Inverse Problems

Challenges when trying to solve inverse problems:

• They may not have a unique solution.
• Often, small changes in data cause arbitrarily large changes in solution.

Definition: (Hadamard, 1902)

If the solution of a problem is not unique, or it does not depend continuously on the data, then the problem is ill-posed.

• Many inverse problems are also ill-posed.
• For realistic problems (containing noise), we cannot hope to compute an exact solution.

Goal: Compute a meaningful approximation.
Approach: Use regularization.