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 Stability - the solution depends continuously on the data

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

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

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

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

Regularisation Theory - compact operator - one to one - For, we would want to solveWe actually know... Problem!

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

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

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

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

Minimization

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

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

Example of a regularisation strategy Regularisation strategy:

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

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

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

Tykhonov Regularisation compact singular system for

Rewrite : Landweber Iteration Iterative process Then where

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

Conclusion -Worst case error - Regularisation strategies - Filtering - Tykhonov Regularisation - Landweber Iteration