Title: "Numerical Solutions of Ill-posed Problems: A Geometric
Perspective"

Zhonggang Zeng
Northeastern Illinois University

Abstract:
Arising frequently in sciences and engineering, ill-posed problems
remain a challenge and a frontier in scientific computing because
their solutions appear to be unstable and infinitely sensitive to
data perturbations.  On the other hand, the hypersensitivity of such
problems may be a "misconception", as argued by W. Kahan. In many
cases, the instability can be effectively removed so that the accurate
solutions become attainable.  In this talk we present a geometric
perspective on the nature of ill-posed problems: They form Riemannian
manifolds of positive codimensions and those manifolds are entangled
in certain stratification structures.  Those geometric properties
lead to a "three-strikes" principle for regularizing the ill-posed
problems with the hypersensitivity eliminated.  We shall also present
a novel two-staged strategy that is proven effective for solving
ill-posed algebraic problems such as matrix rank-revealing, solving
singular equations, and computing the Jordan Canonical Form.