to $n_p=3561$) the meshes stay good.
Samples in (b), with zooms of the necks to illustrate the meshes.

ll(a)   (b)

ll(c)   (d)

Further results from biocyl/cmds1.m,
$(\al,\eps)=(1,1/4)$, continuation of Helfrich cylinder
to $c_0>0$ (blue branch c0, c0r after mesh refinement),
with bifurcation to c1r (red). (a) BDs of area $A$, energy $E$, and
mesh distortion $\delm$. (b) A full sample at c0/pt10,
different colormap (yellow$>$pink)
for better visibility of mesh. (c) further samples, cut
at $x=0$. (d) zoom into solution close to $\Xhem$.
On c0, results after pt33
become somewhat mesh–dependent, but our experiments suggest that the
branch connects to $\Xhem$ at $c_0=1/2$ (with c0r/pt46
shortly before that connection).

In Fig. <ref> we continue to $c_0>0$. This is initially easy,
but after a first fold near $c_0\approx 1.3$ and then decreasing $c_0$
below $c_0\approx 0.15$ it becomes difficult to maintain mesh distortion
$\del<100$. We give the full result of our continuation
but remark that the behavior on c0 (c0r after refinement,
to $n_p=3130$ at pt46)
after pt30, say, becomes mesh dependent.
First, however, at $c_0\approx 0.94$, we find a BP to a non–axisymmetric
branch (c1, red), 3rd sample in (c). We then get a second
fold at $c_0\approx 0.08\pm \eta$, with $\eta\in(-0.02,0.02)$
dependent on the mesh. Relatedly, the further continuation becomes
somewhat non–smooth and quantitatively depends on the mesh,
but qualitatively we get similar behavior for different meshes,
namely: The neck becomes thin and short, and the
solutions seem to approximate a solution

$\Xhem$ consisting of two
hemispheres with radius 1, centered at $(x,y,z)=(\pm 1,0,0)$,

and hence touching at $(0,0,0)$, see c0r/pt46, at $c_0=0.47$.
$\Xhem$ is an exact solution of HC with
$\lam_1=1/4$ at $c_0=1/2$, and given our various experiments with
mesh refinement we believe that the branch c0r connects
to $\Xhem$. However, the third plot in (a), and the samples in
(c) (at $c_0\approx 0.47$, shortly before the supposed connection
to $\Xhem$) show how the mesh quality seriously degrades as we approach
the supposed connection.

ll(a)   (b)

ll(c)   (d)

Further results from biocyl/cmds1.m,
$(\al,\eps)=(1,1/4)$, continuation of Helfrich cylinder
to $c_0>0$ (blue branch c0, c0r after mesh refinement),
with bifurcation to c1r (red). (a) BDs of area $A$, energy $E$, and
mesh distortion $\delm$. (b) A full sample at c0/pt10,
different colormap (yellow$>$pink)
for better visibility of mesh. (c) further samples, cut
at $x=0$. (d) zoom into solution close to $\Xhem$.
On c0, results after pt33
become somewhat mesh–dependent, but our experiments suggest that the
branch connects to $\Xhem$ at $c_0=1/2$ (with c0r/pt46
shortly before that connection).

Also, we have, e.g., $\ind(X)=3$ at
pt33 (with one unstable direction from the fold at $c_0\approx 1.3$,
and two from the double BP at $c_0\approx 0.94$), while the
“almost–$\Xhem$”–solutions near pt46 have $\ind(X)=0$, i.e., are stable.
Hence, since $\ind(X)$ should decrease by one at the (supposed) fold between
pt33 and pt46, we expect another double BP between
pt33 and pt46. However, the behavior of
the branch is quantitatively mesh–dependent also near the left fold,
and while the changes in $\ind(X)$ are detected, the bisection
loops to localize the fold and/or the BPs do not converge.
Thus, altogether the behavior after pt33 is conjectured.
On the other hand, for $c_0\in(0.3,0.5)$, say, and in particular
near pt46, the solutions like pt46 stay stable under
mesh refinement. In any case, a pinch–off (topological change)
which should occur after pt46 as we approach $\Xhem$
cannot be resolved with our methods,
but probably requires the use of some phase field method, see, e.g.,
[For Neumann BCs cBCs, hemispheres
are highly degenerate in the following sense: Setting up HC over
a single circle at wlog at $X_1=1$,
and wlog with $(\al,\lam_1,c_0)=(1,1/4,1/2)$ the single unit hemisphere
is again an exact solution, but we are not able to continue this in
$\lam_1$ or $c_0$, i.e., it seems isolated. This is different for BCs natBCs, see <ref>.]

