Convergence of the discrete dipole approximation. II. An extrapolation
  technique to increase the accuracy
Microsoft Word - DDAextrapolation_preprint.doc
Convergence of the discrete dipole approximation.  
II. An extrapolation technique to increase the accuracy. 
Maxim A. Yurkin 
Faculty of Science, Section Computational Science, of the University of Amsterdam, 
Kruislaan 403, 1098 SJ, Amsterdam, The Netherlands 
Institute of Chemical Kinetics and Combustion, Siberian Branch of the Russian Academy of 
Sciences, Institutskaya 3, Novosibirsk 630090 Russia 
myurkin@science.uva.nl
Valeri P. Maltsev 
Institute of Chemical Kinetics and Combustion, Siberian Branch of the Russian Academy of 
Sciences, Institutskaya 3, Novosibirsk 630090 Russia 
Novosibirsk State University, Pirogova Str. 2, 630090, Novosibirsk, Russia 
Alfons G. Hoekstra 
Faculty of Science, Section Computational Science, of the University of Amsterdam, 
Kruislaan 403, 1098 SJ, Amsterdam, The Netherlands 
alfons@sciene.uva.nl
Abstract 
We propose an extrapolation technique that allows accuracy improvement of the 
discrete dipole approximation computations. The performance of this technique was 
studied empirically based on extensive simulations for 5 test cases using many different 
discretizations. The quality of the extrapolation improves with refining discretization 
reaching extraordinary performance especially for cubically shaped particles. A two 
order of magnitude decrease of error was demonstrated. We also propose estimates of 
the extrapolation error, which were proven to be reliable. Finally we propose a simple 
method to directly separate shape and discretization errors and illustrated this for one 
test case. 
Keywords: discrete dipole approximation, non-spherical particle light scattering, accuracy, 
extrapolation 
OCIS code: 290.5850, 260.2110, 000.4430 
mailto:myurkin@science.uva.nl
mailto:alfons@sciene.uva.nl
1. Introduction 
The discrete dipole approximation (DDA) is a well-known method to solve the light 
scattering problem for arbitrary shaped particles. Since its introduction by Purcell and 
Pennypacker1 it has been improved constantly. The formulation of DDA summarized by 
Draine and Flatau2 more than 10 years ago is still most widely used for different applications,3 
partly due to the publicly available high-quality and user-friendly code DDSCAT.4 
DDA directly discretizes the volume of the scatterer and hence is applicable to arbitrary 
shaped particles. However, the drawback of this discretization is the extreme computational 
complexity of DDA of O(N 2), where N is the number of dipoles. This complexity is 
decreased to O(NlogN) by advanced numerical techniques.2,5 Still the usual application 
strategy for DDA is “single computation”, where a discretization is chosen based on available 
computational resources and some empirical estimates of the expected errors.3,4 These error 
estimates are based on a limited number of benchmark calculations3 and hence are external to 
the light scattering problem under investigation. Such error estimates have evident drawbacks, 
however no better alternative is available. 
Usually errors in DDA are studied as a function of the size parameter of the scatterer x 
(at a constant or few different values of N), e.g. 2,6. Only several papers directly present errors 
versus discretization parameter (e.g. d – the size of a single dipole).7-15 The range of d 
typically studied in those papers is limited to a 5 times difference between minimum and 
maximum values (with the exception of two papers9,10 where it is 15 times). Only two 
papers7,15 use extrapolation (to zero d) to get an exact result of some measured quantity, 
however they use the simplest linear extrapolation without any theoretical foundation nor 
discussion of its capabilities. 
It is acknowledged for a long time that DDA errors are due to two different factors: 
shape (it is not always possible to describe the particle shape exactly by a collection of 
cubical cells) and discretization (finite size of each cell).6 However, the question which of 
them is more important in different cases is still open. A discussion on this issue spanned 
through several papers16-20 that have not reached any definite conclusions yet. The uncertainty 
is due to the indirect methods used that have inherent interpretation problems. 
In accompanying paper,21 that from now on we will refer to as Paper 1, we performed a 
theoretical analysis of DDA convergence when refining the discretization. It provides the 
basis for this paper, where an extrapolation technique is introduced (Section 2) to improve the 
accuracy of DDA computations. We thoroughly discuss all free parameters that influence 
extrapolation performance and provide a step-by-step prescription, which can be used with 
any existing DDA code without any modifications. It is important to note that although 
Paper 1 provides a firm theoretical background, it is not necessary to go through all 
theoretical details to understand and apply the extrapolation technique that we introduce here. 
In Section 3 we present extensive numerical results of DDA computations for 5 different 
scatterers using many different discretizations. These results are discussed in Section 4 to 
evaluate the performance of the extrapolation technique. We also propose a new method to 
directly separate shape and discretization errors of DDA (described and illustrated in 
Section 3.B). The results and possible applications are discussed in Section 4. We formulate 
the conclusions of the paper in Section 5. 
2. Extrapolation 
In this section we describe a straightforward technique to significantly increase the accuracy 
of a DDA simulation with a relatively small increase of computation time. This technique 
does not require any modification of a DDA program but only postprocessing of computed 
data. Therefore it can be easily implemented in any existing DDA code. 
