Articles in Refereed Journals

  • S. Lu, P. Mathé, S. Pereverzyev, Randomized matrix approximation to enhance regularized projection schemes in inverse problems, Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data, 36 (2020), pp. 085013/1-- 085013/20, DOI 10.1088/1361-6420/ab9c44 .
    The authors consider a randomized solution to ill-posed operator equations in Hilbert spaces. In contrast to statistical inverse problems, where randomness appears in the noise, here randomness arises in the low-rank matrix approximation of the forward operator, which results in using a Monte Carlo method to solve the inverse problems. In particular, this approach follows the paradigm of the study N. Halko et al 2011 SIAM Rev. 53 217?288, and hence regularization is performed based on the low-rank matrix approximation. Error bounds for the mean error are obtained which take into account solution smoothness and the inherent noise level. Based on the structure of the error decomposition the authors propose a novel algorithm which guarantees (on the mean) a prescribed error tolerance. Numerical simulations confirm the theoretical findings.

  • A. Rastogi, G. Blanchard, P. Mathé, Convergence analysis of Tikhonov regularization for non-linear statistical inverse problems, Electronic Journal of Statistics, 14 (2020), pp. 2798--2841, DOI 10.1214/20-EJS1735 .
    We study a non-linear statistical inverse problem, where we observe the noisy image of a quantity through a non-linear operator at some random design points. We consider the widely used Tikhonov regularization (or method of regularization) approach to estimate the quantity for the non-linear ill-posed inverse problem. The estimator is defined as the minimizer of a Tikhonov functional, which is the sum of a data misfit term and a quadratic penalty term. We develop a theoretical analysis for the minimizer of the Tikhonov regularization scheme using the concept of reproducing kernel Hilbert spaces. We discuss optimal rates of convergence for the proposed scheme, uniformly over classes of admissible solutions, defined through appropriate source conditions.

  • P. Mathé, M.T. Nair, B. Hofmann, Regularization of linear ill-posed problems involving multiplication operators, Applicable Analysis. An International Journal, published online 28.04.2020, url, DOI 10.1080/00036811.2020.1758308 .
    We study regularization of ill-posed equations involving multiplication operators when the multiplier function is positive almost everywhere and zero is an accumulation point of the range of this function. Such equations naturally arise from equations based on non-compact self-adjoint operators in Hilbert space, after applying unitary transformations arising out of the spectral theorem. For classical regularization theory, when noisy observations are given and the noise is deterministic and bounded, then non-compactness of the ill-posed equations is a minor issue. However, for statistical ill-posed equations with non-compact operators less is known if the data are blurred by white noise. We develop a theory for spectral regularization with emphasis on this case. In this context, we highlight several aspects, in particular, we discuss the intrinsic degree of ill-posedness in terms of rearrangements of the multiplier function. Moreover, we address the required modifications of classical regularization schemes in order to be used for non-compact statistical problems, and we also introduce the concept of the effective ill-posedness of the operator equation under white noise. This study is concluded with prototypical examples for such equations, as these are deconvolution equations and certain final value problems in evolution equations.

  • B. Hofmann, S. Kindermann, P. Mathé, Penalty-based smoothness conditions in convex variational regularization, Journal of Inverse and Ill-Posed Problems, 27 (2019), pp. 283--300, DOI 10.1515/jiip-2018-0039 .
    The authors study Tikhonov regularization of linear ill-posed problems with a general convex penalty defined on a Banach space. It is well known that the error analysis requires smoothness assumptions. Here such assumptions are given in form of inequalities involving only the family of noise-free minimizers along the regularization parameter and the (unknown) penalty-minimizing solution. These inequalities control, respectively, the defect of the penalty, or likewise, the defect of the whole Tikhonov functional. The main results provide error bounds for a Bregman distance, which split into two summands: the first smoothness-dependent term does not depend on the noise level, whereas the second term includes the noise level. This resembles the situation of standard quadratic Tikhonov regularization Hilbert spaces. It is shown that variational inequalities, as these were studied recently, imply the validity of the assumptions made here. Several examples highlight the results in specific applications.

  • S. Lu , P. Mathé, S. Pereverzyev, Analysis of regularized Nyström subsampling for regression functions of low smoothness, Analysis and Applications, 17 (2019), pp. 931--946.

  • P. Mathé, Bayesian inverse problems with non-commuting operators, Mathematics of Computation, 88 (2019), pp. 2897--2912, DOI 10.1090/mcom/3439 .
    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.