The results for c0r up to pt33 are
mesh independent, including the bifurcation of the
non-axisymmetric branch c1 at c0/bpt1.
The mesh then also degrades on the non–axisymmetric branch (c1,
and in a similar way like on the branch c0
but at larger $c_0$), and since next we focus on non–axisymmetric branches in
a slightly different setting we do not further pursue c1 here.

ll(a)   (b)

ll(c)   (d)

Further results from biocyl/cmds1.m,
$(\al,\eps)=(1,1/4)$, continuation of Helfrich cylinder
to $c_0>0$ (blue branch c0, c0r after mesh refinement),
with bifurcation to c1r (red). (a) BDs of area $A$, energy $E$, and
mesh distortion $\delm$. (b) A full sample at c0/pt10,
different colormap (yellow$>$pink)
for better visibility of mesh. (c) further samples, cut
at $x=0$. (d) zoom into solution close to $\Xhem$.
On c0, results after pt33
become somewhat mesh–dependent, but our experiments suggest that the
branch connects to $\Xhem$ at $c_0=1/2$ (with c0r/pt46
shortly before that connection).

ll(a)   (b)

continuation of $\Xc$ in $\lam_1$,

Results for HCd from cmds2.m, continuation
in $\al$.

§.§.§ Intermezzo: Other radii

In cmds2.m we continue $\Xc$ in $\al$, with fixed $(c_0,\lam_1)=(0,1/4)$,
see Fig. <ref>. This
shows no bifurcations.
Numerically, the case $\al\searrow 0$ is challenging due to boundary
layers developing at $x=\pm 1$, see <cit.>. Alternating continuation
and mesh adaptation
based on e2rsshape1 we can reliably continue to $\al=0.05$ and slightly further.

Results for HCd from cmds2.m, continuation
in $\al$.

continuation in $c_0$ for
$(\al,\lam_1)=(1.4, 1/5.6)$ in (a),
and $(\al,\lam_1)=(0.6,2.4)$ in (b–d).

In cmds3.m and Fig. <ref> we repeat the continuation in $c_0$
from Fig. <ref> and Fig. <ref> for different starting cylinders
$\Xc$, namely $\al=1.4$ in Fig. <ref>(a) and $\al=0.6$ in (b–d).
The main differences to the case $\al=1$ are as follows: For both,
$\al=1.4$ and $\al=0.6$, continuing to negative $c_0$ we no longer
have the two folds and associated bistability range as in Fig. <ref>,
and the branches seem to extend to arbitrary large negative $c_0$.
We refrain from plotting this for $\al=1.4$ in (a), and for
$\al=0.6$ in (b) we mainly remark that large negative $c_0$ yields
very long and thin necks, which can be handled with adaptive mesh refinement as
in Fig. <ref>.

continuation in $c_0$ for
$(\al,\lam_1)=(1.4, 1/5.6)$ in (a),
and $(\al,\lam_1)=(0.6,2.4)$ in (b–d).

Continuing to positive $c_0$, for $\al=1.4$ in (a), the main difference
to Fig. <ref> is that the branch c0r only has one fold
(at $c_0\approx 2.3$) and then continues to negative $c_0$, see
b/c0r/pt51. The difference to $\al=1$ is that a solution like
$\Xhem$ no longer exists, i.e., two hemispheres of radius $\al=1.4$
(or any $\al\ne 1$) cannot be near a (genus 1) cylinder type solution.
Additionally, the larger radius makes the continuation of
the red non–axisymmetric branch b/c1qr easier wrt to mesh handling,
see the last sample in (a).
For $\al=0.6$ in (c,d), the smaller radius gives a “pearling” behavior
after the fold on the blue branch, and it seems likely that the branch
s/c0r continues to a solution of type hemisphere–sphere–hemisphere,
which would give similar problems as discussed for $\Xhem$ in Fig. <ref>.
Here we stop the continuation after pt30 (third sample in (d)),
as further continuation requires excessive mesh adaptation and the solutions
are unstable anyway. The pearling shape is also inherited by the
non-axisymmetric branch s/c1 (last sample in (d)).

continuation in $c_0$ for
$(\al,\lam_1)=(1.4, 1/5.6)$ in (a),
and $(\al,\lam_1)=(0.6,2.4)$ in (b–d).

