally consistent, in: Proceedings of the 13th International Conference on Artificial Intelligence and Statistics (AISTATS 2010), Y.W. Teh, M. Titterington, eds., 9 of JMLR Workshop and Conference Proceedings, Journal of Machine Learning Research, Cambridge, MA, USA, 2010, pp. 57--64.
H. Heitsch, N. Strogies, Consequences of uncertain friction for the transport of natural gas through passive networks of pipelines, Preprint no. 2513, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2513 .
Assuming a pipe-wise constant structure of the friction coefficient in the modeling of natural gas transport through a passive network of pipes via semilinear systems of balance laws with associated linear coupling and boundary conditions, uncertainty in this parameter is quantified by a Markov chain Monte Carlo method. Here, information on the prior distribution is obtained from practitioners. The results are applied to the problem of validating technical feasibility under random exit demand in gas transport networks. In particular, the impact of quantified uncertainty to the probability level of technical feasible exit demand situations is studied by two example networks of small and medium size. The gas transport of the network is modeled by stationary solutions that are steady states of the time dependent semilinear problems.
M. Eigel, Adaptive Galerkin FEM for stochastic forward and inverse problems, Optimisation and Numerical Analysis Seminars, University of Birmingham, School of Mathematics, UK, February 15, 2018.
M. Eigel, Adaptive tensor methods for forward and inverse problems, SIAM Conference on Uncertainty Quantification (UQ18), Minisymposia 96, 109, 122 ``Low-Rank Approximations for the Forward- and the Inverse Problems I--III'', April 16 - 19, 2018, Garden Grove, USA, April 19, 2018.
M. Hintermüller, Automated regularization parameter choice rule in image processing, Workshop ``New Directions in Stochastic Optimisation'', August 19 - 25, 2018, Mathematisches Forschungsinstitut Oberwolfach, August 23, 2018.
M. Hintermüller, Bilevel optimisation in automated regularisation parameter selection in image processing, WIAS--PGMO Workshop on Nonsmooth and Stochastic Optimization, June 26, 2018, Humboldt-Universität zu Berlin, June 26, 2018.
M. Hintermüller, Bilevel optimization and some "parameter learning" applications in image processing, SIAM Conference on Imaging Science, Minisymposium MS5 ``Learning and Adaptive Approaches in Image Processing'', June 5 - 8, 2018, Bologna, Italy, June 5, 2018.
M. Marschall, Bayesian inversion with adaptive low-rank approximation, Workshop of the French-German-Italian LIA (Laboratoire International Associe) COPDESC on Applied Analysis ``Analysis, Control and Inverse Problems for PDEs'', November 26 - 30, 2018, University of Naples Federico II and Accademia Pontaniana, Italy, November 29, 2018.
P. Mathé, Bayesian inverse problems with non-commuting operators, Chemnitz September of Applied Mathematics 2018, Chemnitz Symposium on Inverse Problems, September 27 - 28, 2018, Technische Universität Chemnitz, Fakultät für Mathematik, September 28, 2018.
P. Mathé, Complexity of linear ill-posed problems in Hilbert space, Stochastisches Kolloquium, Georg-August-Universität Göttingen, Institut für Mathematische Stochastik, February 7, 2018.
M. Eigel, Adaptive stochastic FE for explicit Bayesian inversion with hierarchical tensor representations, Institut National de Recherche en Informatique et en Automatique (INRIA), SERENA (Simulation for the Environment: Reliable and Efficient Numerical Algorithms) research team, Paris, France, June 1, 2017.
R. Gruhlke, Multi-scale failure analysis with polymorphic uncertainties for optimal design of rotor blades, Frontiers of Uncertainty Quantification in Engineering (FrontUQ 2017), September 6 - 8, 2017, München, September 6, 2017.
M. Hintermüller, Bilevel optimization and applications in imaging, Workshop ``Emerging Developments in Interfaces and Free Boundaries'', January 22 - 28, 2017, Mathematisches Forschungsinstitut Oberwolfach.
