5  Analysis of the Operator-Splitting Method for the Model Problem

In this section the properties of the operator-splitting method of the previous section are considered. In particular the convergence of the Jacobi-like method on the model problem is analysed.

5.1  Properties of General Method

Returning to the general operator-splitting method of section 4.1, since the operator B of the previous section is an approximate inverse of K then it is helpful to write
B K = P - Â
(14)
where Â:X ® Z can be regarded as the residual operator. It follows from (12) that
P f(n) = Â f(n-1) + B g
(15)
and that
P f(n) = ( Á + Â + Â2+ ... + Â(n-1)) Bg
(16)
where Á:Z ® Z is the identity operator. The series (16) is a Neumann series. The iterative method would always converge if ||Â|| < 1, in which case (15) would be a contraction mapping and K f=g would have a unique solution.
If the function g has been obtained through applying the operator K to a function f then substituting K f for g in (16) gives
P f(n) = ( Á + Â + Â2+ ... + Ân-1) BK f.
and hence that
P f(n) = ( Á + Â + Â2+ ... + Ân-1) (P-Â)f

= (P - Ân) f.
It follows that the error d(n) in the approximation to f at in iterate n is given by
P d(n) = P ( f - f(n) ) = Ân f
(17)
and the error e(n) in the corresponding approximation to g is
e(n) = g - g(n) = K P-1 Ân f.
(18)

5.2  Fourier Analysis of the Jacobi-like Method

Returning to the simple diffusion equation of section 3, recall that the eigenvalues are kN = e-N2 p2 T and hence they cluster and tend toward zero as N ® ¥. It follows that the corresponding rN=1-kN lie in [0,1] and tend toward unity as N ® ¥. Thus, as a result of this, Â is not a strict contraction mapping.
Let us now consider the the domain of the diffusion equation to be (x,t) Î [0,1] ×[0,T] with u(x,0)=f(x), u(x,T)=g(x) and u(0,t)=u(1,t)=0. Given any particular eigenfunction fN = sin(N px), it follows from (18) that the error in iterate n is
dN(n) = fN - fN(n) = rNn fN = (1-e-N2 p2 T)n sin(N px)
(19)
and
eN(n) = gN - gN(n) = kN rNn fN = e-N2 p2 T (1-e-N2 p2 T)n sin(N px)
(20)
where X=Y=C¥[0,1] and P is the identity operator. Since rN ® ¥ as N ® ¥ then (19) is a contraction when N is finite. In this case, the Jacobi-like method converges to the original f and the error function for iteration n is given by (19).
The eigenvalues of the model problem converge rapidly to zero. If l is the eigenvalue corresponding to N=1 then the complete set of eigenvalues form the sequence l, l4, l9, l16,..., and these correspond to the eigenfunctions sin(px), sin(2px), sin(3px), sin(4px),... , Hence if the first eigenvalue is 0.9 then the the simple diffusion equation (6) has the eigenvalues 0.9, 0.6561, 0.387420, 0.185302, 0.071790, 0.022528, 0.005726, 0.001179, 0.000197, 0.000026, ... to six decimal places, if the first eigenvalue is 0.5 then the eigenvalues are 0.5, 0.0625, 0.001953, 0.000015,.... Since rN = 1- kN then the sequence rN, N=1,2,...¥ converges quickly to unity with N. It follows that the error term converges much more slowly to zero for the higher eigenfunctions. For example if the eigenvalues are 0.5, 0.0625, 0.001953, 0.000015,... then the rn are 0.5, 0.9373, 0.998047, 0.999985,.... In order to return a relative error of less than 1% the number of iterations required is 6, 71, 2355, and 307009 for the first four eigenfunctions. Test problems demonstrating the convergence rates of the method on the eigenfunctions of the model problem are given in the next section.

5.3  Discussion

The analysis of the previous section suggests that progress towards the inverse solution of the first eigenfunction of K is relatively rapid, with the rate of convergence declining as we pass through the eigenfunctions. The slower rate of convergence of the higher eigenfunctions is a positive property of the method. Potentially, the iteration could be terminated before the higher eigenfunctions (ultimately the noise) affect the solution. For example the iteration can be terminated once iterate f(n) has the property that
||K f(n) - g|| < e
where e represents the uncertainty in g as a result of measurement or numerical error.
It is in this respect that the method is similar to the truncated eigenfunction expansion method described in section 2.3. However, in this method the involvement of the higher eigenfunctions is naturally delayed by the method, rather than being deliberately removed.