In Paper 1 we have proven that the error of any measured quantity is bounded by a 
quadratic function of the discretization parameter mkdy =  (k – free space wave vector, m – 
refractive index of the scatterer): 
( ) ( )yybayybay lnlnδ 11222 φφφφφ −+−≤ , (1)
where φ y is some measured quantity (e.g. extinction efficiency Qext, Mueller matrix elements 
at some scattering angle Sij(θ ), etc.) and δφ y its error (difference between a result of the 
numerical simulation and an exact value). ,  are constants (independent on y), which 
are described in detail in Paper 1. 
Here we proceed and assume that for sufficiently small y, δφ y can in fact be 
approximated by a quadratic function of y (taking the logarithmic term as a constant). The 
applicability of this assumption will be tested empirically in Section 3.B. Introduction of 
higher-order terms is possible but not necessary (contrary to the quadratic term), and we avoid 
it in order to keep our technique as simple and robust as possible. We can now write: 
yy yayaa ζφ +++= 2210 , (2)
where a0, a1, a2 are constants that are chosen such that ζ y – the error of the approximation – is 
minimized. a0 is then an estimate for the exact value of the measured quantity φ 0. A 
procedure to determine a0 is basically fitting of a quadratic function over several points 
, which are obtained by a standard DDA simulation. In the ideal case of  one 
can use any three values of y to obtain the exact value of φ
},{ yy φ 0=yζ
 0. However, in practice different 
fits will always give different results. We limit ourselves to the usual least-square polynomial 
fit of the data. There are three question one should answer before conducting such a fit: 
1) how many and which values of y to use? 
2) how to weight the influence of different calculated values used in the fitting, i.e. what is 
the behavior of expected errors ζ y? (Note that in the polynomial fit we minimize χ2, the 
summation of the squared difference between computed values and the fitting function 
weighted by the inverse of the expected error ζ y.) 
3) how to estimate the difference between a0 and φ 0, i.e. the error of the final result? 
It is important to note that, although there are some theoretical hints, answers to these 
questions are mainly empirical and should be tested. Our approach is based on the test cases 
presented in Section 3.B. These may not be representative for all scattering problems, but they 
do show the potential power of our approach. We do not attempt to choose the most suitable 
fit options, but merely demonstrate the applicability of the technique. 
We start by analyzing the second question, i.e. what is the expected deviation from the 
quadratic model, i.e. what is the functional dependence of ζ y on y, to be used as weighting 
function in the polynomial fitting procedure? For cubically shaped particles, defined in 
Paper 1 as particles whose shape can be exactly discretized using cubical subvolumes, one 
expects a smooth variation of the function φ y, and the error can be attributed as a model error, 
i.e. coming mainly from neglecting higher order terms in the convergence analysis of Paper 1. 
In that case the error ζ y is expected to be a cubical function of y. We have tried cubical, 
quadratic and linear error functions when fitting results for cubically shaped particles and 
found that, although the differences are small, cubical errors generally lead to the best fits 
(data not shown). 
Shape errors, which are present for non-cubically shaped particles, are expected to be 
very sensitive to y, because they depend upon the position of the particle surface inside the 
boundary dipole that changes considerably by a small variation of y (for details see Paper 1). 
Therefore shape errors can be viewed as random noise superimposed upon a smooth variation 
of φ y. The asymptotic behavior of shape errors is linear in y (see Paper 1). Indeed, in certain 
cases we found that using linear errors ζ y results in significantly better fits than when using 
cubical errors. However in other cases linear errors performed significantly worse. In our 
experience, using a cubical error function is in general always more reliable, even in the 
presence of shape errors, because it decreases the influence of points with high values of y, 
where the error is larger and less predictable. Since we want the procedure to be as robust as 
possible and not to use more complex error functions than strictly needed (e.g. polynomial), 
we take a cubical dependence of the error ζ y, both for cubically- and non-cubically shaped 
scatterers. 
The choice of values of y for computation can be described by the interval [ymin,ymax], 
the number of points and their spacing. ymin is usually determined by available computer 
hardware (time or memory bounds), that is the best discretization that can be computed for a 
given resource. The goal of the extrapolation procedure is to increase the accuracy beyond 
this “single DDA boundary”. We will show in Section 3.B that the overall performance of this 
technique strongly depends upon ymin. 
The choice of ymax is governed by two notions: a larger interval of data points generally 
leads to better extrapolation but errors for high values of y are more random and their 
significance is anyway much smaller (since we use a cubical error function). We have found 
that for cubically shaped scatterers a good choice is minmax 2yy = , while for non-cubically 
shaped scatterers increasing the interval to minmax 4yy =  does improve the fits. Probably that 
is due to the fact that the quality of fit for non-cubically shaped scatterers is determined by 
quasi-random shape errors and increasing the range leads to larger statistical significance of 
the result. We will also demand that ymax is less than 1, since otherwise DDA is definitely far 
from its asymptotic behavior. 