M. Hintermüller, Bilevel optimization and applications in imaging, Mathematisches Kolloquium, Universität Wien, Austria, January 18, 2017.
M. Hintermüller, Bilevel optimization and some ``parameter learning'' applications in image processing, LMS Workshop ``Variational Methods Meet Machine Learning'', September 18, 2017, University of Cambridge, Centre for Mathematical Sciences, UK, September 18, 2017.
A. Koziuk, Bootstrap for the regression problem with instrumental variables, Haindorf Seminar 2017, January 24 - 28, 2017, Humboldt-Universität zu Berlin, Wirtschaftswissenschaftliche Fakultät, Hejnice, Czech Republic, January 26, 2017.
M. Marschall, Bayesian inversion using hierarchical tensors, 88th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2017), Section S15 ``Uncertainty Quantification'', March 6 - 10, 2017, Bauhaus Universität Weimar/Technische Universität Ilmenau, Weimar, March 8, 2017.
M. Marschall, Sampling-free Bayesian inversion with adaptive hierarchical tensor representation, Frontiers of Uncertainty Quantification in Engineering (FrontUQ 2017), September 6 - 8, 2017, München, September 7, 2017.
M. Marschall, Sampling-free Bayesian inversion with adaptive hierarchical tensor representation, International Conference on Scientific Computation and Differential Equations (SciCADE2017), MS21 ``Tensor Approximations of Multi-Dimensional PDEs'', September 11 - 15, 2017, University of Bath, UK, September 14, 2017.
P. Mathé, Bayesian inverse problems with non-commuting operators, Statistical Foundations of Uncertainty Quantification for Inverse Problems Workshop, June 19 - 22, 2017, University of Cambridge, Center for Mathematical Sciences, UK, June 21, 2017.
P. Mathé, Complexity of supervised learning, ibc-paris2017 : Information Based Complexity, High-Dimensional Problems, March 14 - 15, 2017, Institut Henri Poincaré, Paris, France, March 15, 2017.
P. Mathé, Numerical integration (mini course), November 20 - December 4, 2017, Fudan University, School of Mathematical Sciences, China.
N. Buzun, Multiplier bootstrap for change point detection, Mathematical Statistics and Inverse Problems, February 8 - 12, 2016, Faculté des Sciences de Luminy, France, February 11, 2016.
N. Buzun, Multiplier bootstrap for change point detection, Spring School ``Structural Inference 2016", March 13 - 18, 2016, DFG Forschergruppe FOR 1735, Lübeck, Germany, March 14, 2016.
M. Eigel, Bayesian inversion using hierarchical tensor approximations, SIAM Conference on Uncertainty Quantification, Minisymposium 67 ``Bayesian Inversion and Low-rank Approximation (Part II)'', April 5 - 8, 2016, Lausanne, Switzerland, April 6, 2016.
T. Wu, Bilevel optimization and applications in imaging sciences, August 24 - 25, 2016, Shanghai Jiao Tong University, Institute of Natural Sciences, China.
P. Dvurechensky, Gradient and gradient-free methods for pagerank algorithm learning, Workshop on Modern Statistics and Optimization, February 23 - 24, 2016, Russian Academy of Sciences, Institute for Information Transmission Problems, Moscow, Russian Federation, February 24, 2016.
P. Dvurechensky, Random gradient-free methods for web-page ranking model learning, 30th annual conference of the Belgian Operational Research Society, January 27 - 29, 2016, Louvain-la-Neuve, Belgium, January 28, 2016.
M. Hintermüller, K. Papafitsoros, C. Rautenberg, A fine scale analysis of spatially adapted total variation regularisation, Imaging, Vision and Learning based on Optimization and PDEs, Bergen, Norway, August 29 - September 1, 2016.