Results for HCd from cmds3.m, $\al=1.4$.

§.§.§ Continuation in surface tension

In cmds4.m we return to fixed $\al=1$ and
continue in $\lam_1$, starting at $\lam_1=0.25$ and
$c_0=0.7$ corresponding to Fig <ref>, and in cmds4b.m
we repeat this for $c_0=0$, starting from the genuine Helfrich
cylinder $\Xc$.
In both cases, for increasing $\lam_1$ the solutions initially
only slightly change shape, but at larger $\lam_1$
(near $\lam_1=8$ for $c_0=0$, and near $\lam_1=4$
for $c_0=0.7$) we get an S–shaped bistability region due to
two consecutive folds, similar to the case of $c_0\approx -6$
in Fig. <ref> (with fixed $\lam_1=1/4$).
[Such folds were already found in <cit.>
for $c_0=0$ in the 1D axisymmetric setting.]
However, we find no bifurcations to non–axissymmetric branches.
The case of decreasing $\lam_1$ is more interesting, and
in Fig. <ref> we only show the basic result for $c_0=0$ and
large $\lam_1>0$ for completeness.

ll(a)   (b)

continuation of $\Xc$ in $\lam_1$,

continuation in $\lam_1$ at $c_0=0.7$. In (a):
axisymmetric branch  l0b (dark blue, sample in (b)),
l0 (blue, sample in (c))), and
four bifurcating branches, l1 (red, refined to l1qr,
samples in (d) and (e), with zoom),
l2 (magenta), l3 (orange), l4 (green), samples in (f).
The kinks of l1–l4
are due to adapative mesh refinement every 5th step (after pt10),
from ${\tt np}=2700$ to ${\tt np}\approx 3600$ at the end of each

Even if $\lam_1<0$ is in general unphysical, see Remark <ref>(b),
to $\lam_1<0$ yields interesting results, in particular for $c_0>0$.
For both, $c_0=0.7$ and $c_0=0$, for $\lam_1<0$ the
solutions start to bulge out in the “middle” (near $x=0$), and
there are folds around $\lam_1=-2$ after which we obtain pronounced
“tire shapes”.
Moreover, we obtain
bifurcations to $D_m$–symmetric branches, with increasing
angular wave numbers $m=1,2,3,4,\ldots$. For $c_0=0.7$,
the bifurcating branches return to $\lam_1>0$, such that
in Fig. <ref>(a–f) we obtain four new solutions at $\lam_1=0.25$.
For $c_0=0$, the $D_m$ symmetric branches initially behave similarly,
but do not reach $\lam_1>0$ and instead asymptote to $\lam_1=0_-$,
with arbitrary large $A$, see Fig. <ref> and again Remark
Therefore we put $c_0=0.7$ first in Fig.<ref>.
Nevertheless, for both $c_0=0$ and $c_0=0.7$, the BCs
do not allow “large portions of planes” as solutions and thus
yield the folds of the blue branches.
Moreover, $E$ stays bounded below
(see Fig. <ref>), and also for $c_0=0$ in Fig. <ref>
and Fig. <ref>(b) it seems that $\lam_1 A\to 0$
as $\lam_1\nearrow 0$, i.e., that the area grows slower than $|1/\lam_1|$.

Results from cmds2.m and cmds2b.m.
Continuation in $\lam_1$ at $c_0=0.7$ (a-f), and at
$c_0=0$ (g,h). In (a):
axisymmetric branch  l0b (dark blue, sample in (b)),
l0 (blue, sample in (c))), and
four bifurcating branches, l1 (red, refined to l1qr,
samples in (d) and (e), with zoom),
l2 (magenta), l3 (orange), l4 (green), samples in (f).
The kinks of l1–l4
are due to adapative mesh refinement every 5th step (after pt10),
from ${\tt np}=2700$ to ${\tt np}\approx 3600$ at the end of each
branch. In (g) ($c_0=0$) we use the same color code for the branches
as in (a),
and have, e.g., np=4400 at 0l1qr/pt40 (i),
i.e. nu=2np=8800 DoF.