Spacing of the sample points depends partly on the problem, especially for cubically 
shaped scatterers (in that case an arbitrary number of dipoles cannot be used). We space 
computational points approximately uniform on a logarithmic scale, acknowledging the fact 
that a relative difference in y is more significant than an absolute. The total number of points 
should be large enough for statistical significance. However, a large number of points 
increases computational time. We have used 5 points for cubically shaped particles (ratio of 
y1  values is 8:7:6:5:4) and 9 points for non-cubically shaped particles (ratio of y1  values is 
16:14:12:10:8:7:6:5:4) or less if minmax 4yy < . 
The estimation of the error of the final result is difficult since this error is due to model 
imperfection and not to some kind of random noise. The standard least-square fitting 
technique22 provides a standard error (SE) for the parameter a0, which we use as a starting 
point. Numerical simulations (Section 3.B) show that for spheres (the only non-cubical shape 
we studied) real errors are less than 2×SE in most cases. That is what one would expect if ζ y 
is considered completely random (which is similar to the expected behavior of the shape 
errors). For cubical shapes, on the contrary, we have to estimate the error as 10×SE to reliably 
describe the real errors. It is important to note that an error estimate based on the SE is the 
simplest one can use. Its drawback is that we have to use a large multiplier (based on the real 
errors obtained in some of our simulations), which may lead to significant overestimation of 
real errors in certain cases. 
We can now formulate the step-by-step extrapolation technique. We use abbreviations 
(c) and (nc) for cubically and non-cubically shaped scatterers respectively. 
1) Select ymin based on your computational resources. 
2) Take ymax to be 2 (c) or 4 (nc) times ymin but not larger than 1. 
3) Choose 5 (c) or 9 (nc) points over the interval [ymin,ymax] approximately uniformly 
spaced on a logarithmic scale. 
4) Perform DDA computations for each y. 
5) Fit the quadratic function (Eq. (2)) over the points  using y},{ yy φ 3 as errors of data 
points; a0 is then the estimate of φ 0. 
Multiply SE of a0 by 10 (c) or 2 (nc) to obtain an estimate of the extrapolation error. 
Results of using this procedure are presented in Section 3, together with computational costs. 
The extrapolation procedure is similar to a Romberg integration method,22 which is 
adaptive. The error estimate, obtained by extrapolation, is an internal accuracy indicator of 
DDA computations that is just as important as the increase in the accuracy itself. Our error 
estimate opens the way to adaptive DDA, i.e. a code that will reach a required accuracy, using 
minimum computational resources. 
3. Numerical simulations 
A.Discrete Dipole Approximation 
The basics of the DDA method were summarized by Draine and Flatau.2 In this paper we use 
the LDR prescription for dipole polarizability,23 which is most widely used nowadays, e.g. in 
the publicly available code DDSCAT 6.1.4 We also employ dipole size correction6 for non-
cubically shaped scatterers to ensure that the cubical approximation of the scatterer has the 
correct volume; this is believed to diminish shape errors, especially for small scatterers.2 We 
use a standard discretization scheme without any improvements for boundary dipoles. 
The main numerical challenge of DDA is to solve a large system of 3N linear equations. 
This is done iteratively using some Krylov-subspace method,22 while the matrix-vector 
products are computed using an FFT-based algorithm.5 Our code – Amsterdam DDA 
(ADDA) – is capable of running on a cluster of computers (parallelizing a single DDA 
computation), which allows us to use practically an unlimited number of dipoles, since we are 
not limited by the memory of a single computer.24,25 We used a relative error of residual 
 as a stopping criterion. Tests suggest that the relative error of the measured quantities 
due to the iterative solver is then  (data not shown) and hence can be neglected (total 
relative errors in our simulations are  – see Section 
810−<
710−<
56 1010 −− ÷> 3.B). All DDA simulations 
were carried out on the Dutch national compute cluster LISA.26 
The execution time of one iteration depends solely on N, it consists of an arithmetic part 
which scales linearly with N and an FFT part which scales as NlnN. The number of iterations 
only slightly depends on the discretization parameter y for fixed geometry of the scatterer. 
Rahola proved this theoretically for any Krylov-subspace method,27 and our own experience 
agrees with this conclusion. Therefore the total computational time scales linearly with N 
( ) or slightly faster (considering logarithm and imperfect optimization), which is 
consistent with our timing results (data not shown). 
3−∝ y
We can now estimate the computational overhead of the extrapolation technique 
compared to a single DDA computation for ymin (time – t(ymin)). Considering the spacing of 
points we used (described in Section 2) the execution time needed for 5 points computation is 
 and for the 9 points computation – )(5.2 min5 ytt < )(7.2 min9 ytt < . Memory requirements are 
the same as for a single computation. For comparison one should note that an 8 times increase 
in computational time and memory requirements (for single DDA computation with 
Fig. 1. Cubical discretization of a sphere using 16 dipoles per diameter (total 2176 dipoles). 
2minyy = ) gives only a 2 to 4 times increase in accuracy (depending in which error regime – 
linear or quadratic – ymin is located). 
B.Results 
We study five test cases: one cube with 8=kD , three spheres with , and a 
particle obtained by a cubical discretization of the 
30,10,3=kD
10=kD  sphere using 16 dipoles per D 
(total 2176 dipoles, see Fig. 1, x equal to that of a sphere). By D we denote the diameter of a 
sphere or the edge size of a cube. All scatterers are homogenous with . Although 
DDA errors significantly depend on m (see e.g. 