  • B. Hofmann, P. Mathé, Tikhonov regularization with oversmoothing penalty for non-linear ill-posed problems in Hilbert scales, Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data, 34 (2018), pp. 015007/1--015007/14.

  • S. Lu, P. Mathé, S. Pereverzyev, Balancing principle in supervised learning for a general regularization scheme, Applied and Computational Harmonic Analysis. Time-Frequency and Time-Scale Analysis, Wavelets, Numerical Algorithms, and Applications, 48 (2020), pp. 123--148, DOI 10.1016/j.acha.2018.03.001 .

  • 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.

  • L.T. Ding, P. Mathé, Minimax rates for statistical inverse problems under general source conditions, Computational Methods in Applied Mathematics, pp. published online on 05.12.2017, url, DOI 10.1515/cmam-2017-0055 .
    We describe the minimax reconstruction rates in linear ill-posed equations in Hilbert space when smoothness is given in terms of general source sets. The underlying fundamental result, the minimax rate on ellipsoids, is proved similarly to the seminal study by D. L. Donoho, R. C. Liu, and B. MacGibbon, it Minimax risk over hyperrectangles, and implications, Ann.  Statist. 18, 1990. These authors highlighted the special role of the truncated series estimator, and for such estimators the risk can explicitly be given. We provide several examples, indicating results for statistical estimation in ill-posed problems in Hilbert space.

  • P. Mathé, S.V. Pereverzev, Complexity of linear ill-posed problems in Hilbert space, Journal of Complexity, 38 (2017), pp. 50--67.

  • 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.

  • S. Becker, P. Mathé, A different perspective on the Propagation-Separation approach, Electronic Journal of Statistics, 7 (2013), pp. 2702--2736.
    The Propagation-Separation approach is an iterative procedure for pointwise estimation of local constant and local polynomial functions. The estimator is defined as a weighted mean of the observations with data-driven weights. Within homogeneous regions it ensures a similar behavior as non-adaptive smoothing (propagation), while avoiding smoothing among distinct regions (separation). In order to enable a proof of stability of estimates, the authors of the original study introduced an additional memory step aggregating the estimators of the successive iteration steps. Here, we study theoretical properties of the simplified algorithm, where the memory step is omitted. In particular, we introduce a new strategy for the choice of the adaptation parameter yielding propagation and stability for local constant functions with sharp discontinuities.

  • A. Wilms, P. Mathé, F. Schulze, Th. Koprucki, A. Knorr, U. Bandelow, Influence of the carrier reservoir dimensionality on electron-electron scattering in quantum dot materials, Phys. Rev. B., 88 (2013), pp. 235421/1--235421/11.
    We calculated Coulomb scattering rates from quantum dots (QDs) coupled to a 2D carrier reservoir and QDs coupled to a 3D reservoir. For this purpose, we used a microscopic theory in the limit of Born-Markov approximation, in which the numerical evaluation of high dimensional integrals is done via a quasi-Monte Carlo method. Via a comparison of the so determined scattering rates, we investigated the question whether scattering from 2D is generally more efficient than scattering from 3D. In agreement with experimental findings, we did not observe a significant reduction of the scattering efficiency of a QD directly coupled to a 3D reservoir. In turn, we found that 3D scattering benefits from it?s additional degree of freedom in the momentum space.

  • 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.

  • Q. Jin, P. Mathé, Oracle inequality for a statistical Raus--Gfrerer-type rule, SIAM ASA J. Uncertainty Quantification, 1 (2013), pp. 386--407.

  • S. Lu, P. Mathé, Heuristic parameter selection based on functional minimization: Optimality and model function approach, Mathematics of Computation, 82 (2013), pp. 1609--1630.

  • A. Wilms, D. Breddermann, P. Mathé, Theory of direct capture from two- and three-dimensional reservoirs to quantum dot states, physica status solidi (c), 9 (2012), pp. 1278--1281.

  • 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.

  • 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.

  • G. Blanchard, P. Mathé, Conjugate gradient regularization under general smoothness and noise assumptions, Journal of Inverse and Ill-Posed Problems, 18 (2010), pp. 701--726.

  • 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, S. Mathew, J.G.M. Schoenmakers, Multiple stochastic volatility extension of the Libor market model and its implementation, Monte Carlo Methods and Applications, 15 (2009), pp. 285-310.
    In this paper we propose a Libor model with a high-dimensional specially structured system of driving CIR volatility processes. A stable calibration procedure which takes into account a given local correlation structure is presented. The calibration algorithm is FFT based, so fast and easy to implement.

