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- ItemInverse learning in Hilbert scales(Dordrecht [u.a.] : Springer Science + Business Media B.V, 2023) Rastogi, Abhishake; Mathé, PeterWe study linear ill-posed inverse problems with noisy data in the framework of statistical learning. The corresponding linear operator equation is assumed to fit a given Hilbert scale, generated by some unbounded self-adjoint operator. Approximate reconstructions from random noisy data are obtained with general regularization schemes in such a way that these belong to the domain of the generator. The analysis has thus to distinguish two cases, the regular one, when the true solution also belongs to the domain of the generator, and the ‘oversmoothing’ one, when this is not the case. Rates of convergence for the regularized solutions will be expressed in terms of certain distance functions. For solutions with smoothness given in terms of source conditions with respect to the scale generating operator, then the error bounds can then be made explicit in terms of the sample size.
- ItemConvergence analysis of Tikhonov regularization for non-linear statistical inverse problems(Ithaca, NY : Cornell University Library, 2020) Rastogi, Abhishake; Blanchard, Gilles; Mathé, PeterWe 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.
- ItemMultiband k $cdot$ p model and fitting scheme for ab initio-based electronic structure parameters for wurtzite GaAs(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Marquardt, Oliver; Caro, Miguel A.; Koprucki, Thomas; Mathé, Peter; Willatzen, MortenWe develop a 16-band k · p model for the description of wurtzite GaAs, together with a novel scheme to determine electronic structure parameters for multiband k · p models. Our approach uses low-discrepancy sequences to fit k · p band structures beyond the eight-band scheme to most recent ab initio data, obtained within the framework for hybrid-functional density functional theory with a screened-exchange hybrid functional. We report structural parameters, elastic constants, band structures along high-symmetry lines, and deformation potentials at the Γ point. Based on this, we compute the bulk electronic properties (Γ point energies, effective masses, Luttinger-like parameters, and optical matrix parameters) for a ten-band and a sixteen-band k · p model for wurtzite GaAs. Our fitting scheme can assign priorities to both selected bands and k points that are of particular interest for specific applications. Finally, ellipticity conditions can be taken into account within our fitting scheme in order to make the resulting parameter sets robust against spurious solutions.
- ItemParameter choice methods using minimization schemes(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Bauer, Frank; Mathé, PeterIn 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.
- ItemInfluence of the carrier reservoir dimensionality on electron-electron scattering in quantum dot materials(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Wilms, Alexander; Mathé, Peter; Schulze, Franz; Koprucki, Thomas; Knorr, Andreas; Bandelow, UweWe 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
- ItemCalibration methods for gas turbine performance models(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Borchardt, Jürgen; Mathé, Peter; Printsypar, GalinaThe WIAS software package BOP is used to simulate gas turbine models. In order to make accurate predictions the underlying models need to be calibrated. This study compares different strategies of model calibration. These are the deterministic optimization tools as nonlinear least squares (MSO) and the sparsity promoting variant LASSO, but also the probabilistic (Bayesian) calibration. The latter allows for the quantification of the inherent uncertainty, and it gives rise to a surrogate uncertainty measure in the MSO tool. The implementation details are accompanied with a numerical case study, which highlights the advantages and drawbacks of each of the proposed calibration methods.
- ItemInterpolation in variable Hilbert scales with application to inverse problems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Mathé, Peter; Tautenhahn, UlrichFor 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.
- ItemSimple Monte Carlo and the metropolis algorithm(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Mathé, Peter; Novak, ErichWe 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.
- ItemSharp converse results for the regularization error using distance functions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Flemming, Jens; Hofmann, Bernd; Mathé, PeterIn 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
- ItemAnalysis of profile functions for general linear regularization methods(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Mathé, Peter; Hofmann, BerndThe 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 ...