Partial Differential Equations and Mathematical Physics
Many objects of physical interest cannot be studied directly. Examples include, imaging the interior of the body, the determination of cracks within solid objects, and material parameters such as the conductivity of inaccessible objects. Translated into mathematical terms we obtain partial differential equations containing unknown coefficients or boundary conditions. We attempt to solve for these by means of further measurements. Closely related to such problems are issues of optimal design. Given some physical system described by a partial differential equation model and subject to unknown design parameters, can one determine these in such a way that the system behaves in some prescribed way? We are interested in the question of when a unique determination can be made, as well as designing algorithms for the efficient numerical recovery of the unknown or optimal configuration.
Multiple solutions with different performance indices exist in many nonlinear partial differential equations and dynamic systems. Finding multiple critical points in a stable numerical way is interesting and important to both theory and applications. We are using Morse theory and other nonlinear functional analysis tools to establish some new local critical point theory from which local minimax numerical methods can be developed to solve multiple solution problems.
There are many overlaps with the Numerical Analysis and Scientific Computation research group.