Extramural Research

##### 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:**Broadway, Virginia

**Project Period:**January 1, 1995 through January 1, 1996

**Project Amount:**$102,000

**RFA:**STAR Graduate Fellowships (1995)

**Research Category:**Fellowship - Mathematics , Environmental Statistics , Academic Fellowships

### Description:

**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 {u1*j*(0)}, corresponding to the source terms
{*fj*}, which constitute a basis for *L*^{2}(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* *L*^{2}(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 {*f*n}^{∞}n
= 1 comprising a basis of *L*^{2} uniquely
determine an *L*^{2} 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