Application of Numerical Algebraic Geometry to Partial Differential Equations

October 27, 2006
  • Polynomial systems
  • 13P15
In Numerical Algebraic Geometry, solution components of polynomial systems are characterized by witness points. Such nice points are computed efficiently by continuation methods. In this talk, which is joint work with Wenyuan Wu and Jan Verschelde, I will outline progress on extending these methods to Partial Differential Equations. I will describe the jet geometric picture of PDE in their jet space, to which the methods of Numerical Algebraic Geometry are applied. In particular witness points in jet geometry are cut out by the intersection of random linear spaces with submanifolds in jet space (there will be nice pictures). Applications to constrained mechanical systems; overdetermined systems for symmetries of differential equations will also be described.