5.1=m
12), we limit ourselves to one single value and 
study the effects of size and shape of the scatterer. 
The maximum number of dipoles per D (nD) was 256. The values of nD that we used are 
of the form  (p is an integer), except for the discretized sphere, where all np2}7,6,5,4{ ⋅ D are 
multiples of 16 (this is required to exactly describe the shape of the particle composed from a 
number of cubes – see Fig. 1). The minimum values for nD were 8 for the  sphere, 16 
for the cube, the  sphere, and the discretized sphere, and 40 for the  sphere. 
10=kD 30=kD
Typical computation time for the finest discretization (for the cube with , 
resulting in ) currently is 2.5 hours on a cluster of 64 P4-3.4 GHz processors. We 
expect that it can be improved by an order of magnitude by using modern FFT routines (e.g. 
fastest Fourier transform in the West – FFTW
047.0=y
7107.1 ⋅=N
28) and a faster iterative solvers (bi-conjugate 
gradient stabilized or quasi-minimal residual that were shown to be clearly superior to 
CGNR29,30 that we still use). We are currently improving our code along these lines. 
All computations use a direction of incidence parallel to one of the principal axes of the 
cubical dipoles. The scattering plane is parallel to one of the faces of the cubical dipoles. In 
this paper we show results only for the extinction efficiency Qext (for incident light polarized 
parallel to one of the principal axes of the cubical dipoles) and phase function S11(θ ) as the 
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.
most commonly used in applications. However, the extrapolation technique is equally 
applicable to any measured quantity. For instance, we have also applied it for other Mueller 
matrix elements (data not shown). 
Reference (exact) results of S11(θ ) and Qext for spheres are obtained by Mie theory (the 
relative accuracy of the code we use31 is at least ). Unfortunately, no analytical theory 
is available for the cube and the discretized sphere, which could provide us with exact results. 
Instead, we use extrapolation over the 5 finest discretizations as reference results for these 
shapes. 
610−<
To justify this choice we discuss, as an example, simulation results of Qext for the cube. 
Instead of showing values of Qext itself, we show in Fig. 2a ( )10ext −aQ , with a0 obtained 
through fitting the 5 finest discretizations. The extrapolation through these 5 best points 
( , ) is also shown. The deviation of the fit from the five best points 
(that overlap on 
047.0min =y 094.0max =y
Fig. 2(a)) is very small indeed. This is also characterized by a small estimate 
of the extrapolation error  (see 6108.1 −× Table 1). In Paper 1 we proved that DDA converges 
to the exact solution, therefore the result of the best extrapolation should be close to the exact 
result. The relative difference between the best discretization and the best extrapolation is 
only , therefore it does not make a big difference which one to use as a reference 
when evaluating, for instance, the error of the extrapolation through the 5 worst 
5100.9 −×
 cube kD=8
 fit (5 best points)
 discretized sphere kD=10
 fit (5 best points)
(b) sphere kD=3
 fit (9 best points)
 sphere kD=10
 fit (9 best points)
 sphere kD=30
 fit (9 best points)
y=kd·m  
Fig. 2. Signed relative errors of Qext versus y and their fits by quadratic functions for (a) kD = 8 cube 
and discretized kD = 10 sphere, (b) 3 spheres. 5 and 9 best points are used for fits in (a) and (b) 
respectively. 
Table 1. Extrapolation errors of Qext. Estimate of the extrapolation errors is 10×SE for first two 
particles and 2×SE for spheres. 
Extrapolation ymin ymax Points Error for ymin Estimate Real 
kD = 8 cube 
0.047 0.094 5 9.0×10-5 1.8×10-6 –––– 
0.094 0.19 5 1.6×10-4 6.6×10-6 4.6×10-6
0.19 0.38 5 2.2×10-4 5.3×10-5 4.0×10-5
0.38 0.75 5 1.1×10-4 3.7×10-4 3.2×10-4
Discretized kD = 10 sphere 
0.058 0.12 5 1.0×10-4 2.4×10-5 –––– 
0.12 0.23 5 2.0×10-4 9.0×10-6 7.9×10-6
0.23 0.93 4 4.3×10-4 1.2×10-3 5.9×10-4
kD = 3 sphere 
0.018 0.070 9 2.2×10-4 1.0×10-5 4.1×10-6
0.035 0.14 9 4.0×10-4 5.9×10-5 4.8×10-5
0.070 0.28 9 6.8×10-4 8.7×10-5 5.7×10-6
0.14 0.54 9 9.0×10-4 3.7×10-4 7.0×10-4
0.28 0.54 5 2.4×10-4 4.3×10-3 1.8×10-3
kD = 10 sphere 
0.059 0.23 9 2.7×10-4 2.0×10-4 2.7×10-5
0.12 0.47 9 5.5×10-4 5.5×10-4 3.7×10-4
0.23 0.93 9 1.5×10-3 3.1×10-3 2.1×10-3
kD = 30 sphere 
0.18 0.70 9 3.8×10-4 1.3×10-3 1.4×10-3
0.18 0.35 5 3.8×10-4 3.3×10-3 6.9×10-4
Table 2. Comparison of shape and discretization errors of Qext for kD = 10 sphere discretized with 
y = 0.93. All errors are relative to the best extrapolation result for the discretized sphere. 
