Prerequisites & Notation

Prerequisites for This Chapter

This chapter develops the mathematical theory of ill-posed problems and regularization. It builds directly on the functional analysis foundations of Chapter 1 and draws on linear algebra from the Telecom book.

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RFI Chapter 1 — Functional Analysis Foundations. Banach and Hilbert spaces, bounded linear operators, compact operators and their spectral theory (DCompact Operator, TSpectral Theorem for Compact Self-Adjoint Operators), and the singular value decomposition for operators (DSingular System of a Compact Operator). The Picard condition for solvability is introduced there and used throughout this chapter.

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Telecom Chapter 1 — Linear Algebra Foundations. Matrix SVD (TSVD Existence Theorem), condition number, the normal equation for least squares. Needed for the finite-dimensional analogues of the operator results here.

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Telecom Chapter 2 — Probability and Random Variables. Gaussian distributions, expectation, and variance. Needed for the Bayesian interpretation of Tikhonov regularization and the noise models in inverse problems.

Notation for Chapter 2

Symbol	Meaning	Introduced
$\mathcal{A}$	Forward operator mapping model parameters to data; $\mathcal{A} \colon \mathcal{X} \to \mathcal{Y}$	s01
$\mathcal{A}^*$	Adjoint of $\mathcal{A}$	s02
$\mathcal{A}^\dagger$	Moore--Penrose pseudoinverse of $\mathcal{A}$	s02
$\sigma_k$	$k$ -th singular value of $\mathcal{A}$ (ordered: $\sigma_1 \geq \sigma_2 \geq \cdots$ )	s02
$v_k,\, u_k$	Right and left singular vectors (or functions) of $\mathcal{A}$	s02
$\alpha$	Regularization parameter (controls trade-off between data fidelity and stability)	s03
$R_\alpha$	Regularized reconstruction operator; $R_\alpha \colon \mathcal{Y} \to \mathcal{X}$	s03
$\delta$	Noise level: $\\|y^\delta - y\\| \leq \delta$	s01
$y^\delta$	Noisy data; $y^\delta = \mathcal{A}x^\dagger + \eta$ with $\\|\eta\\| \leq \delta$	s01
$x^\dagger$	True (minimum-norm least-squares) solution	s02
$x_\alpha^\delta$	Regularized solution: $x_\alpha^\delta = R_\alpha y^\delta$	s03
$\mathcal{N}(\mathcal{A})$	Null space (kernel) of $\mathcal{A}$	s01
$\mathcal{R}(\mathcal{A})$	Range of $\mathcal{A}$	s01
$F_\alpha$	Spectral filter function of a regularization method	s04
$\mu$	Source condition order; also qualification of a regularization method	s03
$R(x)$	Regularization penalty functional (e.g., $\\|x\\|^2$ , $\\|x\\|_1$ , $\mathrm{TV}(x)$ )	s06
$\lambda$	Regularization parameter in variational form $\min \mathcal{D} + \lambda R$	s06
$\mathcal{F}$	Nonlinear forward operator in $\mathcal{F}(x) = y$	s07
$\mathcal{F}'(x)$	Fréchet derivative (Jacobian) of $\mathcal{F}$ at $x$	s07

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