6 Test Problems and Results

The Jacobi-like operator-splitting method described in the previous section is applied to a range of test problems in this section. To begin with the method is employed to solve the model diffusion problem (6) for Fourier components, then for a general function. However, the method is applicable to non-linear diffusion problems and results from the application of the method to such a problem are also given.

As explained in the previous section, the operating splitting methods require the numerical modelling of the forward operator K. For the test problems of this section the parabolic partial differential equations that represent the forward operator are discretised and solved using the Crank-Nicolson finite difference method.

6.1 Inverse Solution of Model Equation for Fourier Components

Figure 2 shows the convergence of the inversion method on the first eigenfunctions sin(px) of the model diffusion equation (6) where the total diffusion is such that the amplitude of is exactly halved (ie l = 0.5). Recall that in the method the first estimate of f is g.

Figure 3 shows the results from the application of the method to f(x)=[1/2] ( sin(px) + sin(3 px)). In this case g is not visibly distinguishable from the g in figure 2 once the factor [1/2] is taken into account. The results from the iteration show how the method initially pursues the solution of the first eigenfunction and then goes on to modify this with the third eigenfunction.

6.2 Inverse Solution of Model Equation for General Functions

In figure 4 the method is applied to the inverse solution of the model problem with the original f being a sin² function and the total diffusion being one tenth that in the tests of figures 2 and 3. In this case, as in the case of functions that are not finite eigenfunction expansions, convergence to the original f is not observed.

In figure 5 the method is applied to a problem where the original f is a higher frequency sin² function. The total diffusion in obtaining the g (=f⁽⁰⁾) is just one fortieth of the total diffusion applied in figures 1-4. Even so, as has been observed earlier, higher frequencies decay much faster in the forward operator and most of the characteristics of f are apparently lost in g. However, the iterative solutions in the inversion method demonstrates that the much of the quality of the original f can be recovered.

In figure 6 the inversion method is applied to a square wave. In this test problem the total diffusion is again one tenth of that applied in figures 2 and 3. The figure shows that the original f is discontinuous but g is a smooth function. As discussed earlier, the solution of the operator-splitting method mimics a truncated eigenfunction expansion and their is a similarity between the iterative solutions to those observed when fitting a square wave with a Fourier series.

6.3 Inverse Solution of Non-linear Diffusion Equation

In figures 7 and 8 the examples of figures 4 and 5 are repeated but this time with K a non-linear operator. In these cases D=D(u)=[3/2] - u. The same size and number of time steps are taken in these examples as are taken in figures 4 and 5 respectively. Hence the total diffusion in the corresponding examples is approximately the same.