 Shape Discretization Total 
Error 3.1×10-3 8.3×10-3 5.2×10-3
discretizations ( 38.0min =y , 75.0max =y ). Hence all conclusions with respect to the 
reliability of the error estimates (as discussed in Section 4) do not depend on the choice of 
reference if ymin is large enough. We also apply this reasoning to smaller ymin and assume that 
using the reference value obtained by extrapolation of the finest discretizations is a good 
enough estimate of the exact value. 
The same justification is valid for the discretized sphere (see Table 1 for Qext results). 
Comparison of errors of different extrapolations results of S11(θ ) (shown in Fig. 3 and Fig. 4) 
is even more convincing. Reference results themselves (both of Qext and S11(θ )) can be found 
in Paper 1. 
Next we show the results obtained by the extrapolation technique. The dependence of 
the signed relative errors of Qext on y for all 5 test cases are shown in Fig. 2. Fig. 2(a) depicts 
results for the cube and the discretized sphere. The 5 best points for each scatterer are fitted 
by a quadratic function, using the method described in Section 2. Fig. 2(b) depicts 
extrapolation results for spheres, using the 9 best points for each of them (cf. Section 2). Since 
the exact Mie solution is available, intersection of a fit with a vertical axis is a measure of the 
accuracy of extrapolation result. Table 1 summarizes the parameters (ymin, ymax, number of 
points) of all the extrapolations, which were carried out, and their performance for Qext. 
0 30 60 90 120 150 180
Scattering angle θ, deg
 y = 0.75
 y = 0.38
 extrapolation
 estimate
 y = 0.19
 y = 0.094
 extrapolation
 estimate
 y = 0.094
 y = 0.047
 extrapolation
          (estimate)
Fig. 3. Errors of S11(θ ) in logarithmic scale for extrapolation using 5 values of y in the intervals (a) 
[0.047,0.094], (b) [0.094,0.19], and (c) [0.38,0.75] for kD = 8 cube. Estimate of the extrapolation error 
is 10×SE. 
Next we present some of the extrapolations results for S11(θ ). Results for the cube are 
shown in Fig. 3. Each subfigure shows real (compared to the best extrapolation – reference) 
and estimated errors together with the errors of the finest and crudest discretizations used. 
Only the estimate of the error is shown for the best extrapolation – Fig. 3(a). Fig. 3(b) and (c) 
show extrapolation results using 5 points in the intervals [0.094,0.19] and [0.38,0.75] 
respectively. The performance of the extrapolation for the discretized sphere is shown in Fig. 
4: (a) – best extrapolation, (b) and (c) – results for extrapolation using 5 and 4 points in the 
intervals [0.12,0.23] and [0.23,0.93] respectively. The broad spacing of points for 
0 30 60 90 120 150 180
Scattering angle θ, deg
 y = 0.93
 y = 0.23
 extrapolation
 estimate
 y = 0.23
 y = 0.12
 extrapolation
 estimate
 y = 0.12
 y = 0.058
 extrapolation
          (estimate)
Fig. 4. Errors of S11(θ ) in logarithmic scale for extrapolation using 5 values of y in the intervals (a) 
[0.058,0.12], (b) [0.12,0.23] ((c): 4 values of y in the interval [0.23,0.93]) for the discretized kD = 10 
sphere. Estimate of the extrapolation error is 10×SE. 
extrapolation depicted in Fig. 4(c) is, as was noted above, due to the complex shape of the 
discretized sphere that limits possible values of y to be 0.93 divided by an integer (total time 
for computing these 4 points is ). It is important to note once more that we use 
10×SE as an estimate of extrapolation error for the cube and discretized sphere and 2×SE for 
spheres (cf. Section 
)(6.1 minyt<
Extrapolation results for the 3=kD  sphere are summarized in Fig. 5: (a) shows the best 
extrapolation (using 9 points in the interval [0.018,0.070]), and (b) shows the worst, but still 
satisfactory result, i.e. one that shows definite improvement of accuracy over most of the θ 
0 30 60 90 120 150 180
Scattering angle θ, deg
 y = 0.55
 y = 0.14
 extrapolation
 estimate
 y = 0.070
 y = 0.018
 extrapolation
 estimate
Fig. 5. Errors of S11(θ ) in logarithmic scale for extrapolation using 9 values of y in the intervals (a) 
[0.018,0.070], (b) [0.14,0.55] for kD = 3 sphere. Estimate of the extrapolation error is 2×SE. 
range. The extrapolation using 5 points from the interval [0.28,0.54] is no longer satisfactory 
(data not shown). Errors of the two best extrapolations for the 10=kD  sphere (using 9 points 
from the intervals [0.059,0.23] and [0.12,0.47]) are shown in Fig. 6(a) and (b) respectively. A 
third extrapolation for  sphere is not satisfactory (data not shown). Both 
extrapolations for the  sphere show similar controversial results, only one of them (9 
points from the interval [0.18,0.70]) that is overall slightly better is shown in 
10=kD
30=kD
Fig. 7. The 
estimate of the extrapolation error is overall slightly higher than the real errors of the 
extrapolation (data not shown). 