The main issue in both, cmds4.m and cmds4b.m (which is essentially
a copy of cmds4.m, with a different start (at $c_0=0$)
of the blue branch and hence directory names starting with 0), is the
mesh adaptation for the strong budding of the $D_m$ symmetric branches.
In detail, to compute the branch, e.g., l1 ($m=1$,
sample in Fig. <ref>(d))
with refinement (sample in (e)) we proceed as follows.
After branch switching at l0/bpt1 (double, by rotational
symmetry, but again we just choose the first of the two kernel vectors)
we do a few steps without rotational PC, which we then switch on.
Subsequently, we continue with mesh adaptation based on area
every five continuation steps. This works very robustly,
also on the branches with $m=2,3,4$ and similarly for $c_0=0$ in Fig. <ref>,
and allows continuation to large buds.
This shows again that bulging out (i.e., expanding)
is typically not problematic wrt meshes.
The only unbeautiful  effect are the kinks on the $m=1,2,3,4$
branches in Fig. <ref>(a), which occur at mesh–refinement.
To have smoother branches we would need more frequent but less strong
refinement, but this is clearly not critical.

continuation in $\lam_1$ at $c_0=0.7$. In (a):
axisymmetric branch  l0b (dark blue, sample in (b)),
l0 (blue, sample in (c))), and
four bifurcating branches, l1 (red, refined to l1qr,
samples in (d) and (e), with zoom),
l2 (magenta), l3 (orange), l4 (green), samples in (f).
The kinks of l1–l4
are due to adapative mesh refinement every 5th step (after pt10),
from ${\tt np}=2700$ to ${\tt np}\approx 3600$ at the end of each

continuation in $\lam_1$ at $c_0=0$;
color code as in Fig.<ref>,
and, e.g., np=4400 at 0l1qr/pt40.

ll(a)   (b)

Energies $E$ and the part $\lam_1A$ of $E$ along the branches
of Fig. <ref>; (a) $c_0=0.7$, (b) $c_0=0$.

The branches in Fig. <ref>
can be continued further by alternating cont and refineX,
but this requires some fine–tuning of the
cont–refineX loop parameters, and eventually the
continuation of the branches l*qr fails again due to
bad triangles in the necks, which apparently cannot easily be fixed.

Thus,  to keep
the script cmds4.m simple we stop at the given points.

Finally we note that the stabilities
in Fig. <ref> and Fig. <ref> are as expected,
namely $\ind(X)=m$ on the $D_m$ branch(es).

Results from cmds2.m and cmds2b.m.
Continuation in $\lam_1$ at $c_0=0.7$ (a-f), and at
$c_0=0$ (g,h). In (a):
axisymmetric branch  l0b (dark blue, sample in (b)),
l0 (blue, sample in (c))), and
four bifurcating branches, l1 (red, refined to l1qr,
samples in (d) and (e), with zoom),
l2 (magenta), l3 (orange), l4 (green), samples in (f).
The kinks of l1–l4
are due to adapative mesh refinement every 5th step (after pt10),
from ${\tt np}=2700$ to ${\tt np}\approx 3600$ at the end of each
branch. In (g) ($c_0=0$) we use the same color code for the branches
as in (a),
and have, e.g., np=4400 at 0l1qr/pt40 (i),
i.e. nu=2np=8800 DoF.

ll(a)   (b)

Energies $E$ and the part $\lam_1A$ of $E$ along the branches
of Fig. <ref>; (a) $c_0=0.7$, (b) $c_0=0$.

§.§ Biomembrane caps

In the demo biocaps we consider BCs natBCs for
the circle $\pa X{=}\{x^2+y^2{=}\al^2\}$ in the $x$–$y$ plane. Thus,

on $X$, and, on $\pa X$,
u=0 (since we keep $\al$ fixed throughout),

In our experiments we fix $\al=1$ and $\lam_1=1/4$, and first $c_0=1/2$
and vary $b$, and want to start with the upper unit hemisphere. Then $H=K=1$
(choosing the inner normal for the hemisphere) and
$\kap_n=1$ and hence cabc2 requires $b=-1/2$. The files in biocaps
are as usual, with the main adaption in hcapbra.m to compute
$\int_X K\dd S$ for the energy $E$, see Remark <ref>c).