M. Hintermüller, Bilevel optimization and applications in imaging, Imaging, Vision and Learning based on Optimization and PDEs, August 29 - September 1, 2016, Bergen, Norway, August 30, 2016.
M. Hintermüller, Bilevel optimization for a generalized total-variation model, SIAM Conference on Imaging Science, Minisymposium ``Non-Convex Regularization Methods in Image Restoration'', May 23 - 26, 2016, Albuquerque, USA, May 26, 2016.
M. Hintermüller, Optimal selection of the regularisation function in a localised TV model, SIAM Conference on Imaging Science, Minisymposium ``Analysis and Parameterisation of Derivative Based Regularisation'', May 23 - 26, 2016, Albuquerque, USA, May 24, 2016.
P. Mathé, Complexity of linear ill-posed problems in Hilbert space, IBC on the 70th anniversary of Henryk Wozniakowski, August 29 - September 2, 2016, Banach Center, Bedlewo, Poland, August 31, 2016.
P. Mathé, Complexity of linear ill-posed problems in Hilbert space, Chemnitz Symposium on Inverse Problems, September 22 - 23, 2016, Technische Universität Chemnitz, Fakultät für Mathematik, September 22, 2016.
P. Mathé, Discrepancy based model selection in statistical inverse problems, Mathematical Statistics and Inverse Problems, February 8 - 12, 2016, Centre International de Rencontres Mathématiques (CIRM), Luminy, France, February 11, 2016.
V. Spokoiny, Gradient and gradient-free methods for pagerank algorithm learning, Workshop on Modern Statistics and Optimization, February 23 - 24, 2016, Russian Academy of Sciences, Institute for Information Transmission Problems, Moscow, Russian Federation.
P. Dvurechensky, Semi-supervised pagerank model learning with gradient-free optimization methods, Traditional Youth School ``Control, Information and Optimization'', June 14 - 20, 2015, Moscow, Russian Federation, June 17, 2015.
P. Mathé, Minimax signal detection in statistical inverse problems, Algorithms and Complexity for Continuous Problems, September 21 - 25, 2015, Schloss Dagstuhl, September 25, 2015.
P. Mathé, A random surfer in the internet, German-Estonian Academic Week ``Academica'', October 13 - 15, 2014, University of Tartu, Faculty of Mathematics and Computer Science, Estonia, October 14, 2014.
P. Mathé, Bayesian analysis of statistical inverse problems, Colloquium of the Faculty of Mathematics and Computer Science, University of Tartu, Estonia, October 13, 2014.
P. Mathé, Bayesian regularization of statistical inverse problems, Rencontres de Statistique Mathématique: Nouvelles Procédures pour Nouvelles Données, December 15 - 19, 2014, Centre International de Rencontres Mathématiques (CIRM), Marseille, France, December 17, 2014.
P. Mathé, Convergence of Bayesian schemes in inverse problems, Kolloquium über Angewandte Mathematik, Georg-August-Universität Göttingen, Fakultät für Mathematik und Informatik, June 10, 2014.
P. Mathé, Merging regularization theory into Bayesian inverse problems, Chemnitz Symposium on Inverse Problems, September 15 - 19, 2014, Universität Chemnitz, Fachbereich Mathematik, September 18, 2014.
P. Mathé, Oracle-type posterior contraction rates in Bayesian inverse problems, International Workshop ``Advances in Optimization and Statistics'', May 15 - 16, 2014, Russian Academy of Sciences, Institute of Information Transmission Problems (Kharkevich Institute), Moscow, May 16, 2014.
P. Mathé, Parameter estimation by thresholding, Mathematical Modelling and Simulation Workshop, WIAS Berlin, April 15, 2014.
P. Mathé, Smoothness beyond differentiability, Seminar for Doctoral Candidates, University of Tartu, Faculty of Mathematics and Computer Science, Estonia, October 15, 2014.
