Research on Inverse Problems for Differential Equations
The general nature of an inverse problem is to deduce a cause from an
effect. Consider a physical system, depending on a collection of
parameters, in which one can speak of inputs to the system and outputs
from the system. If all of the parameters were known perfectly then for
a given input we could predict the output. It may happen, however, that
some of the parameters characterizing the system are not known, being
inaccessible to direct measurement. If it is important to know what
these parameters are, in order that the system be understood as
completely as possible, we might try to infer them by observing the
outputs from the system corresponding to special inputs. Thus we seek
the cause (the system parameters) given the effect (the output of the
system for a given input).
An important example is the inverse problem of geophysics, in
which we seek to investigate the structure of the interior of the
earth.
Elastic waves may propagate through the earth in a manner which depends
on the material properties of the earth. A concentrated source of
energy at the surface of
causes waves to penetrate into the earth which are then partially
reflected
back to the surface. If the material properties of the earth's interior
were known completely then we could predict the nature of the reflected
wave from knowledge of the source. Since in fact we cannot measure
these
properties directly we seek to infer them by observing the reflected
waves in response to a collection of known sources.
In formulating such problems mathematically, we typically find that
the
problem amounts to that of determining one or more coefficients in a
differential equation, or system of differential equations, given
partial knowledge of certain special solutions of the equation(s). In
the seismology problem just discussed, the propagation of waves in the
earth
is governed by the equations of elasticity, a system of partial
differential equations in which the material properties of the earth
manifest themselves as coefficient functions in the equations. The
measurements we can make amount to the knowledge of special solutions
of the equations at special points, e.g. those points on the surface of
the earth in this example.
Inverse problems in differential equations have this general
character.
One has a certain definite kind of differential equation (or system of
equations) containing one or more unknown (or partially known)
coefficient functions. From some limited knowledge about certain
special
solutions of the equations we seek to determine the unknown coefficient
functions. Problems of this type arise in a variety of important
applications areas, such as geophysics, optics, quantum mechanics,
astronomy, medical imaging and materials testing.
Research activities of P. Sacks on inverse problems
- Inverse problems for acoustic media in N-dimensional
space. (Items 9,10,16,21,25,27,28,37,44,50,51 in publication list)
The governing differential equation here is the acoustic wave
equation, a single hyperbolic equation. This work includes uniqueness
theorems and analysis of various operators relating coefficients to
data.
- Inverse problems for layered acoustic media. (Items
14,20, 43)
Here it is assumed that the material properties characterizing the
acoustic medium vary in only one direction.
- Inverse problems for layered elastic media. (Items
11,13,15,29,49) The governing equations are the equations of linear
elasticity, a hyperbolic system. In comparison to the acoustic case,
there is more than one wave speed. Current projects include extension
of
this work to some cases of anisotropic elasticity.
- Computational methods for classical inverse scattering
problem of
quantum mechanics and the inverse Sturm-Liouville problem. (Items
17,32,33,35,36,40,54,55,56) `Time domain' methods are applied to
develop new
numerical methods for `frequency domain' problems.
- Phase identification problems. (Items
34,38,41,42,45,46,48,52,53) This is an auxiliary problem arising in a
number of inverse scattering problems: The scattering data is
naturally described in terms of a complex valued function, but only
the amplitude is available for measurement. Thus we need to infer
phase information in some indirect way, and/or analyze to what extent
the inverse problem can be solved using only the amplitude
information.
Inverse problems at Iowa State University
Research of a more applied nature on inverse problems may be found in
several other places at ISU. See for example