Results of S11(θ ) for all extrapolations (see Table 1) support the following trend: the 
quality of the extrapolation (defined as decrease of error compared to a single DDA 
computation for ymin) rapidly degrades with increasing ymin. The ratio of estimated to real 
errors increase with increasing ymin (that can be considered as a degradation of the estimate 
quality). 
Computation of exact results for both the 10=kD  sphere and its cubical discretization 
( ) allows us for the first time to directly separate and compare shape and 
discretization error of single DDA computations. The shape error is the difference between 
some measured quantity for a discretized sphere (calculated to a high accuracy) and that for 
the exact sphere. The discretization error is difference between calculation using a limited 
number of dipoles (2176) and exact (very accurate) solution for the cubical discretization of 
the sphere (first curve in 
93.0=y
Fig. 4(c)). The total error is just the sum of the two. These three 
0 30 60 90 120 150 180
Scattering angle θ, deg
 y = 0.47
 y = 0.12
 extrapolation
 estimate
 y = 0.23
 y = 0.059
 extrapolation
 estimate
Fig. 6. Errors of S11(θ ) in logarithmic scale for extrapolation using 9 values of y in the intervals (a) 
[0.059,0.23], (b) [0.12,0.47] for kD = 10 sphere. Estimate of the extrapolation error is 2×SE. 
types of errors for S11(θ ) are shown in Fig. 8, all relative to the exact value for discretized 
sphere. Errors of Qext are shown in Table 2. 
4. Discussion 
In their review Draine and Flatau2 gave the condition 1<y  for applicability of DDA. Usually 
 (10 dipoles per wavelength in the medium) is used in applications.6.0=y 3 Smaller y are 
used only in studies of DDA errors2,12,13 or of light scattering by particles much smaller than a 
wavelength (then d is determined by a shape of a scatterer, and y, being proportional to 
scatterer size, can be arbitrarily small).32 However, if one wishes to achieve better (than usual) 
accuracy of a DDA simulation, smaller y must be used. 
The best extrapolation for the cube (Fig. 3(a)) shows a large improvement compared to 
the best single DDA calculation (it should be noted, however, that this result is based on the 
empiric error estimate). Maximum errors are decreased more than 2 orders of magnitude. This 
would be impossible to reach by a single DDA calculation as it will require over 6 orders of 
magnitude increase in execution time and memory, since there is only linear convergence for 
such small y. Even for  the extrapolation can be called satisfactory because the 
maximum error is decreased almost two times when considering the estimate of the error (the 
real errors are even less). It is important to note that an estimate of the error is important by 
38.0min =y
0 30 60 90 120 150 180
Scattering angle θ, deg
 y = 0.18
 extrapolation
Fig. 7. Errors of S11(θ ) in logarithmic scale for extrapolation using 9 values of y in the interval 
[0.18,0.70] for kD = 30 sphere. 
0 30 60 90 120 150 180
Scattering angle θ, deg
 discretization
 shape
 total
Fig. 8. Comparison of discretization and shape errors of S11(θ ) for kD = 10 sphere discretized using 16 
dipoles per D ( y = 0.93). 
itself (even when it is not less than the error of a single DDA computation) because it does not 
require an exact solution (that is usually unavailable in real applications). In general, the 
extrapolation decreases large errors better than those that are already small, i.e. it may 
significantly decrease maximum errors but prove less satisfactory for certain measured 
quantity (e.g. S11 for certain θ). This conclusion holds true for all the extrapolations we 
performed (Fig. 3 – Fig. 7 and those not shown). 
Extrapolation results for the discretized sphere (Fig. 4) are similar to those for the cube. 
Extrapolations for  and 0.12 are very good (more than an order of magnitude 
decrease of maximum errors), while for 
058.0min =y
23.0min =y  it is on the edge of being satisfactory. 
The latter is strongly influenced by the fact that only 4 points in a broad interval are used 
(hence it does not fully comply with the procedure specified in Section 2). 
The best extrapolation for the 3=kD  sphere (Fig. 5(a)) shows results comparable to 
cubically shaped scatterers, however it uses an extremely small 018.0min =y . Already for 
 (14.0min =y Fig. 5(b)) it only decreased the maximum errors by a factor of two. A similar 
boundary value of ymin for satisfactory extrapolation is observed for  sphere (10=kD Fig. 
6(b)), while the best extrapolation (Fig. 6(a)) does show good results (4 times decrease of 
maximum error), although significantly worse than the analogous results for cubically shaped 
scatterers. Unfortunately we are currently not able to reach sufficiently small y for the 
 sphere and the best extrapolation (30=kD Fig. 7) uses rather large , resulting in 
almost negligible improvement of accuracy. 
18.0min =y
We have also studied a  porous cube that was obtained by dividing a cube into 
27 smaller cubes and then removing randomly 9 of them. All the conclusions are the same as 
those reported for the cube, but with slightly higher overall errors (data not shown). 
