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 2^{L}×2^{L}×2^{L}cells
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 2^{l}×2^{l}×2^{l} cells. Thus, the
coarsest gridG0 contains only one hexahedron and consequently has only 8 vertices.

A hexahedral grid cellE_{l} ∈ 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={x_{i}|i= 1, . . . , n_{l}}, where n_{l} = (2^{l}+ 1)^{3}.
Finally, the minimal edge length of the cells on grid levell is denoted by h_{l}.

Now, in order to perform a multi-level optimization, one starts on a coarse grid
of level l_{0} ∈ {0, . . . , L − 1} and optimizes the probe location on Gl0 yielding a
set of parameters ¯u_{l}_{0}. 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 ¯u_{L} 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

R_{l}^{l}_{+1} :V_{l}^{h}_{+1} →V_{l}^{h} forl = 0, . . . , L−1 ,

whereV_{l}^{h} 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 χ^{l}_{t}^{+1}, χ^{l}_{v}^{+1} ∈V_{l}^{h}_{+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 R_{l}^{l}_{+1} = (P_{l}^{l}_{+1})^{T} where P_{l}^{l}_{+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 restrictionR^{l}_{l}_{+1} to
the characteristic ﬁnite element functions χ^{l}_{t}^{+1} and χ^{l}_{v}^{+1}:

χ^{l}_{t} =R^{l}_{l}_{+1}χ^{l}_{t}^{+1} , χ^{l}_{v} =R^{l}_{l}_{+1}χ^{l}_{v}^{+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 usingχ^{l}

t

and χ^{l}_{v}.

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 probeD_{pr} on coarse grids is needed. If the probe is not
resolved on G_{l}, its radius has to be adapted to the resolution h_{l}. More precisely,
the probe radius has to be greater than or equal to h_{l}√

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-h_{l}
h^{l}

√ 2 Figure 3.13: Intersection of the probe with a slice of the grid orthogonal to the probe.

Here,h_{l} is the minimal voxel size of the grid
G_{l}, so the diameter of the probe must be at
leasth_{l}√

2.

+

++ + + ++

+ + + + + ++ + + + +

Iteration count lnf

−375

−370

−365

−360

−355

−350

−345

−340

−335

−330

−325

−320

Iteration count lnf

−375

−370

−365

−360

−355

−350

−345

−340

−335

−330

−325

−320

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 χ^{l}_{t} and χ^{l}_{v}.

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
χ^{l}

t(x_{j}) =

1 if

R^{l}_{l}_{+1}χ^{l}^{+1}

(x_{j})≥0.5 ,

0 else , (3.35)

and analog for χ^{l}_{v}. 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 u^{0} 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.