The minimization of the objective function can be accelerated significantly by per-forming pre-optimization runs on coarse grids and taking the optimal probe location of each pre-optimization as the start positioning for the optimization on the next finer grid.
In more detail, to be able to provide such pre-optimizations, we first have to assume that we work on an octree grid, i. e. a hexahedral grid with 2L×2L×2Lcells for some L ∈ N. Note that this is in fact no restriction, since every Cartesian grid can be embedded into an octree grid. Because the number of grid cells in each coordinate direction is a power of two, we implicitly have a hierarchy of grids GL ⊂ · · · ⊂ G0 such that for 0≤l ≤L each grid Gl has 2l×2l×2l cells. Thus, the coarsest gridG0 contains only one hexahedron and consequently has only 8 vertices.
A hexahedral grid cellEl ∈ Gl is split into 8 child elements in the next grid of finer resolutionGl+1. In the following, the numberl∈ {0, . . . , L}is referred to as thelevel of the gridGl. Moreover, the vertices of the grids are ordered lexicographically, and the set of vertices ofGl is denoted by Nl={xi|i= 1, . . . , nl}, where nl = (2l+ 1)3. Finally, the minimal edge length of the cells on grid levell is denoted by hl.
Now, in order to perform a multi-level optimization, one starts on a coarse grid of level l0 ∈ {0, . . . , L − 1} and optimizes the probe location on Gl0 yielding a set of parameters ¯ul0. Then, this parameter-set is used as the initial guess for an optimization on the following finer grid Gl0+1. This has to be continued until the finest grid GL is reached and an optimal set ¯uL has been obtained.
This approach provides the possibility to descent much faster in the energy profile
of the objective function. Moreover, one can expect that with decreasing levell in the grid hierarchy the energy profile is regularized and existing local minima are smoothed. Thus, already on coarse levels the probe location and orientation are led to a dominant minimum and the algorithm is not likely to end up in a suboptimal local minimum. Finally, as already mentioned above, the multi-scale approach reduces the computational time by minimizing the number of iteration steps on the finest grid.
Obviously the multi-scale method requires the definition of the geometrical shape of the tumor and the vascular system on coarser grids. To be more precise, we need a restriction
Rll+1 :Vlh+1 →Vlh forl = 0, . . . , L−1 ,
whereVlh is the finite element space of globally continuous, piecewise trilinear func-tions on the gridGl(cf. Sect. 2.4.2). This means we need a restriction that transports the characteristic finite element functions χlt+1, χlv+1 ∈Vlh+1 (cf. Sect. 3.3.1), on the next coarser level. A straightforward approach would use the trivial restriction which directly multiplies the nodal values of a fine-grid function with the coarse-grid basis function. However, this leads to unsatisfactory results, since on coarse grids it does not preserve details of the tumor that have important influence on the choice of the optimal probe positioning. Therefore, it is much more advisable to use the classical restriction of multigrid approaches for the solution of PDEs [42]. Let us set Rll+1 = (Pll+1)T where Pll+1 is the trilinear interpolation from grid Gl to gridGl+1
involving the weights {1/8,1/4,1/2,1}. Then one can apply the restrictionRll+1 to the characteristic finite element functions χlt+1 and χlv+1:
χlt =Rll+1χlt+1 , χlv =Rll+1χlv+1 . (3.34) This restriction accumulates the nodal values of fine-grid points into the coarse-grid nodes such that important details cannot be lost. In fact, it is mass conserving and leads to fuzzy boundaries of the tumor and vessel. So on coarse grids we encounter grid cells whose nodal values lie in the interval [0,1]. We can interpret the nodal values as the proportions of the coarse grid cells which lie inside the tumor or vessel.
Still the quadrature presented in Sect. 3.3.1 can be applied on coarse levels usingχl
t
and χlv.
Depending on the elements’ size on coarse levels, it can happen that the probe radius is too small to be resolved on a coarse grid. Therefore, a special construction of the approximation of the probeDpr on coarse grids is needed. If the probe is not resolved on Gl, its radius has to be adapted to the resolution hl. More precisely, the probe radius has to be greater than or equal to hl√
2/2. This adaptation of the radius guarantees that each slice of the probe contains at least one grid node (see Fig. 3.13).