Initial results for H2,natBCs from biocaps/cmds1.m. In (a)
we continue in $b$ starting at b/pt1 from the unit hemisphere
with $(\al,\lam_1,c_0,b)=(1,1/4,1/2,-1/2)$,
to increasing $b$ (branch b, black) and decreasing $b$ (branch
bb, grey). On b we go to the flat disk (last sample),
while on bb the hemisphere bulges out. This is mainly intended
for later continuation in $c_0$, and in (b) we do so starting from bb/pt9
at $b\approx -1.66$. This gives the double well shape for $E$, with a short
unstable segment between the two folds at $c_0\approx \pm 0.33$.
See Fig. <ref> for the cases of $b\approx -3.4$ (bb/pt20) and
$b\approx -4$.

Fig.<ref>(a) shows the continuation in $b$.
This is mainly intended for subsequent continuation in $c_0$ at
negative $b$, and (b) shows the case of $b\approx -1.66$. The problem
is symmetric under $(c_0,X_3)\mapsto -(c_0,X_3)$, and in particular at
$c_0=0$ we have the flat disk as an exact solution (for any $b$).
See c00b/pt11 for a nearby solution with $c_0\approx  -0.07$,
which lies between two folds with exchange of stability. This unstable
part will feature interesting bifurcations to non–axisymmetric
branches at more negative $b$,
see Fig. <ref>, while the remainder(s) of the axisymmetric
branches are all stable, with the samples c00b/pt34 and c00/pt14
in  <ref>(b) showing the typical behavior at strongly negative or
positive $c_0$, respectively.

The main  numerical challenges and used tricks for Hcb

a) To have a mesh as regular as possible for the initial hemisphere
we use a classical subdivision and projection algorithm,
see subdiv_hemsphere.m, modified from subdivided_sphere.m
from the . Subsequently, we do one mesh–refinement with
e2rsbdry as selector, as a good resolution near $\pa X$ turns
out helpful later. The initial mesh then has np=2245 nodes,
which for instance at c01b-1q/pt24 is refined (by triangle
area, i.e., e2rsA, followed by retrigX) to np=4475.
The mesh quality in all our solutions stays quite good, i.e.,
$\delm<20$ for all solutions, and mostly $\delm<10$.

b) The boundary $\ga=\pa X$ is parameterized by arclength as
$\ga(\phi)=\al(\cos(\phi/\al),\sin(\phi/\al),0)$. Then
$\kap=\ga''=-\ga/\al$ and the normal curvature on
$\pa X$ reads  $\ds \kap_n=-\frac 1 \al \spr{N,X}$, which
is used to implement the BCs cabc2.

c) The “integral” sum(K) over the
discrete Gaussian curvature ${\tt K}$ always evaluates to $2\pi\chi(X)$,
cf. Footnote <ref>.

Thus we once more use
Gauss-Bonnet $\ds\int_X K\dd S=2\pi\chi(X)-\int_{\pa X}\kap_g\dd s$, where
$\kap_g=\sign(N_3)\frac 1 \al \|N\times \ga\|$,
to compute the energy $E$,
and in hcapbra.m we evaluate $\ds\int_{\pa X}\kap_g\dd s$
by a trapezoidal rule.
Another alternative is to zero out
$K|_{\pa X}$ before evaluating  sum(K), and for comparison
we also put that into hcapbra.m. In our computations we find a
relative error $<0.01$ (usually much smaller) between the two methods,
when doing the initial refinement step near $\pa X$ from (a).

