Inverse Problems for Nonlinear Partial Differential Equations

This project will address three topics in the general area of inverse problems and those are the amount of data needed to recover the desired unknown, the stability of the result in terms of the data measurements, and the existence or not of an algorithm to go from the measured data to the desired unknowns. Each of these will be an essential component of the work for this proposal.

The first of these is critical and from a mathematical viewpoint typically the most challenging. Without such a result we have no guarantee that even finding a solution to the mathematical problem allows us to correlate this with the actual physical solution. In each of the works we anticipate this will require the greatest effort and challenges.

Answering the stability question will be essential for us to determine: "given a tolerance level between our constructed solution and the actual one, what is the allowed maximal error in the data measurements that will allow us to achieve this." Of course, to carry this out one needs a reconstruction method and each algorithm, even if it provides a solution, may require a different error bound on the data. Thus in some sense the question we have to answer is not only if a computational algorithm can be found, but in what sense is it near to being optimal?

This latter question is one where the work of a mathematically strong undergraduate student can be engaged. Mentoring of such students will be an aspect of this work. This project will support 3 undergraduate students each year of the 3-year grant.

Specifically, the recovery of the nonlinear terms in nonlinear reaction-diffusion equations and systems of parabolic type is sought; that is, coefficients such as the conductivity or the reaction or interaction terms that depends on the solution itself. An example here is a (spatially or environment variable) rate coefficient in a complex inter-species interaction term that itself has to be determined as is typical in sophisticated epidemic models.

Also considered are nonlinear hyperbolic equations occurring in, for example, medical imaging. Nonlinear acoustics has a term that essentially represents the object to be reconstructed and this is coupled to a second term that arises from the nonlinear model and appears as a coefficient in the leading term of the partial differential operator. The simplest model that retains the nonlinear effects is to take this to be the identity operator but a more realistic case is to assume this is more complex and additionally seek its recovery.

The damped or attenuated wave equation occurs in many areas of physics and engineering. The usual assumption is the damping mechanism is proportional to velocity so that a time-derivative term is incorporated into the basic equation. It is often been observed in applications such as acoustics, viscoelasticity, structural vibration and seismic wave propagation, that the magnitude of the damping is frequency dependent and obeys a power law behavior.

A typical formulation involves operators of nonlocal type and these are usually based on fractional derivatives or fractional powers of differential operators. The aim is to explore these effects with particular emphasis on asking whether the inverse problems are more tractable (that is, in terms of ill-conditioning and convergence of numerical methods) for both types of damping.

In all cases, the analysis of iterative schemes to recover the unknown terms is an essential feature of the work.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.