A. Andresen, Finite sample analysis of maximum likelihood estimators and convergence of the alternating procedure, 29th European Meeting of Statisticians (EMS), July 20 - 25, 2013, Eötvös Loránd University, Budapest, Hungary, July 20, 2013.
H. Mai, Estimating a subordinators density, DynStoch 2013, April 17 - 19, 2013, University of Copenhagen, Department of Mathematical Sciences, Denmark, April 17, 2013.
P. Mathé, Bayes analysis for model calibration, Alstom Workshop, WIAS, May 7, 2013.
P. Mathé, Signal detection in inverse problems, PreMoLab Workshop on: Advances in predictive modeling and optimization, May 16 - 17, 2013, WIAS-Berlin, May 16, 2013.
P. Mathé, Signal detection in inverse problems, Mathematical Modelling and Analysis ( MMA2013 ) and Approximation Methods and Orthogonal Expansions ( AMOE2013 ), May 27 - 31, 2013, University of Tartu, Institute of Mathematics, Estonia, May 29, 2013.
P. Mathé, Statistical Inverse Problems, Applied Math Seminar, University of Warwick, Mathematics Institute, Coventry, UK, October 18, 2013.
S. Becker, Image processing via orientation scores, Workshop ``Computational Inverse Problems'', October 23 - 26, 2012, Mathematisches Forschungsinstitut Oberwolfach, October 25, 2012.
A. Andresen, Non asymptotic Wilks phenomenon in semiparametric estimation, PreMoLab: Moscow-Berlin Stochastic and Predictive Modeling, May 31 - June 1, 2012, Russian Academy of Sciences, Institute for Information Transmission Problems (Kharkevich Institute), Moscow, May 31, 2012.
A. Andresen, Nonasymptotic Wilks phenomenon in semiparametric estimation, 2. Structural Inference Day, WIAS, April 23, 2012.
A. Andresen, Nonasymptotic Wilks phenomenon in semiparametric estimation, Haindorf Seminar 2012 (Klausurtagung des SFB 649), February 9 - 12, 2012, Humboldt-Universität zu Berlin, Wirtschaftswissenschaftliche Fakultät, Hejnice, Czech Republic, February 10, 2012.
P. Mathé, An oracle-type bound for a statistical RG-rule, Nonparametric and High-dimensional Statistics, December 17 - 21, 2012, Centre International de Rencontres Mathématiques (CIRM), Marseille, France, December 20, 2012.
P. Mathé, Diskrepanz-basierte Parameterwahl in statistischen inversen Problemen, Technische Universität Chemnitz, Fakultät für Mathematik, September 19, 2012.
P. Mathé, Regularization of statistical inverse problems in Hilbert space, Journées Statistiques du Sud 2012, June 20 - 22, 2012, Université Toulouse, Institut National des Sciences Appliquées, France, June 20, 2012.
P. Mathé, Statistische Datenanalyse unter BOP, Workshop on Simulation in Industrial Process Engineering, WIAS, September 6, 2012.
P. Mathé, Using the discrepancy principle in statistical inverse problems, Regularisation symposium, Australian National University, Mathematical Sciences Institute, Canberra, November 22, 2012.
V. Spokoiny, Basics of modern parametric statistics, February 13 - 28, 2012, Independent University of Moscow, Center for Continuous Mathematical Education, Russian Federation.
V. Spokoiny, Bernstein--von Mises theorem for quasi posteriors, International Workshop on Recent Advances in Time Series Analysis (RATS 2012), June 8 - 12, 2012, University of Cyprus, Department of Mathematics and Statistics, Protaras, June 9, 2012.
V. Spokoiny, Bernstein--von Mises theorem for quasi posteriors, Workshop ``Frontiers in Nonparametric Statistics'', March 11 - 17, 2012, Mathematisches Forschungsinstitut Oberwolfach, March 12, 2012.
