Preface -- 1 An introduction using classical examples -- 1.1 Numerical differentiation. First look at the problem of regularization. The balancing principle -- 1.1.1 Finite-difference formulae -- 1.1.2 Finite-difference formulae for nonexact data. A priori choice of the stepsize -- 1.1.3 A posteriori choice of the stepsize -- 1.1.4 Numerical illustration -- 1.1.5 The balancing principle in a general framework -- 1.2 Stable summation of orthogonal series with noisy coefficients. Deterministic and stochastic noise models. Description of smoothness properties -- 1.2.1 Summation methods -- 1.2.2 Deterministic noise model -- 1.2.3 Stochastic noise model -- 1.2.4 Smoothness associated with a basis -- 1.2.5 Approximation and stability properties of -methods -- 1.2.6 Error bounds -- 1.3 The elliptic Cauchy problem and regularization by discretization -- 1.3.1 Natural linearization of the elliptic Cauchy problem -- 1.3.2 Regularization by discretization -- 1.3.3 Application in detecting corrosion -- 2 Basics of single parameter regularization schemes -- 2.1 Simple example for motivation -- 2.2 Essentially ill-posed linear operator equations. Least-squares solution. General view on regularization -- 2.3 Smoothness in the context of the problem. Benchmark accuracy levels for deterministic and stochastic data noise models -- 2.3.1 The best possible accuracy for the deterministic noise model -- 2.3.2 The best possible accuracy for the Gaussian white noise model -- 2.4 Optimal order and the saturation of regularization methods in Hilbert spaces -- 2.5 Changing the penalty term for variance reduction. Regularization in Hilbert scales -- 2.6 Estimation of linear functionals from indirect noisy observations -- 2.7 Regularization by finite-dimensional approximation.

2.8 Model selection based on indirect observation in Gaussian white noise -- 2.8.1 Linear models given by least-squares methods -- 2.8.2 Operator monotone functions -- 2.8.3 The problem of model selection (continuation) -- 2.9 A warning example: an operator equation formulation is not always adequate (numerical differentiation revisited) -- 2.9.1 Numerical differentiation in variable Hilbert scales associated with designs -- 2.9.2 Error bounds in L2 -- 2.9.3 Adaptation to the unknown bound of the approximation error -- 2.9.4 Numerical differentiation in the space of continuous functions -- 2.9.5 Relation to the Savitzky-Golay method. Numerical examples -- 3 Multiparameter regularization -- 3.1 When do we really need multiparameter regularization? -- 3.2 Multiparameter discrepancy principle -- 3.2.1 Model function based on the multiparameter discrepancy principle -- 3.2.2 A use of the model function to approximate one set of parameters satisfying the discrepancy principle -- 3.2.3 Properties of the model function approximation -- 3.2.4 Discrepancy curve and the convergence analysis -- 3.2.5 Heuristic algorithm for the model function approximation of the multiparameter discrepancy principle -- 3.2.6 Generalization in the case of more than two regularization parameters -- 3.3 Numerical realization and testing -- 3.3.1 Numerical examples and comparison -- 3.3.2 Two-parameter discrepancy curve -- 3.3.3 A numerical check of Proposition 3.1 and use of a discrepancy curve -- 3.3.4 Experiments with three-parameter regularization -- 3.4 Two-parameter regularization with one negative parameter for problems with noisy operators and right-hand side -- 3.4.1 Computational aspects for regularized total least squares -- 3.4.2 Computational aspects for dual regularized total least squares.

3.4.3 Error bounds in the case B D I -- 3.4.4 Error bounds for B ¤ I -- 3.4.5 Numerical illustrations. Model function approximation in dual regularized total least squares -- 4 Regularization algorithms in learning theory -- 4.1 Supervised learning problem as an operator equation in a reproducing kernel Hilbert space (RKHS) -- 4.1.1 Reproducing kernel Hilbert spaces and related operators -- 4.1.2 A priori assumption on the problem: general source conditions -- 4.2 Kernel independent learning rates -- 4.2.1 Regularization for binary classification: risk bounds and Bayes consistency -- 4.3 Adaptive kernel methods using the balancing principle -- 4.3.1 Adaptive learning when the error measure is known -- 4.3.2 Adaptive learning when the error measure is unknown -- 4.3.3 Proofs of Propositions 4.6 and 4.7 -- 4.3.4 Numerical experiments. Quasibalancing principle -- 4.4 Kernel adaptive regularization with application to blood glucose reading -- 4.4.1 Reading the blood glucose level from subcutaneous electric currentmeasurements -- 4.5 Multiparameter regularization in learning theory -- 5 Meta-learning approach to regularization - case study: blood glucose prediction -- 5.1 A brief introduction to meta-learning and blood glucose prediction -- 5.2 A traditional learning theory approach: issues and concerns -- 5.3 Meta-learning approach to choosing a kernel and a regularization parameter -- 5.3.1 Optimization operation -- 5.3.2 Heuristic operation -- 5.3.3 Learning at metalevel -- 5.4 Case-study: blood glucose prediction -- Bibliography -- Index.