4 Operator-Splitting Methods

In this section methods based on splitting the operator K are developed. Just as the Tikhonov method (5) is analagous to the minimum norm solution to a rank deficient matrix, operator-splitting methods are similar to the Jacobi and related methods for the iterative solution of linear systems of equations. The operator-splitting methods are also similar to the defect correction method [13] and is more commonly known as Landweber iteration, as considered in Chjapter 6 of Engl et al [5]..

4.1 General Derivation of Operator-Splitting Method

In general, the method is initiated by determining that the solution belongs to a particular space of functions Z Í X. Let the operator B:YŽ Z be a mapping that forms an approximation inverse to the operator K. Applying the operator B to both sides of (1) gives

B K f = B g .

(9)

Let us split the operator BK in (9), giving the following

{ P + ( BK - P ) } f = B g .

(10)

The P: X Ž Z in (10) is a projection operator such that P f^* Î Z is equivalent to the function f^* Î X and P is surjective.

The following iterative solution method follows from the operator-splitting:

P f⁽⁰⁾ Ź B g ,

(11)

P f⁽ⁿ⁾ Ź B g + (P - BK) f^(n-1) .

(12)

The right hand sides of the (12) can always be evaluated, provided the action of the operators B and K can be simulated. Since P is surjective its inverse exists and the solutions f⁽⁰⁾, f⁽¹⁾, ... can be obtained from (11)-(12). It should also be borne in mind that the operator K could be non-linear (as could the operator B), in such cases the operator(s) should exhibit their dependence on f.

In the method (12), a priori information about the nature of the solution is communicated to the successive approximations through the choice of the operator B and in particular its range Z. The selection of Z is crucial in determining the nature of inverse solution, ensuring that the iteration (12) returns a physically acceptable solution in a pre-determined form. It is through the specification of Z that the nearby problem is determined in the operator splitting method; the solution is not sought in its original space X, but in the subspace Z. This technique can include that of regularization by projection (see Kress [5], for example), in which Z is a space of finite-dimensional approximations.

4.2 Jacobi-like method

For most of the remainder of the paper, a particular example of the inversion method of the previous section will be investigated. The approximate inverse operator B is taken to be the identity operator Á:YŽ Y (i.e. Z = Y). It will be demonstrated that this approximate inverse leads to an effective inverse solution method.

Substituting Á for B in (11)-(12) gives rise to the following method

P f⁽⁰⁾ Ź g ,

P f⁽ⁿ⁾ Ź g + (P - K) f^(n-1) .

(13)

Since Á:YŽ Y then the successive solutions P f⁽ⁿ⁾, n=0,1,2,... all lie in the space Y, the same space as g. Recall from the discussion in subsection 3.3 that (ignoring the possibility of noise) the function g lies in a smoother space than the function f. Hence choosing B=Á biases the method toward returning a smooth solution.

In setting B = Á the method exploits the property that the diffusion operator does not in itself greatly affect the general trends in the function f; g is similar to f apart from the fact that it has been spread out and smoothed. In operator terms, choosing Á as the approximate inverse is equivalent to approximating the Gaussian-like kernel of the diffusion operator by the Dirac delta. For the simple diffusion equation model problem (6), the method effectively uses the approximate inverse

g(x) = K f =

ó
ő

1

0

exp{

-(x-y)²

4 T

} f(y) dy ť Á f(x) =

ó
ő

1

0

d(x-y) f(y) dy .

4.3 Comparison with Jacobi Iteration

The operator-splitting method with B = Á is analagous to the Jabobi method for the solution of a linear system of equations. Reconsider the discrete form of the model problem (8). Let us split the matrix K into its diagonal and non-diagonal components so that

K = D + L + U

where D is a diagonal matrix, L is lower-triangular and U is upper-triangular. Provided the diagonal elements are non-zero (as they must be) then by dividing each row of K by its diagonal term on that row the matrix D can be replaced by the identity matrix I without loss of generality. The Jabobi method then involves the iteration

D f⁽ⁿ⁾ = g - (L+ U) f^(n-1) .

Hence the application of the Jacobi method to the linear system of equations arising from the solution of the operator equation is equivalent to the operator-splitting method with B = Á.