V. Spokoiny, Bernstein--von Mises theorem for quasi posteriors, Workshop II on Financial Time Series Analysis: High-dimensionality, Non-stationarity and the Financial Crisis, June 19 - 22, 2012, National University of Singapore, Institute for Mathematical Sciences, June 21, 2012.
V. Spokoiny, Bernstein--von Mises theorem for quasi posteriors, Workshop on Recent Developments in Statistical Multiscale Methods, July 16 - 18, 2012, Georg-August-Universität Göttingen, Institut für Mathematische Stochastik, July 17, 2012.
V. Spokoiny, Bernstein--von Mises theorem for quasi posteriors, PreMoLab Seminar, Russian Academy of Sciences, Institute for Information Transmission Problems (Kharkevich Institute), Moscow, March 15, 2012.
V. Spokoiny, Parametric estimation: Modern view, PreMoDay I, February 24, 2012, Russian Academy of Sciences, Institute for Information Transmission Problems (Kharkevich Institute), Moscow, February 24, 2012.
V. Spokoiny, Some methods of modern statistics, Information Technology and Systems 2, August 19 - 25, 2012, Russian Academy of Sciences, Institute for Information Transmission Problems (Kharkevich Institute), Petrozavodsk, August 20, 2012.
P. Mathé, Conjugate gradient iteration with noisy data, Foundations of Computational Mathematics (FoCM'11), July 4 - 14, 2011, Budapest, Hungary, July 6, 2011.
V. Spokoiny, Alternating and semiparametric efficiency, École Nationale de la Statistique et de l'Analyse de l'Information (ENSAI), Rennes, France, September 16, 2011.
V. Spokoiny, Modern parametric theory, September 13 - 16, 2011, École Nationale de la Statistique et de l'Analyse de l'Information (ENSAI), Rennes, France.
P. Mathé, Conjugate gradient iteration under white noise, International Conference on Scientific Computing (SC2011), October 10 - 14, 2011, Università di Cagliari, Dipartimento di Matematica e Informatica, Cagliari, Italy, October 14, 2011.
V. Spokoiny, Semiparametric estimation alternating and efficiency, École Nationale de la Statistique et de l'Administration Économique (ENSAE), Paris, France, December 12, 2011.
M. Becker, Self-intersection local times: Exponential moments in subcritical dimensions, Excess Self-Intersections and Related Topics, December 6 - 10, 2010, Centre international de rencontres mathématiques (CIRM), Luminy, France, December 6, 2010.
S. Becker, Regularization of statistical inverse problems and the Bakushinskii veto, Rencontres de Statistiques Mathématiques 10, December 13 - 17, 2010, Centre International de Rencontres Mathématiques (CIRM), Luminy, France, December 17, 2010.
N. Krämer, Conjugate gradient regularization --- A statistical framework for partial least squares regression, 4th Workshop on Partial Least Squares and Related Methods for Cutting-edge Research in Experimental Sciences, May 10 - 11, 2010, École Supérieure d'Électricité, Department of Information Systems and Decision Sciences, Gif-sur-Yvette, France, May 11, 2010.
D. Belomestny, Estimating the distribution of jumps in regular affine models: Uniform rates of convergence, Leipziger Stochastik-Tage, March 1 - 5, 2010, Universität Leipzig, Fakultät für Mathematik und Informatik, March 2, 2010.
D. Belomestny, Statistical inference for multidimensional timechanged Levy processes based on low--frequency data, 28th European Meeting of Statisticians, August 19 - 23, 2010, University of Piraeus, Department of Statistics and Insurance Science, Greece, August 23, 2010.
G. Blanchard, N. Krämer, Kernel partial least squares is universally consistent, AI & Statistics 2010, Sardinia, Italy, May 13 - 15, 2010.
G. Blanchard, N. Krämer, Optimal rates for conjugate gradient regularization, AI & Statistics 2010, Sardinia, Italy, May 13 - 15, 2010.