  • P. Mathé, S.V. Pereverzev, The use of higher order finite difference schemes is not dangerous, Journal of Complexity, 25 (2009), pp. 3--10.
    PDF (125 kByte)

  • 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.
    PDF (276 kByte)

  • R. Krämer, P. Mathé, Modulus of continuity of Nemytskiĭ operators with application to a problem of option pricing, Journal of Inverse and Ill-Posed Problems, 16 (2008), pp. 435--461.
    PDF (424 kByte)

  • P. Mathé, B. Hofmann, Direct and inverse results in variable Hilbert scales, Journal of Approximation Theory, 154 (2008), pp. 77--89.
    PDF (228 kByte)

  • 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.
    PDF (150 kByte)

  • P. Mathé, N. Schöne, Regularization by projection in variable Hilbert scales, Applicable Analysis. An International Journal, 2 (2008), pp. 201-- 219.
    PDF (293 kByte)

  • 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.
    For solving linear ill-posed problems with noisy data regularization methods are required. In the present paper regularized approximations are obtained by a general regularization scheme in Hilbert scales. We exploit operator monotonicity of certain functions for deriving order optimal error bounds that characterize the accuracy of the regularized approximations. These error bounds are obtained under general smoothness conditions

  • 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.

  • P. Mathé, E. Novak, Simple Monte Carlo and the Metropolis algorithm, Journal of Complexity, 23 (2007), pp. 673--696.
    Abstract, PDF (289 kByte)
    We study the integration of functions with respect to an unknown density. Information is available as oracle calls to the integrand and to the non-normalized density function. We are interested in analyzing the integration error of optimal algorithms (or the complexity of the problem) with emphasis on the variability of the weight function. For a corresponding large class of problem instances we show that the complexity grows linearly in the variability, and the simple Monte Carlo method provides an almost optimal algorithm. Under additional geometric restrictions (mainly log-concavity) for the density functions, we establish that a suitable adaptive local Metropolis algorithm is almost optimal and outperforms any non-adaptive algorithm.

  • F. Bauer, P. Mathé, S.V. Pereverzev, Local solutions to inverse problems in geodesy. The impact of the noise covariance structure upon the accuracy of estimation, Journal of Geodesy. Springer-Verlag, Berlin. English, English abstracts., 81 (2006), pp. 39--51.
    In many geoscientific applications, one needs to recover the quantities of interest from indirect observations blurred by colored noise. Such quantities of interest often correspond to the values of bounded linear functionals acting on the solution of some observation equation. For example, various potential quantities are derived from harmonic coefficients of the Earth's gravity potential. Each such coefficient is the value of the corresponding linear functional. The goal of the paper is to discuss a new way to use information about noise covariance structure which allows estimation of the functionals of interest with order optimal rates and does not involve a covariance operator directly in the estimation process. It is done on the base of a balancing principle for the choice of regularization parameter which is new in geoscientific applications. A number of tests demonstrate its applicability. In particular we could find appropriate regularization parameters by knowing a small part of the gravitational field on the Earth's surface with high precision and reconstructing the rest globally by downward continuation from satellite data.

  • P. Mathé, S.V. Pereverzev, Regularization of some linear ill-posed problems with discretized random noisy data, Mathematics of Computation, 75 (2006), pp. 1913--1929.
    For linear statistical ill-posed problems in Hilbert spaces we introduce an adaptive procedure to recover the unknown solution from indirect discrete and noisy data.. This procedure is shown to be order optimal for a large class of problems. Smoothness of the solution is measured in terms of general source conditions. The concept of operator monotone functions turns out to be an important tool for the analysis.

  • P. Mathé, S.V. Pereverzev, The discretized discrepancy principle under general source conditions, Journal of Complexity, 22 (2006), pp. 371--381.
    We discuss adaptive strategies for choosing regularization parameters in Tikhonov- Phillips regularization of discretized linear operator equations. Two rules turn out to be entirely based on the underlying regularization scheme. Among them only the discrepancy principle allows to search for the optimal regularization parameter from the easiest problem. This possible advantage cannot be used with the standard projection scheme. We present a modified scheme, in which the discretization level varies with the successive regularization parameters, and which allows to use the advantage, mentioned before.

  • P. Mathé, The Lepskiĭ principle revisited, Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data, 22 (2006), pp. L11--L15.
    Abstract, PDF (178 kByte)
    Recently a new parameter choice strategy, often called the Lepskii-type strategy, became attractive for regularizing ill-posed problems. Here we present this construction in a simple and unified way. We emphasize that previous modifications can be seen to choose the same parameter, albeit drawing different conclusions from it. Finally we exhibit the application of this a posteriori parameter choice for Tikhonov regularization.