Further results from cmds1.m. (a,b)
Continuation in $c_0$ from bb/pt20, $(\al,\lam_1,b)=(1,0.25,-3.4)$,
starting from $c_0=0.5$, branches c01b (black, to decreasing $c_0$) and
c01 (grey, to increasing $c_0$). There are two BPs on the unstable
part of c01b for $c_0<0$, and the symmetric BPs for $c_0>0$.
The blue branch c01b-1q has azimuthal wave number $m=1$
and is stable after its fold.
The red branch c01b-2q has $m=2$
and connects to the symmetric BP at $c_0>0$. The 2nd plot in (a)
shows where the part $b\int K\dd S$ of $E$ becomes dominant,
taking into account the rather large $|b|$.
(c) Similarly starting at bb/pt24 with $b\approx -4$; zoom of BD
near upper left fold of the branch c02b (black)
similar to c01b from (a). The blue branch is
qualitatively as in (a),  but now the $m=2$ branch c02b-2q (red)
also folds back giving stable solutions, and there is a secondary BP on it,
giving the green branch c02b-2q-1.

In Fig. <ref> we repeat the continuation in $c_0$ from
Fig. <ref>(b) at more negative $b$, namely $b\approx -3.4$ in (a)
and $b\approx -4$ in (b). For lower $b$, the unstable part of the
$c_0$ continuation expands, and we find two (or more, for even lower $b$)
BPs between the left fold and $c_0=0$, with azimuthal wave numbers
$m=1$ and $m=2$. As before, these bifurcations are double by $S^1$ symmetry,
and to continue the bifurcating branches we set the usual rotational PC
after two steps. The blue $m=1$ branch then behaves similarly
in (a) and (b), i.e., it becomes stable after a fold at $c_0\approx -0.2$
($b=-3.4$) resp. $c_0\approx -0.27$ ($b=-4$).

However, the $m=2$ branch behaves differently: For $b=-3.4$ it connects
to the symmetric BP at $c_0>0$. For $b=-4$, the red branch c02b-2q
first shows
a secondary BP to a branch (c02b-2q-1, green) with broken
$\Z_2$ symmetry, and then shows a fold at
$c_0\approx -0.21$ where it becomes stable.
The branch c02b-2q-1 also shows a fold, at $c_0\approx -0.11$, after
which however one unstable eigenvalue remains, i.e., $\ind(X)=1$ at, e.g.,
c02b-2q-1/pt20 (last sample in (c)). The somewhat non–smooth shape
of the red and green branches is due to repeated
and heavy mesh refinement, to, e.g., $n_p=5560$ at c02b-2/pt30.

Further results from cmds1.m. (a,b)
Continuation in $c_0$ from bb/pt20, $(\al,\lam_1,b)=(1,0.25,-3.4)$,
starting from $c_0=0.5$, branches c01b (black, to decreasing $c_0$) and
c01 (grey, to increasing $c_0$). There are two BPs on the unstable
part of c01b for $c_0<0$, and the symmetric BPs for $c_0>0$.
The blue branch c01b-1q has azimuthal wave number $m=1$
and is stable after its fold.
The red branch c01b-2q has $m=2$
and connects to the symmetric BP at $c_0>0$. The 2nd plot in (a)
shows where the part $b\int K\dd S$ of $E$ becomes dominant,
taking into account the rather large $|b|$.
(c) Similarly starting at bb/pt24 with $b\approx -4$; zoom of BD
near upper left fold of the branch c02b (black)
similar to c01b from (a). The blue branch is
qualitatively as in (a),  but now the $m=2$ branch c02b-2q (red)
also folds back giving stable solutions, and there is a secondary BP on it,
giving the green branch c02b-2q-1.

In Fig. <ref> we continue some $m=1$ solutions from the blue
branch in Fig. <ref>(b) in $b$ and in $\lam_1$, with fixed
$c_0$. The unstable black solutions with $c_0\approx -0.35$ in (a)
continue to $b\approx -1.8$, where the branch bifurcates from
the axisymmetric branch. This is in contrast to Fig. <ref>
where for the continuation in $b$ at $c_0=1/2$ no bifurcations from
the axisymmetric branch were found.
The blue branch in Fig. <ref>(a) with $c_0\approx -0.24$
shows a fold at $b\approx -3.11$.
From these and similar experiments, $m=1$ solutions do not seem to
exist for $b$ somewhat larger than $-2$, and no stable
ones for $b$ larger than $-3$. In Fig. <ref>(b), the black unstable branch
continues to large $\lam_1$, but the stable blue branch again shows a
fold, at moderate $\lam_1\approx 0.44$.
This indicates that solutions of this type also do
not continue to “large” $\lam_1$.
while the physical (biological) significance of the parameter regimes
and solutions in Figs. <ref>–<ref> certainly
needs to be discussed,
mathematically we have obtained some stable
non–axisymmetric solutions.