P. Mathé, Analysis of inverse problems under general smoothness assumptions, 5th International Conference on Inverse Problems: Modeling and Simulation, May 24 - 29, 2010, Izmir University, Department of Mathematics and Computer Sciences, Antalya, Turkey, May 25, 2010.
P. Mathé, Regularization under general noise assumptions, Chemnitz Symposium on Inverse Problems 2010, September 23 - 24, 2010, Chemnitz University of Technology, Department of Mathematics, September 23, 2010.
P. Mathé, Warum und wie rechnen wir mit allgemeinen Quelldarstellungen?, Seminar Inverse Probleme (Fr. C. Böckmann), Universität Potsdam, Institut für Mathematik, January 12, 2010.
D. Belomestny, Estimation of the jump activity of a Lévy process from low frequency data, Haindorf Seminar 2009, February 12 - 15, 2009, Humboldt-Universität zu Berlin, CASE -- Center for Applied Statistics and Economics, Hejnice, Czech Republic, February 12, 2009.
D. Belomestny, Spectral estimation of the fractional order of a Lévy process, Workshop ``Statistical Inference for Lévy Processes with Applications to Finance'', July 15 - 17, 2009, EURANDOM, Eindhoven, Netherlands, July 16, 2009.
G. Blanchard, Convergence du gradient conjugué fonctionnel pour l'apprentissage statistique, École Normale Supérieure, Paris, France, March 16, 2009.
P. Mathé, Approximation theoretic aspects in variable Hilbert scales, 2nd Dolomites Workshop on Constructive Approximation and Applications, September 3 - 10, 2009, Università degli Studi di Verona, Dipartimento di Informatica, Italy, September 5, 2009.
P. Mathé, Discretization under general smoothness assumptions, Monday Lecture Series, Johann Radon Institute for Computational and Applied Mathematics (RICAM), Linz, Austria, June 29, 2009.
P. Mathé, Inverse problems in different settings, Seminar ``Algorithms and Complexity for Continuous Problems'', September 20 - 25, 2009, Schloss Dagstuhl, September 25, 2009.
P. Mathé, On some minimization-based heuristic parameter choice in inverse problems, Chemnitz-RICAM Symposium on Inverse Problems, Johann Radon Institute for Computational and Applied Mathematics (RICAM), Linz, Austria, July 14, 2009.
P. Mathé, Typical behavior in heuristic parameter choice, Kick-Off Meeting of Mini Special Semester on Inverse Problems, May 18 - 22, 2009, Johann Radon Institute for Computational and Applied Mathematics (RICAM), Linz, Austria, May 18, 2009.
V. Spokoiny, Saddle point model selection for inverse problems, Workshop ``Challenges in Statistical Theory: Complex Data Structures and Algorithmic Optimization'', August 23 - 29, 2009, Mathematisches Forschungsinstitut Oberwolfach, August 26, 2009.
P. Mathé, Average case analysis of inverse problems, Symposium on Inverse Problems 2008, September 25 - 26, 2008, TU Chemnitz, Fachbereich Mathematik, September 25, 2008.
A. Kroshnin, D. Dvinskikh, P. Dvurechensky, N. Tupitsa, C. Uribe, On the complexity of approximating Wasserstein barycenter, Preprint no. . arXiv:1901.08686, Cornell University Library, arXiv.org, 2019.
C. A. Uribe, D. Dvinskikh, P. Dvurechensky, A. Gasnikov, A. Nedić, Distributed computation of Wasserstein barycenters over networks, Preprint no. arXiv:1803.02933, Cornell University Library, arXiv.org, 2018.
P. Dvurechensky, D. Dvinskikh, A. Gasnikov, C.A. Uribe, A. Nedić, Decentralize and randomize: Faster algorithm for Wasserstein barycenters, Preprint no. arXiv:1806.03915, Cornell University Library, arXiv.org, 2018.
