Computational and Applied Math Proseminar

Friday, November 3, 1:40 p.m. GWC 604

Alexandra Smirnova

Dept. Math. & Stats, Georgia State University

On Iteratively Regularized Numerical Procedures for Nonlinear Ill-Posed Problems

slides Abstract In this research we construct numerical algorithms for solving nonlinear ill-posed (unstable) problems. This area is extremely difficult, since solutions to ill-posed problems are very sensitive to small variation in input data. For that reason ill-posed problems cannot be solved by classical methods of Computational Mathematics: the corresponding numerical procedures for them turn out to be divergent. However ill-posed problems are frequently encountered in many branches of natural sciences and engineering.

Examples of specific applications that will benefit from this research include, for example, chaos theory, as we are hopeful to achieve a more precise computation of the Feigenbaum constants, parameters that are important in understanding the behavior of nonlinear, or chaotic, systems.

One of the principal goals of nonlinear studies is the development of ways to prevent chaos in systems, such as those designed to control aircraft. As the control parameters of a nonlinear system are changed, the system moves from regular to chaotic behavior. Feigenbaum constants indicate where this phase change will take place. Knowing the precise point at which chaos will ensue will allow system designers to constrain the values of the control parameters accordingly.

Examples of nonlinear systems for which more accurate knowledge of the Feigenbaum constants will improve functionality include industrial robots, wind turbines for electric power generation, chemical reactors, aircraft and spacecraft vibration suppression systems, and heart rate control systems.

For further information please contact: mittelmann@asu.edu