Further results from cmds1.m. (a,b)
Continuation in $c_0$ from bb/pt20, $(\al,\lam_1,b)=(1,0.25,-3.4)$,
starting from $c_0=0.5$, branches c01b (black, to decreasing $c_0$) and
c01 (grey, to increasing $c_0$). There are two BPs on the unstable
part of c01b for $c_0<0$, and the symmetric BPs for $c_0>0$.
The blue branch c01b-1q has azimuthal wave number $m=1$
and is stable after its fold.
The red branch c01b-2q has $m=2$
and connects to the symmetric BP at $c_0>0$. The 2nd plot in (a)
shows where the part $b\int K\dd S$ of $E$ becomes dominant,
taking into account the rather large $|b|$.
(c) Similarly starting at bb/pt24 with $b\approx -4$; zoom of BD
near upper left fold of the branch c02b (black)
similar to c01b from (a). The blue branch is
qualitatively as in (a),  but now the $m=2$ branch c02b-2q (red)
also folds back giving stable solutions, and there is a secondary BP on it,
giving the green branch c02b-2q-1.

cmds2.m, experiments with continuation of
$m=1$ solutions from Fig. <ref> in $b$ (a), and in $\lam_1$ (b).
The starting points in (a) are from $c\approx -0.35$ for the black branch
bc1, and from $c\approx -0.24$ for the blue branch bc2 (this is
the same solution as c01b-1q/pt16).
The continuation in $\lam_1$ in (b) has the same
starting points and the same colors. The unstable branch (now
with fixed $(c_0,b)=(-0.35,-3.4)$) continues to large $\lam_1$,
but the blue branch with fixed  $(c_0,b)=(-0.24,-3.4)$ has a fold at
$\lam_1\approx 0.44$.

§ SUMMARY AND OUTLOOK

We explained the basic setup of the  extension library Xcont
for continuation of 2D submanifolds $X$ (surfaces) of $\R^3$,
and gave a number of examples. These were partly introductory,
e.g. the spherical caps in <ref>, partly classical, e.g.,
the Enneper minimal surface in <ref>, the nodoids in
and <ref>, and the TPS in <ref>,
and partly rather specific such as the Plateau problem in <ref>
or the 4th order Helfrich type cylinders and caps in
<ref> and <ref>. Besides
<cit.>, and to some extent <cit.>,
there seem to be few numerical continuation
and bifurcation experiments for such geometric problems for 2D surfaces,
i.e., without imposing some axial symmetry,
and we are not aware of a general
software for such tasks.

The basic setup for all our problems (except H2, which is of
4th order) is similar: We consider CMC surfaces, which mainly differ
wrt constraints and/or boundary conditions. Along the way we explained
a number of techniques/tricks which we expect to be crucial in
many applications.

A major problem for
continuation (over longer parameter regimes) is the mesh handling
as $X$ changes and hence the mesh distorts.
We explained how this (often) can be abated via moving of mesh points
(moveX), refinement (refineX which sometimes should be
combined with re–triangulation by retrigX) and
coarsening (coarsenX),
and coarsening of degenerate triangles (degcoarsenX), although
the choice of the parameters controling these functions often
requires some trial and error.  In any case, $X$ bulging out (increasing area)
is usually
harmless, but bulging in (the development of necks) is more challenging.

This is a first step.
With the demos we hope to give a pool of applications which
users can use as templates for their own problems, and we are
curious what other applications users will consider, and of course are happy
to help if problems occur. As indicated above,
our own further research, to be presented elsewhere, mostly since bifurcation
diagrams become complicated, but also since some of the numerics
require further tricks, includes:

* Further classical minimal surfaces (and CMC companions) such
as Schwarz H
and Scherk surfaces (surface families);

* Bifurcations of closed vesicles in Helfrich type problems.

* Coupling of membrane curvature and reaction–diffusion equations
for proteins as in, e.g., <cit.>.

