14 Numerical Methods for Coupled Population Balance Systems …
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with a scalar S
−,ν
agg as the result of the summation. The computation of (3.7) is of
complexity O(kN ) and results in a piecewise constant function (in the form of a
tensor with N entries) corresponding to the number of disappearing particles in each
cell.
3.2.2 Efficient Evaluation of a Discrete Convolution via Fourier
Transform
This section deals with the efficient evaluation of
Q
+
agg (j) =
j 1
k 1 =0
· · ·
j d int
k d int =0
φ j−k · ψ k ,
(3.8)
which is required in order to compute the source term F
+
agg (f , m) in (3.6). Since the
computation is analogous for all kernel factors, the index ν has been dropped.
It is well known that a discrete convolution (3.8) can be evaluated simultaneously
for all j using the multi-dimensional convolution theorem [50],
Q
+
agg = F
−1
(F(φ) F(ψ)),
(3.9)
where F and F
−1 denote the Fourier transform and its inverse and denotes the
elementwise (or Hadamard) product.
The result of a convolution of a tensor of size n × · · · × n is a tensor of size
2n × · · · × 2n with an index j ∈ {0, . . . , 2n − 1}
d int . However, one is only interested
in the n × · · · × n subtensor since all other entries go beyond the computational
domain (properties larger than m max ). In order to calculate this full convolution result
via a sequence of uni-variate Fourier transforms, the input tensors φ and ψ need to
be enlarged to this size by adding zeros. One then obtains tensors
φ,
ψ ∈ R
2n×···×2n
with entries
φ j ,
ψ j =
φ j , ψ j if j ∈ {0, . . . , n − 1}
d int ,
0
else,
(3.10)
in a process called zero-padding.
The multi-variate Fourier transform (3.9) is defined by
F : R
2n×···×2n
→ C
2n×···×2n
,
φ → F(
φ)
with
(3.11)
(F(
φ)) j :=
2n−1
s=0
φ s ·
d int
q=1
e
iπs q j q /n
.