We study the problem of decentralized distributed computation of a discrete approximation for regularized Wasserstein barycenter of a finite set of continuous probability measures distributedly stored over a network. Particularly, we assume that there is a network of agents/machines/computers where each agent holds a private continuous probability measure, and seeks to compute the barycenter of all the measures in the network by getting samples from its local measure and exchanging information with its neighbors. Motivated by this problem, we develop and theoretically analyze a novel accelerated primal-dual stochastic gradient method for general stochastic convex optimization problems with linear equality constraints. Then, we apply this method to the decentralized distributed optimization setting to propose a new algorithm for the distributed semi-discrete regularized Wasserstein barycenter problem. The proposed algorithm can be executed over arbitrary networks that are undirected, connected and static, using the local information only. Moreover, we show explicit non-asymptotic complexity in terms of the problem parameters. Finally, we show the effectiveness of our method on the distributed computation of the regularized Wasserstein barycenter of univariate Gaussian and von Mises distributions, as well as on some applications to image aggregation.
P. Mathé, Bayesian inverse problems with non-commuting operators, Preprint no. arXiv:1801.09540, Cornell University Library, arXiv.org, 2018.
The Bayesian approach to ill-posed operator equations in Hilbert space recently gained attraction. In this context, and when the prior distribution is Gaussian, then two operators play a significant role, the one which governs the operator equation, and the one which describes the prior covariance. Typically it is assumed that these operators commute. Here we extend this analysis to non-commuting operators, replacing the commutativity assumption by a link condition. We discuss its relation to the commuting case, and we indicate that this allows to use interpolation type results to obtain tight bounds for the contraction of the posterior Gaussian distribution towards the data generating element.
M. Hintermüller, N. Strogies, On the identification of the friction coefficient in a semilinear system for gas transport through a network, Preprint, DFG SFB Transregio 154 ``Mathematical Modelling, Simulation and Optimization using the Example of Gas Networks'', 2017.
L. Bogolubsky, P. Dvurechensky, A. Gasnikov, G. Gusev, Y. Nesterov, A. Raigorodskii, A. Tikhonov, M. Zhukovskii, Learning supervised PageRank with gradient-based and gradient-free optimization methods, Preprint no. arXiv:1603.00717, Cornell University Library, arXiv.org, 2016.
A. Chernov, P. Dvurechensky, A. Gasnikov, Fast primal-dual gradient method for strongly convex minimization problems with linear constraints, Preprint no. arXiv:1605.02970, Cornell University Library, arXiv.org, 2016.
S.W. Anzengruber, B. Hofmann, P. Mathé, Regularization properties of the discrepancy principle for Tikhonov regularization in Banach spaces, Preprint no. 12, Technische Universität Chemnitz, Fakultät für Mathematik, 2012.
B. Hofmann, P. Mathé, Parameter choice in Banach space regularization under variational inequalities, Preprint no. 05, Technische Universität Chemnitz, Fakultät für Mathematik, 2012.
V. Spokoiny, Parametric estimation. Finite sample theory, Preprint no. arXiv:1111.3029, Cornell University Library, arXiv.org, 2012.
V. Spokoiny, Roughness penalty, Wilks phenomenon, and Bernstein--von Mises theorem, Preprint no. arXiv:1205.0498, Cornell University Library, arXiv.org, 2012.
R.I. Boţ, B. Hofmann, P. Mathé, Regularizability of ill-posed problems and the modulus of continuity, Preprint no. 17, Technische Universität Chemnitz, Fakultät für Mathematik, 2011.
B. Hofmann, P. Mathé, Some note on the modulus of continuity for ill-posed problems in Hilbert space, Preprint no. 07, Technische Universität Chemnitz, Fakultät für Mathematik, 2011.
G. Blanchard, P. Mathé, Discrepancy principle for statistical inverse problems with application to conjugate gradient iteration, Preprint no. 07, Universität Potsdam, Institut für Mathematik, 2011.

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