In the left graph of Fig. 3.14, the progression of the value of the objective func-tion (3.10) during the optimizafunc-tion for a configurafunc-tion shown in Sect. 3.8 (cf. Fig. 3.16) is depicted.13 The transitions between different grid levels are marked by vertical
13Remember, that using the objective function (3.10) in fact means to use the numerical
modifica-hl hl
√ 2 Figure 3.13: Intersection of the probe with a slice of the grid orthogonal to the probe.
Here,hl is the minimal voxel size of the grid Gl, so the diameter of the probe must be at leasthl√
2.
+
++ + + ++
+ + + + + ++ + + + +
Iteration count lnf
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Iteration count lnf
−375
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Figure 3.14:Left: The progression of the objective function value is shown for the multi-scale optimization algorithm (•) and the standard algorithm (+). In contrast to the standard algorithm, the multi-scale optimization finds a slightly better minimum with a considerably smaller number of iterations on the finest-grid. Right: The progression of the objective function value is shown for the definition of a coarse-grid tumor and vascular domains which involve a thresholding.
dotted lines in the energy plot. At these stages of the algorithm, the probe placement u∈ U thas been found on a coarse level is re-interpreted on the finer grid. Conse-quently, the energy jumps at the transition points in the graph. The progression of the energy during the iterations of the standard algorithm is shown in the left graph of Fig. 3.14 as well. The minimum found by the multi-scale algorithm is slightly bet-ter (lnf ≈ −356.9) than the one found by the standard algorithm (lnf ≈ −356.3).
Moreover, the multi-scale optimization needs significantly less fine-grid steps than the standard algorithm.
In the numerical examples presented in Sect. 3.8, a further speedup of the multi-level algorithm is obtained by thresholding the restricted values of χlt and χlv.
tion (3.11), which for a comparison of different objective function values has to be transformed as described in (3.12). Therefore, in Fig. 3.14 the logarithm lnf of the objective function f is shown.
Thereby, it is set χl
t(xj) =
1 if
Rll+1χl+1
(xj)≥0.5 ,
0 else , (3.35)
and analog for χlv. Here, the speedup of the algorithm can be explained by the following considerations: Without a thresholding, for all voxels that belong to the tumor, too low temperatures would be penalized approximately equal (due to the deep slope of the exponential function), even if the respective voxels belong to the tumor only by a minor portion. Hence, for a pre-optimization on a coarse grid, the active tumor region that influences the objective function value would be sig-nificantly larger than the real tumor region which influences the objective function value in the main optimization. Thus, the result of a pre-optimization without thresholding the tumor would yield a worse initial value for the main optimization than the result of a pre-optimization that uses a thresholding. The progression of the corresponding objective function value is shown in the right graph of Fig. 3.14.
Since here the masses of the tumor and the vessels are not conserved on the coarse levels, the values of the objective function increase at the transition stages.
The accelerated algorithm stops already after four fine-grid iterations and from the graphs we see that the minimal value of the objective function is only slightly larger (lnf ≈ −356.6) than the one obtained by the non-thresholded restriction variant. For the configuration shown in Fig. 3.16, the optimal probe positions p of both multi-scale variants differ by at most 0.5 mm and the orientations differ by at most 5◦. Since this lies below the accuracy that can be achieved in practice, here the faster algorithm is preferable.
3.7.1 The Multi-Level Optimization Algorithm
In Alg. 3.2 the extension of the basic Alg. 3.1 (see Sect. 3.4.4) for the optimization of the RF probe placement to a multi-level approach as described above is depicted.
For each level l (see lines 3–10 of Alg. 3.2), the optimization is performed as seen in Sect. 3.4.4, but with the difference that the initial start positioning u0 is set to the optimal solution ¯u of the previous optimization. Thus, it can be expected that the algorithm needs only very few iteration steps on the finest grid, since therefore the initial start positioning can be expected to be already very close to the optimal solution.