Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 326 / 1 / 1 / 3 / a - Codex (@codex) - OurBigBook.com

Table of contents
- Solution a

Solution

Repeated substitution in Landweber iteration from

u_{0} = 0

gives

u_{k} = τ \sum_{j = 0}^{k - 1} (I - τ A^{*} A)^{j} A^{*} f .

On the

i

th singular vector,

A^{*} A

σ_{i}^{2}

and

A^{*} y_{i} = σ_{i} x_{i}

. The finite geometric series therefore gives

u_{k} = i \sum \frac{1 - ( 1 - τ σ _{i}^{2} ) ^{k}}{σ _{i}} ⟨ f, y_{i} ⟩ x_{i} .

This is a spectral regularization method with filter

g_{k} (λ) = \frac{1 - ( 1 - τ λ ) ^{k}}{λ}, u_{k} = g_{k} (A^{*} A) A^{*} f .

 Ancestors (12)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version