This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: 1 Lecture 10 Recall, a problem is said to be well-posed if 1. there exists a solution 2. the solution is unique 3. the solution depends continuously on the data. In terms of an operator equation A = f where A : U V, U, V subsets of normed space X and Y , respectively, this means A : U V is bijective and A- 1 : V U is continuous. So there are 3 types of ill-posedness 1. If A is not surjective (onto), then A = f is not solvable for all f V (nonexistence of a solution). 2. If A is not injective (one to one), then A = f may have more than one solution (nonuniqueness). 3. If A- 1 exists, but is not continuous, then the solution does not depend continuously on the data (instability). These properties are not completely independent. Theorem 1 If A : X Y is a bounded linear operator mapping a Banach space X bijectively into a Banach space Y , then A- 1 : Y X is bounded and therefore continuous....
View Full Document
This note was uploaded on 03/14/2011 for the course MATH 676 taught by Professor Staff during the Fall '08 term at Colorado State.
- Fall '08