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regularization:tikhonov_regularization [Inverse Problems and Data Assimilation Wiki and International Community]

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regularization:tikhonov_regularization

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 regularization:tikhonov_regularization [2012/12/27 14:33]potthast created regularization:tikhonov_regularization [2012/12/27 14:40] (current)potthast 2012/12/27 14:40 potthast 2012/12/27 14:33 potthast created 2012/12/27 14:40 potthast 2012/12/27 14:33 potthast created Line 1: Line 1: ===== 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 [[ip:​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: ​ + + - Groetsch: Inverse problems in the mathematical sciences, 1993 + - Rainer Kress: Linear Integral Equations, Springer, 1989 + - David Colton and Rainer Kress: Inverse Acoustic and Electromagnetic Scattering Theory, 1993 + - Engl, Hanke and Neugebauer: Regularization of Inverse Problems, 1996 + - Andreas Kirsch: Introduction into the mathematical theory of inverse problems, Springer 1996 +