Regularization of Inverse Problems

By Heinz W. Engl, Martin Hanke, Andreas Neubauer. 1996

Subfields: Optimization & Training, Numerical Methods & Scientific Computing

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Citations in the corpus

Learning Theory for Distribution Regression (2014, methodology): “For the validity of the lemma see Engl et al. (1996) and Sriperumbudur et al.”
Kernel Mean Estimation via Spectral Filtering (2014, methodology): “Unlike in earlier works [15–18] where the spectral filtering approach has been used in supervised learning problems, we here deal with unsupervised setting and only leverage spectral filtering as a way to construct a shrinkage estimator of the kernel mean.”
Kernel Bayes' Rule (2010, methodology): “2β+1 2β+2α}), (n→ ∞). Theorem 5 shows that the best rate is O(n−2α/3) for β≥ 1/2. The idea of the proof is essentially the same as [7, 10], which uses the standard techniques in regularization theory [8]. The condition π/pX ∈ R(AXC β XX) requires smoothness on the prior. The consistency of the KBR m”
Optimal Rates for the Regularized Least-Squares Algorithm (2007, methodology): “In order to avoid confusion we adopt the nomenclature of inverse problems [13].”
An alternating iterative procedure for the Cauchy problem for the Helmholtz equation (2014, methodology): “Different regularization methods have been suggested in the literature [7, 10, 23].”
A simplified Tikhonov regularization method for determining the heat source (2010, unknown): “It is easy to see that K is a linear compact operator (see [21]), and the singular values frng n1⁄41 of K satisfy rn 1⁄4 1 e n2p2 n2p2 ; ð2:5Þ and ðg; xnÞ 1⁄4 1 e n2p2 n2p2 ðf ;XnÞðXn;XnÞ; ð2:6Þ i.”
The GLT class as a generalized Fourier analysis and applications (2006, methodology): “The eigen and singular value asymptotic information is also available in the variable coefficient case, but more work has to be done for understanding the eigenvector structure in order to have a precious information to be used in devising effective regularizing methods [25] for large systems as in ”
A tutorial on inverse problems for anomalous diffusion processes (2015, unknown): “d the statistical nature (Gaussian, Poisson, Laplace ...) of the contaminating noise in the data, which will depend very much on the speciﬁc application. We refer interested readers to the monographs [26, 92, 41] and the survey [49] for updated accounts on regularization methods for constructing sta”
Semi‐Nonparametric IV Estimation of Shape‐Invariant Engel Curves (2003, unknown): “Kress (1999), Korostelev and Tsybakov (1993), Donoho (1995), Engl, et al. (1996) , Cavalier, et al. (2002) and Cohen, et al. (2004).”
Inverse transport theory and applications (2009, background): “Because the inverse problem is ill-posed, the parameters we want to reconstruct need to be regularized to avoid over-fitting of the data [53].”
Bayesian inverse problems for functions and applications to fluid mechanics (2009, background): “For the Hilbert spaces setting see the book [ 21 ] and the references therein; for the Banach space setting see the recent papers [42, 52, 34] and the references therein.”
Reduced basis approximation and a posteriori error bounds for 4D-Var data assimilation (2018, methodology): “We refer to Engl et al. (1996) for regularization of inverse problems in general and to Puel (2009) for Tichonov regularization in data assimilation.”
Convergence Analysis of Modified Hybrid Steepest-Descent Methods with Variable Parameters for Variational Inequalities (2007, background): “20 ). The K-constrained pseudoinverse of A (symbol ˆ”
Simultaneous derivation of intensities and weighting functions in a general pseudo-spherical discrete ordinate radiative transfer treatment (2002, methodology): “... functions (intensity partial derivatives with respect to atmospheric parameters to be retrieved), and Yn =F(Xn) is the synthetic measurement vector corresponding to state vector estimate Xn at iteration step n. The inversion is carried out using cost-function minimization techniques such as non-”
Integral equations of the first kind, inverse problems and regularization: a crash course (2007, background): “Relatively elementary introductions to the fleld may be found in [99] and [40]; much more comprehensive accounts are available in the highly recommended references [59], [ 32 ] and [57].”