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Consider the ill-posed system $$ \mathbf A \mathbf x= \mathbf b.$$

One method to regularize the solution is the Tikhonov method which effectively minimizes $ ||\mathbf A \mathbf x - \mathbf b ||^2 + || \mathbf \Gamma \mathbf x||^2$.

Letting the Tikhonov matrix $\mathbf \Gamma = \lambda \mathbf I$ favours solutions with smaller norms, where $\mathbf I$ is the identity matrix, and the parameter $\lambda$ is chosen empirically of a given system. Singular value decomposition ($\mathbf A = \mathbf{U \Sigma V}^H$) may then be used to calculate the solution via $\mathbf x = \mathbf{V \hat{\Sigma} U}^H \mathbf b$, where

$$\hat{\Sigma}_{ii} = \frac{\Sigma_{ii}}{\Sigma_{ii}^2 + \lambda^2}.$$

My question is how to find $\mathbf x$ which instead favours low frequency solutions?

i.e. I wish to minimize something like $ ||\mathbf A \mathbf x - \mathbf b ||^2 + \lambda^2 \sum_{high \, frequencies} FFT \{ \mathbf x \}^2$.

xyz
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1 Answers1

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Well, it seems like you said how to do it: Let $ T $ be the discrete fourier transform matrix, ie. something like: $$ T_{a,b} = e^{\frac{2 \pi i}{N} ab} $$ or whatever is appropriate for your situation. Then let $ P $ be the projection on to high frequency modes, or more generally something like $ P = \text{diag}(0,0,\cdots,w_{n-2},w_{n-1},w_{n}) $. For some weights $ w_i $ you can choose. Then let $ \Gamma = PT $.

Maybe there is some natural choice for the weights. For example, it might be computationally easier if you take $ \Gamma $ to be some finite difference matrix, eg: $$ \Gamma = \begin{pmatrix} -1 & 1 & 0 & \cdots \\ 0 & -1 & 1 & \cdots \\ && \ddots & \end{pmatrix} $$

user174456
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  • Thanks, the comment about a finite difference matrix was especially helpful. I also forgot that I can use $x=(A^HA +\Gamma^H \Gamma)^{-1} A^H b$ in place of SVD. – xyz Sep 10 '14 at 04:15