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Uniqueness and Numerical Recovery of a Potential on the Real Line
EPA Grant Number: U914813Title: 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: Broadway, Virginia
Project Period: January 1, 1995 through January 1, 1996
Project Amount: $102,000
RFA: STAR Graduate Fellowships (1995)
Research Category: Fellowship - Mathematics , Academic Fellowships , Environmental Statistics
Description:
Objective:The objective of this research project is to consider the recovery of the potential q(x) in the singular problem:
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) = 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.
fellowship, coefficient, flux data, source terms, numerical recovery, infinite-dimensional least-squares minimization. , Ecosystem Protection/Environmental Exposure & Risk, Scientific Discipline, RFA, Mathematics, Monitoring/Modeling, infinite dimensional least squares minimization, coefficient, numerical recovery