Many physical phenomena such as fluid flow, heat conduction and
semiconductor growth include a diffusion process. The action of the
diffusion is a continuous process that has an increasing effect with
time and typically manifests itself as a spreading and smoothing of the
original distribution. Let the full effect of a diffusion
process be written in operator notation as follows:
K f = g
(1)
where f is the original distribution, K: X ® Y
represents the operation of diffusion over the time interval
and g is the final distribution.
Physical problems that include diffusion processes, such as those stated,
can often be modelled by a parabolic partial differential equation of
the following general form
¶u(x,t)
¶t
= D Ñ2 u(x,t) + firstdirectionalderivativesof u.
(2)
The first term on the right hand side of the equation is the diffusion
term and the coefficient D ( ³ 0) represents the rate of diffusion.
Note that D need not be constant,
it may for example be a function of position or
vary with u as in material-dependent
diffusion, the latter resulting in a non-linear equation: K = K(f).
Often the first term is written Ñ.(D Ñu), but by product
differentiation the equation would again take the form of (2).
In the terminology of equation (1), u(x,0) = f(x) and u(x,T) = g(x).
In general, equations of the form (2)
that model physical phenomena cannot be solved analytically.
Numerical methods such as finite difference and finite element methods have
been developed for their solution, simulating the relevant physical
process through stepping forward in time t and re-computing
the function u at the each sucessive time level ( see ref. [10],
[12], for example).
The inverse problem of solving diffusion equations backwards in time
- determining the initial distribution f from final distribution
g - is fundamental to much scientific enquiry. The classical
example of inverse diffusion problems is the backward heat conduction
problem in which the original temperature distribution in
a material is to be determined from the distribution after an
elapsed time. Such a problem is considred in Beck [1], although this book is more concerned
with the inverse heat conduction problem, the estimation of the surface flux history of a heat conducting biody.
The authors' interest in the inverse diffusion problem arises from their
work in atomic mixing modelling [7],[8],[14] and the inverse problem of recovering the
original material distribution [6].
In the forward solution of parabolic equations, it is generally
found that the presence of the diffusion term (the first term on the
right hand side of (2)) improves the stability
properties of the numerical method as it tends to damp away the
numerical oscillation that may otherwise arise.
However, in the backward solution of equations of the form
(2), it is the diffusion term that causes the
difficulty: simply employing the finite difference or finite element
methods with negative time steps is numerically unstable.
The other terms on the right hand side of (2)
simply translate the solution and their effects are easily reversible.
Inverse diffusion problems are examples of ill-posed problems.
Such problems - where the conditions placed on an equation are
not classical and do not determine a unique solution - are
also termed Cauchy problems (see Hadamard [4]).
For diffusion problems,
the smoothing that is inherent in the diffusion process results in
a loss of resolution in the details of the original distribution.
The inverse problem effectively suffers from insufficient
information in g to determine a unique f.
A further consideration is that in all practical applications
the input to the inverse problem g will generally be measured
data. It follows that g will be affected by noise
and g could then easily fall outside of the set Y,
the range of K, and there
will then be no f such that K f=g, f Î X.
The inversion method can only succeed if the original inverse
problem - which will generally have no unique solution -
is replaced by
one that always has a unique solution. The
substitution of a nearby problem for the original
inverse problem is often termed
(or is equivalent to) the regularization of the operator
K.
The inversion method must converge to the solution
of the nearby problem.
In section 2 the general properties of the inverse problem and
approaches to its solution are considered.
The purpose of this paper is to introduce an operator-splitting
or defect correction
method for the iterative solution of
inverse diffusion problems.
The convergence of the resultant
method is analysed, particular in regard to observing Fourier
solutions of
the one-dimensional simple diffusion equation.
The operator-splitting methods are applicable to general
non-linear diffusion
problems. Results from the application of the method
to Fourier components of the simple diffusion equation to illustrate
the points of the analysis. The method is also applied
to problems arising from selecting a general original f
on the simple diffusion equation and it is applied to
a typical non-linear diffusion equation to illustrate
the usefulness of the method.
An Excel spreadsheet demonstration of the inversion method can be downloaded
free from the www.scientific-computing.info web site.