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regularization:tikhonov_regularization

Tikhonov Regularization

1. Operator Approach to Tikhonov Regularization

Let $H$ be a mapping from some normed space $X$ into a normed space $Y$. When solving an equation of the form $$H x = y$$ with an ill-posed operator $H$, Tikhonov Regularization replaces the unstable inverse $H^{-1}$ by a family of stable mappings $R_{\alpha}$ define dy $$\label{Ralpha def} R_{\alpha} := (\alpha I + H^{\ast}H)^{-1}H,$$ where $\alpha>0$ is known as regularization parameter. Here, the unbounded operator $(H^{\ast}H)^{-1}$ which appears in the Moore-Penrose Pseudo-Inverse $(H^{\ast}H)^{-1}H^{\ast}$ is replaced by the bounded operator $(\alpha I + H^{\ast}H^{\ast})^{-1}$. For $y = Hx$ we have $$\begin{array}{cc} (\alpha I + H^{\ast}H)^{-1}H^{\ast} y & = &(\alpha I + H^{\ast}H)^{-1}H^{\ast} Hx \\ & = & (\alpha I + H^{\ast}H)^{-1}( \alpha I + H^{\ast} H - \alpha I ) x \\ & = & x - \alpha (\alpha I + H^{\ast}H)^{-1} x \\ & \rightarrow & x \end{array}$$ for $\alpha \rightarrow 0$, where the convergence $$\alpha (\alpha I + H^{\ast}H)^{-1} x \rightarrow 0, \;\; \alpha \rightarrow 0,$$ is shown by spectral arguments. Please note that this convergence is a pointwise convergence in $X$ and does not hold in norm!

The invertibility of $\alpha I + H^{\ast}H$ is obtained by $$\begin{array}{cc} \langle x, (\alpha I + H^{\ast}H) x \rangle & = & \alpha \langle x,x \rangle + \langle Hx, Hx \rangle \\ & \geq & \alpha || x ||^2 \end{array}$$ according to the Lax-Milgram Lemma, which also yields $$|| (\alpha I + H^{\ast}H)^{-1} || \leq \frac{1}{\alpha}.$$ The sharper result $$|| (\alpha I + H^{\ast}H)^{-1}H^{\ast} || \leq \frac{1}{2\sqrt{\alpha}}$$ is again shown by spectral arguments and the arithmetic-geometric mean $$\frac{\mu}{\alpha + \mu^2} \leq \frac{1}{2\sqrt{\alpha}}.$$

2. Optimization Approach to Tikhonov Regularization

We can reformulate the equation $Hx=y$ into minimizing $$J(x) = || y - Hx ||^2, \;\; x \in X,$$ which is the Moore-Penrose pseudo inverse. A stabilization is given by adding a term to the functional $$J(x) := \alpha || x ||^2 + ||y - Hx||^2, \;\; x \in X.$$ First order optimality conditions for the minimizer lead to $$0 = \nabla_x J(x) = 2\alpha x + 2 H^{\ast}(y - H x),$$ i.e. $$(\alpha I + H^{\ast}H )x = H^{\ast} y.$$ Thus, the minimizer is obtained by $$x_{\alpha} := (\alpha I + H^{\ast}H)^{-1} H^{\ast} y,$$ which coincides with (\ref{Ralpha def}).

3. Spectral Approach to Tikhonov Regularization

The operator $H^{\ast}H$ is a self-adjoint operator. If $H$ is compact, then $H^{\ast}H$ is compact as well, and there is an orthonormal system $\varphi_{j}, j \in \mathbb{N},$ of eigenvectors of $H^{\ast}H$ with eigenvalues $\mu_{j}^2$, such that $$H^{\ast}H \varphi_{j} = \mu_{j}^{2} \varphi_{j}, \;\; j \in \mathbb{N}.$$ Then, the pseudo inverse of $H$ can be written as $$(H^{\ast}H)^{-1}H^{\ast} y = \sum_{j=1}^{\infty} \frac{1}{\mu_{j}} \langle\varphi_{j},y\rangle$$ When $H$ is compact and $\mu_j$ is sorted in descending order, we know that $\mu_j \rightarrow 0$ for $j\rightarrow \infty$. Here, the ill-posedness of $H^{\ast}H$ is reflected by the unboundedness of $1/\mu_j^2$. Stabilization can be achieved by bounding this unbounded term. A spectral damping scheme is achieved by $$R_{\alpha} y := \sum_{j=1}^{\infty} \frac{\mu_j}{\alpha + \mu_{j}^2} <\varphi_{j},y>,$$ which for $\alpha \rightarrow 0$ tends to $H^{-1}y$ for every fixed $y \in H(X)$. Using the spectral representation, this is readily to be identical to the above inverse $(\alpha I + H^{\ast}H)^{-1} H^{\ast}$.

Literature

We refer to the following literature for more detail about Tikhonov Regularization:

1. Groetsch: Inverse problems in the mathematical sciences, 1993
2. Rainer Kress: Linear Integral Equations, Springer, 1989
3. David Colton and Rainer Kress: Inverse Acoustic and Electromagnetic Scattering Theory, 1993
4. Engl, Hanke and Neugebauer: Regularization of Inverse Problems, 1996
5. Andreas Kirsch: Introduction into the mathematical theory of inverse problems, Springer 1996