  • P. Mathé, What do we learn from the discrepancy principle?, Zeitschrift fur Analysis und ihre Anwendungen. Journal for Analysis and its Applications, 25 (2006), pp. 411--420.
    The author analyzes the discrepancy principle when smoothness is given in terms of general source conditions. As it turns out, this framework is particularly well suited to reveal the mechanism under which this principle works. For general source conditions there is no explicit way to compute rates of convergence. Instead arguments must be based on geometric properties. Still this approach allows to generalize previous results. The analysis is accomplished with a result showing why this discrepancy principle inherently has the early saturation for a large class of regularization methods of bounded qualification.

  • 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.

  • P. Mathé, G. Wei, Quasi-Monte Carlo integration over $mathbbR^d$, Mathematics of Computation, 73 (2004), pp. 827-841.
    In this paper we show that a wide class of integrals over $mathbbR^d$ with a probability weight function can be evaluated using a QMC algorithm based on a proper decomposition of the domain and arranging low discrepancy points over a series of hierarchical hypercubes. For certain classes of power/exponential decaying weights the algorithm is of optimal order.

  • P. Mathé, Numerical integration using V-uniformly ergodic Markov chains, Journal of Applied Probability, 41 (2004), pp. 1104-1112.
    We study numerical integration based on Markov chains. Focus is on error bounds uniformly on classes of integrands. Since on general state space the concept of uniform ergodicity is too restrictive to cover important cases we analyze the error of V-uniformly ergodic Markov chains. Emphasis is on the interplay between ergodicity properties of the transition kernel, the initial distributions and the classes of integrands. The analysis is based on arguments from interpolation theory.

  • P. Mathé, Saturation of regularization methods for linear ill-posed problems in Hilbert spaces, SIAM Journal on Numerical Analysis, 42 (2004), pp. 968--973.
    We prove saturation of methods for solving linear ill-posed problems in Hilbert spaces for a wide class regularization methods. It turns out, that under a certain convexity assumption, saturation is necessary. We provide easy to verify assumptions, which allow to calculate the rate, at which saturation occurs.

  • 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, Geometry of 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. 789--804.
    The authors study the best possible accuracy of recovering the solution from linear ill-posed problems in variable Hilbert scales. A priori smoothness of the solution is expressed in terms of general source conditions, given through index functions. Emphasis is on geometric concepts. The notion of regularization is appropriately generalized, and the interplay between qualification of regularization and index function becomes visible. A general adaptation strategy is presented and its optimality properties are studied.

  • P. Mathé, Asymptotic constants for multivariate Bernstein polynomials, Studia Scientiarum Mathematicarum Hungarica. A Quarterly of the Hungarian Academy of Sciences, 40 (2003), pp. 59-69.
    We study the rate of convergence of multivariate Bernstein polynomials on the class of Hölder continuous functions. Using a proper probabilistic representation we are able to derive the asymptotic constant.

  • 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, S.V. Pereverzev, Moduli of continuity of operator valued functions, Numerical Functional Analysis and Optimization. An International Journal, 23 (2002), pp. 623--631.
    We shall study the modulus of continuity of non-negative functions $f$ defined for non-negative self-adjoint operators $A,B$ in some Hilbert space and taking values there. More precisely, we establish the validity of the following type of inequalities $$ normf(A) - f(B)leq C f(normA-B) + C' normA-B, $$ where it is natural to assume, that $f$ is continuous and $f(0)=0$. Such inequalities are valid for non-negative operator monotone functions, as well as for a certain more general class of operator valued ones. We show, that this class of functions is rich enough to cover virtually all examples, for which similar inequalities have been proven before, mainly in the context of discretizations of ill-posed operator equations.

  • 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.

  • P. Mathé, Hilbert space analysis of Latin hypercube sampling, Proceedings of the American Mathematical Society, 129 (2001), pp. 1477--1492.
    Latin Hypercube sampling (LHS)is a specific Monte Carlo estimator for numerical integration of functions on $mathbbR^d$ with respect to some product probability distribution function. Previous analysis established, that LHS is superior to independent sampling, at least asymptotically. Especially, if the function to be integrated allows a good additive fit. We propose a different approach to LHS, based on orthogonal projections in an appropriate Hilbert space, which allows a rigorous error analysis. Moreover, we indicate why convergence cannot be uniformly superior to independent sampling on the class of square integrable functions. We establish a general condition under which uniformity can be achieved, thereby indicating the rôle of certain Sobolev spaces.