Extrapolation of Qext (Table 1) shows similar results as discussed above, however the 
improvement of accuracy is generally less than for maximum errors in S11(θ ) (which is in 
agreement with what we stated above, since errors in Qext are already small). Moreover, one 
should take into account that errors of a single DDA calculation for some ymin are 
unexpectedly small (e.g. the last extrapolations for the cube and the 3=kD  sphere), but these 
are just “lucky hits” near the points where the function  crosses the horizontal axis 
(cf. 
)(δ ext yQ
Fig. 2). 
Summarizing all results we can conclude that shape errors significantly degrade the 
extrapolation performance, because of its abrupt behavior, and therefore the extrapolation 
technique is much more suited for cubically shaped particles. One may expect satisfactory 
extrapolation for non-cubically shaped particles only when 15.0min <y , while for cubically 
shaped particles the condition is 4.0min <y . It is important to note though that extrapolation 
can be used for any ymin. The estimate of the error coming from the fitting procedure (SE) can 
then be used to decide whether this extrapolation was satisfactory or not. The quality of the 
extrapolation significantly increases with decreasing ymin, hence extrapolation is of biggest 
value for obtaining (very accurate) benchmark results. The size of the particle for which the 
extrapolation technique provides significant improvement is mainly determined by available 
computational resources that are required to reach small enough ymin. However, further testing 
is required to evaluate the quality of extrapolation for scatterers large compared to the 
wavelength. 
It is important to note that the linear extrapolation that was applied in two papers7,15 may 
lead to completely erroneous results (e.g. if points on the right branch of the parabolas for the 
cube and  sphere in 3=kD Fig. 2 are used). Quadratic extrapolation, as proposed in this 
paper, is much more reliable. 
Throughout all the extrapolations we have used error estimates as specified in Section 2: 
10×SE and 2×SE for cubically and non-cubically shaped scatterers respectively. All the 
results show that these estimates are reliable, i.e. in most cases real errors are less than the 
estimates. There are only two exceptions, both for the 3=kD  sphere: the fourth extrapolation 
of Qext (Table 1) – real error 1.8 times larger than estimate – and second of S11 – real error 
1.5-2 times larger than estimate for broad range of θ (data not shown). The existence of such 
exceptions is acceptable since the estimates have a statistical nature of a confidence interval. 
However, these estimates, though reliable, are definitely not optimal, i.e. they often 
significantly overestimate the real errors (e.g. Fig. 5(a)). It also seems to be sensitive to the 
spacing of y values used for extrapolation – cf. Fig. 4(c), where unusually broad spacing was 
used. Generally this overestimation increases with increasing ymin. We can conclude that the 
error estimate should be improved, and this is subject of future research. However, the current 
estimate is already suitable for practical applications since they mainly require reliability of 
the error estimate, which is demonstrated empirically in this paper. 
It is important to note that we limited ourselves to a single value of m. While bounds of 
ymin to obtain satisfactory extrapolation definitely dependent on m, other conclusions, such as 
the reliability of the error estimate, are expected to hold true for a broad range of m. This can 
be easily tested for specific values of m of interest using the methodology put forward in this 
paper. 
Finally we discuss the results presented in Fig. 8. One cannot conclude that shape errors 
dominate over discretization errors (or the other way around): for some θ shape errors are 
much larger than discretization, for others – vice versa. However, maximum errors occurring 
in backscattering directions are definitely due to shape errors (ratio of maximum shape to 
maximum discretization errors is about 4). Errors in Qext (Table 2) are, on the contrary, mostly 
due to discretization (although they are almost two orders of magnitude smaller than 
maximum errors of S11). One may expect shape errors to become even more important for 
smaller values of y, since the linear component of discretization errors is significantly smaller 
than that of shape errors (hence for large values of y shape errors scale linearly and 
discretization – almost quadratically). Our single result principally shows different angle 
dependence of shape and discretization errors of S11: shape errors have a clear tendency to 
significantly increase towards backscattering, while the general trend of discretization errors 
is uniform over the whole θ range. 
We have presented a simple method to directly separate shape and discretization errors 
and only one result for illustration. All previous comparisons of shape and discretization 
errors had significant inherent interpretation problems that caused a lot of discussions about 
their conclusions.16-20 Our method is free of such problems and therefore can be used for 
rigorous study of shape errors in DDA. For instance, it can help to directly evaluate the 
performance of different techniques to reduce such errors, e.g. weighted discretization (WD).9 
Discretization errors are then the limit one can achieve by drastically reducing shape errors. 
We have used a traditional DDA formulation2 to show that the extrapolation technique 
can be used with current DDA codes (e.g. DDSCAT4) without any modifications. However, 
as we showed in Paper 1 several modern improvements of DDA (namely integration of 
Green’s tensor (IT)33 and WD) should significantly change the convergence behavior of DDA 
computations and hence influence the performance of the extrapolation technique. IT should 
completely eliminate the linear term for cubically shaped scatterers. This will improve the 
accuracy especially for small y, and probably also improve the quality of the extrapolation for 
such scatterers. WD should significantly decrease shape and hence total errors for non-
cubically shaped particles, moreover it should significantly decrease the amplitude of quasi-
random error oscillations because it takes into account the location of the interface inside the 
boundary dipoles. Therefore WD should improve the quality of the extrapolation for non-
cubically shaped scatterers. Testing of extrapolation performance of DDA using IT and WD is 
a subject of a future study. 
