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Inverse Problems

Inverse Problems. Example. Direct problem given polynomial find zeros. Inverse problem given zeros find polynomial. Well-posedness. A problem is well posed if. Existence - there exists a solution of the problem Uniqueness - there is at most one solution of the problem

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Inverse Problems

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  1. Inverse Problems

  2. Example Direct problem given polynomial find zeros Inverse problem given zeros find polynomial

  3. Well-posedness A problem is well posed if Existence - there exists a solution of the problem Uniqueness - there is at most one solution of the problem Stability - the solution depends continuously on the data

  4. Inverse problem Given , compute such that , ie., Norm problem is not stable Perturb by Example (ill-posed problem) Operator error in data error in solution

  5. Banach stronger norm Inverse problem Given , compute such that The worst-case error linear bounded In general, we do not have the data … … but the perturbed data …

  6. Banach stronger norm Assume - - - extra information for solutions and The worst-case error linear bounded Worst case error:

  7. The worst-case error (example) stronger norm It can then be shown:

  8. Regularisation Theory • - compact operator • one to one For , we would want to solve We actually know ... Problem!

  9. Find an approximation for Aim - small error (hopefully not much worse than the worst case error!) - depends continuously on Idea: Construct a suitable bounded approximation of Approximation

  10. Regularisation Strategy Idea: Construct a suitable bounded approximation of Definition: A regularisation strategy is a family of linear and bounded operators such that Theorem: (due to being compact) 1- is not uniformly bounded 2- Convergence is not uniform, but point wise

  11. Error End problem... Perturbed problem... approximations of

  12. Error End problem... Perturbed problem... approximations of When

  13. Minimization

  14. Regularisation Strategy Idea: Construct a suitable bounded approximation of Definition: A regularisation strategy is a family of linear and bounded operators such that

  15. The worst-case error (example) stronger norm It can then be shown:

  16. Example of a regularisation strategy Regularisation strategy:

  17. Example of a regularisation strategy It can be shown, for a priori information Choose Then… asymptotically optimal

  18. Filtering compact singular system for singular values of orthonormal systems such that and It can be shown is the solution of

  19. Filtering is the solution of Regularisation strategy (Filtering): regularizing filter : when

  20. Tykhonov Regularisation compact singular system for

  21. Landweber Iteration Rewrite : Iterative process Then where

  22. Landweber Iteration compact and It can be shown… defines a regularization strategy Choices for accuracy of : large stability of : small an optimal choice can be made…

  23. Conclusion • Worst case error • Regularisation strategies • Filtering • Tykhonov Regularisation • Landweber Iteration

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