Reverse differentiation and the inverse diffusion problem
Section snippets
INTRODUCTION
The finite element method can be used to solve the direct diffusion photon transport problem in human tissues.[1] In that problem the optical properties of the tissue are assumed known and the outward flux on the boundary calculated for a given isotropic point light source. In the reverse problem measurements of the output flux are made and the problem is to calculate the values of the optical properties that predict outputs that most closely match the measurements. The reverse problem is posed
THE DIRECT PROBLEM
To solve the direct problem by the finite element method the region A is covered by a mesh of elements. We choose to use simple triangular elements in which all variables u, D and μ are assumed to be linear functions of their values at the nodes of the triangles. In the direct problem the values of q, D and μ are assumed known at each node and the values of u at each node required. The special form of the boundary condition Eq. (2)then determines the boundary flux at the boundary nodes.
The
THE INVERSE PROBLEM
The inverse problem is to be solved as an optimisation problem. The optimisation variables are the 2n unknowns D and μ at each node. To evaluate these, more than 2n items of data are required. In each experiment the flux D∂u/∂n is measured at each of the nB boundary nodes. If the measurements are taken for nQ different distributions of the light source q then we require If A is a square with nS nodes on each side then n = nS2 and nB = 4(nS − 1) so we may take For a more
THE GRADIENT CALCULATION
The gradient vector ΔE consists of 2n terms as it involves differentiating with respect to each of the unknowns D and μ at each of the n nodes. It is therefore essential that this be done efficiently. Examination of the steps of the function evaluation in Section 3indicates clearly that performing the Choleski decomposition once and solving all nQ sets of equations are roughly equal operations and dominate the operations in the other steps. Using numerical approximations by central differences
OPTIMISATION ALGORITHM
The optimisation problem Eq. (6)corresponding to the relatively coarse mesh in Example 2 already has 96 variables and 380 terms in the sum of squares. Both the number of variables and the number of terms will increase as the mesh is refined. Algorithms that require the storage of the Hessian or the Jacobian matrices or their approximations are therefore not appropriate. In these circumstances two popular algorithms would be a preconditioned conjugate gradient method and a trust region method
CONCLUSION
In this paper a very efficient method for calculating the gradient vector of the inverse diffusion problem has been presented based on an analytic form of the reverse differentiation method. It enables the full gradient vector to be obtained in fewer than three times the number of operations needed to evaluate the function without the storage overheads associated with the automatic approach.
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