5. Conclusion 
Based on the theoretical convergence analysis as presented in Paper 1, we proposed an 
extrapolation technique together with a step-by-step prescription, which allows accuracy 
improvement of DDA computations. The performance of this technique was studied 
empirically and we showed that it significantly suppresses maximum errors of S11(θ ) when 
 and 0.15 for cubically and non-cubically shaped scatterers respectively (for 
). The quality of the extrapolation improves with decreasing y
4.0min <y
5.1=m min reaching 
extraordinary performance especially for cubically shaped particles – more than two order of 
magnitude decrease of error when 05.0min ≈y  for wavelength-sized scatterers with 5.1=m  
(total computational time for extrapolation is less than 2.7 times that for a single DDA 
computation). 
The proposed estimates of the extrapolation error were proven to be reliable, although 
they can be improved to decrease overestimation of the errors in some cases. This error 
estimate is completely internal, and hence can be used to create adaptive DDA – a code that 
will automatically refine discretization to reach a required accuracy. 
We also proposed a simple method to directly separate shape and discretization errors. 
Maximum errors of S11(θ ) for the 10=kD  sphere with 5.1=m , discretized using 16 dipoles 
per diameter ( ) are mostly due to shape errors, however the same is not true for all 
measured quantities. This method can be employed to rigorously study fundamental 
properties of these two types of errors and to directly evaluate the performance of different 
techniques aimed at reducing shape errors. 
93.0=y
Our theory predicts that modern DDA improvements (namely IT and WD) should 
significantly change the performance of the extrapolation technique, however numerical 
testing of these predictions is left for future research. 
Acknowledgements
We thank Gorden Videen and Michiel Min for valuable comments on earlier version of this 
manuscript and Denis Shamonin for help with 3D graphics. Our research is supported by the 
NATO Science for Peace program through grant SfP 977976. 
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    /JPN <FEFF9ad854c18cea306a30d730ea30d730ec30b951fa529b7528002000410064006f0062006500200050004400460020658766f8306e4f5c6210306b4f7f75283057307e305930023053306e8a2d5b9a30674f5c62103055308c305f0020005000440046002030d530a130a430eb306f3001004100630072006f0062006100740020304a30883073002000410064006f00620065002000520065006100640065007200200035002e003000204ee5964d3067958b304f30533068304c3067304d307e305930023053306e8a2d5b9a306b306f30d530a930f330c8306e57cb30818fbc307f304c5fc59808306730593002>
    /KOR <FEFFc7740020c124c815c7440020c0acc6a9d558c5ec0020ace0d488c9c80020c2dcd5d80020c778c1c4c5d00020ac00c7a50020c801d569d55c002000410064006f0062006500200050004400460020bb38c11cb97c0020c791c131d569b2c8b2e4002e0020c774b807ac8c0020c791c131b41c00200050004400460020bb38c11cb2940020004100630072006f0062006100740020bc0f002000410064006f00620065002000520065006100640065007200200035002e00300020c774c0c1c5d0c11c0020c5f40020c2180020c788c2b5b2c8b2e4002e>
    /NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
    /NOR <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>
    /PTB <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>
    /SUO <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>
    /SVE <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>
    /ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing.  Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
  /Namespace [
    (Adobe)
    (Common)
    (1.0)
  /OtherNamespaces [
    <<
      /AsReaderSpreads false
      /CropImagesToFrames true
      /ErrorControl /WarnAndContinue
      /FlattenerIgnoreSpreadOverrides false
      /IncludeGuidesGrids false
      /IncludeNonPrinting false
      /IncludeSlug false
      /Namespace [
        (Adobe)
        (InDesign)
        (4.0)
      ]
      /OmitPlacedBitmaps false
      /OmitPlacedEPS false
      /OmitPlacedPDF false
      /SimulateOverprint /Legacy
    >>
    <<
      /AddBleedMarks false
      /AddColorBars false
      /AddCropMarks false
      /AddPageInfo false
      /AddRegMarks false
      /ConvertColors /ConvertToCMYK
      /DestinationProfileName ()
      /DestinationProfileSelector /DocumentCMYK
      /Downsample16BitImages true
      /FlattenerPreset <<
        /PresetSelector /MediumResolution
      >>
      /FormElements false
      /GenerateStructure false
      /IncludeBookmarks false
      /IncludeHyperlinks false
      /IncludeInteractive false
      /IncludeLayers false
      /IncludeProfiles false
      /MultimediaHandling /UseObjectSettings
      /Namespace [
        (Adobe)
        (CreativeSuite)
        (2.0)
      ]
      /PDFXOutputIntentProfileSelector /DocumentCMYK
      /PreserveEditing true
      /UntaggedCMYKHandling /LeaveUntagged
      /UntaggedRGBHandling /UseDocumentProfile
      /UseDocumentBleed false
    >>
>> setdistillerparams
  /HWResolution [2400 2400]
  /PageSize [612.000 792.000]
>> setpagedevice
