报告人：曾钟钢 (Bernard J. Brommel Distinguished Research Professor, Northeastern Illinois University)

题目：Numerical Solutions of Ill-posed Problems: A Geometric Perspective

时间：2017年6月14日15：00 – 16：00

地点：华中科技大学恩明楼813

摘要：

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 complex analytic manifolds of positive codimensions and those manifolds are entangled in certain stratification structures. As a result, the hypersensitivity is in one direction so that tiny perturbations can only decreases and never increase singularities. 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.