Grantee Research Project Results

Uniqueness and Numerical Recovery of a Potential on the Real Line

EPA Grant Number: U914813
Title: Uniqueness and Numerical Recovery of a Potential on the Real Line
Investigators: Raschko-Mueller, Jennifer L.
Institution: University of Nebraska at Lincoln
EPA Project Officer: Hahn, Intaek
Project Period: January 1, 1995 through January 1, 1996
Project Amount: $102,000
RFA: STAR Graduate Fellowships (1995) RFA Text |  Recipients Lists
Research Category: Fellowship - Mathematics , Environmental Statistics , Academic Fellowships

Objective:

The objective of this research project is to consider the recovery of the potential q(x) in the singular problem:

-u"j(x) + q(x)uj(x) = fj(x)      -∞ < x < ∞.

Under suitable conditions, the coefficient is uniquely determined from the set of flux data at the origin {u1j(0)}, corresponding to the source terms {fj}, which constitute a basis for L²(R). To facilitate numerical recovery, the problem is formulated as an infinite-dimensional least-squares minimization problem. Tikhonov regularization is employed, and the resulting problem is discretized via sine collocation.

Approach:

Consider the inverse problem of determining the potential q in the singular differential equation:

-u"(x) + (q(x) – )u(x) = f(x)      -∞ < x < ∞

where u, f L²(R), and may be real or complex. The case where f(x) = 0 and the spectral data l for the Schrodinger operator Lu = -u"+ q(x)u is given, and is known as the inverse spectral problem. It has been widely studied on the finite interval, the half-line, and whole line. Our perspective stems from a somewhat different approach taken by Lowe and Rundell. These authors were interested in recovering a time-independent coefficient in a parabolic partial differential equation by imposing forcing functions f(x) on the system and observing steady-state flux at a boundary. Specifically, they showed that for the above second-order differential equation with Dirichlet boundary conditions, the boundary flux measurements {u'n(0)}^∞n = 1 induced by a family of forcing functions {fn}^∞n = 1 comprising a basis of L² uniquely determine an L² potential q. Moreover, they produced an algorithm for computing sufficiently small q. Related uniqueness results based on input sources can be found for parabolic and elliptic problems as well as the determination of multiple coefficients. This work is posed on an infinite interval, yielding a singular differential equation and new difficulties in numerical recovery. Also, the flux data is measured at one point in the interior, as opposed to boundary measurements. We will show that the potential q is uniquely determined by measurements of flux data {u'n (xo)}n=1 at an arbitrary fixed.

Supplemental Keywords:

fellowship, coefficient, flux data, source terms, numerical recovery, infinite-dimensional least-squares minimization., RFA, Scientific Discipline, Ecosystem Protection/Environmental Exposure & Risk, Mathematics, Monitoring/Modeling, Tikhonov regularization, infinite dimensional least squares minimization, numerical recovery

Progress and Final Reports:

Final