§ SPHERES, HEMISPHERES, VPMCF, AND AN ALTERNATIVE SETUP

§.§ Spheres

The demo spheres, containing only the (somewhat minimally necessary)
files sphereinit.m, sG.m, getM.m,  and
is mostly meant to illustrate volume preserving mean curvature
flow (VPMCF) near  spheres, see Fig. <ref>.
[Additionally, the demo contains convtest.m for convergence
tests, see Fig. <ref>.]
We refer to sphereinit.m and
cmds1.m for comments (and to geomflow.m and
vpmcff.m from libs/Xcont for the VPMCF)
and here only note:

Results from sphere/cmds1.m. (a) Continuation
of a sphere in $V$, with comparison of  $A$ and $V$.
(b) An IC for VPMCF from a perturbation of S3/pt30,
with (c) solutions at $t=1$ and $t=5$, and a time–series for $V,A$;
$V$ is conserved to within $0.5\%$.

* The comparison between $rA/3$ and $V$ in Fig. <ref>(a) shows
a very small error which indicates that the solutions are good approximations
of spheres.

* For convex closed initial $X$ (meaning that $X=\pa\Om$ for
a convex domain $\Om\subset\R^3$), the VPMCF converges to a sphere.
See also, e.g., <cit.> for theoretical background.
This also holds for “slightly” non–convex initial $X$ as
in Fig. <ref>(b).
[In detail, $X(0)$ here is obtained as $X(0)=S_{r_0}+0.4(\sin(\vt)(|x|-{r_0})+\xi)N$,
where $\vt$ is the azimuth, and $\xi=0.2({\tt rand}-0.5)$ with ${\tt rand}\in[0,1]$ a
random variable on each node.]

* Our (explicit Euler) implementation of the VPMCF does not
conserve $V$ exactly, but with “reasonable accuracy”, i.e.:
Even for quite “non–spherical” initial $X(0)$, the error
$|1-V_\infty/V_0|{<}0.01$, where $V_\infty=\lim_{t\to\infty}V(t)<V_0$
in all our tests, i.e., $V_\infty$ is always slightly smaller than
[This “volume–accuracy” of geomflow for VPMCF
depends weakly on the Euler–stepsize dt, and more strongly
on the fineness of the discretization of $X$. This can be checked
by changing sw in spheres/cmds1.m for initializing $X$.]

* During continuation, the position of X is not fixed, i.e.,
we have the threefold (in the discrete setting approximate)
translational invariance in $x,y,z$, and hence always a three-dimensional
(approximate) kernel. This could be removed by suitable translational
PCs (see the demo hemispheres). However, the approximate
kernel is not a problem for the continuation here since
in the Newton loops the right hand sides are orthogonal to this kernel.
As here we are mostly interested in the VPMCF, for which the
translational invariances are irrelevant, we refrain from
these PCs to keep the demo sleek and simple.

§.§ Hemispheres

In the demo hemispheres we continue  in volume $V$ hemispheres $X$
sitting orthogonally on the $z=0$–plane, i.e.,

$\pa_r X_3=0$ where $r=\sqrt{x^2+y^2}$,

and test VPMCF for perturbations of such hemispheres, see Fig. <ref>.
Additionally we use the PCs

∫_X N_1S=∫_X N_2S=0,   X=X_0+uN_0,

to fix the translational invariance.

Results from hemisphere/cmds1.m. (a) Continuation
of hemisphere in $V$, with sample at end.
(b) Comparison of $A$ and $V$ for (a).
(c) A perturbation of hsr1/pt50 as IC for VPMCF.
(d) Solutions at $t=1$ and $t=2$, and time–series for $V,A$
from (c,d); $V$ conserved to within $0.8\%$.

Files in pde2path/demos/geomtut/hemispheres.

cmds1.m   continuation in $V$, and VPMCF flow test.

hsinit.m, sGhs.m   init and rhs

qf2.m, qjac2.m   PC hspc, and derivative

getN.m   mod of getN, correction at $z=0$

diskpdeo2.m   mod of (–)default  diskpdeo2.m to have
a finer mesh at $r=1$.

Compared to the spheres in <ref> this requires a few more files,
listed in Table <ref>. The PCs hspc (and $u$–derivatives)
are implemented in qf2, qjac2, and the rhs $G(u)=H-H_0$ in
sGhs is augmented to $\wt G(u)=H-H_0+s_x N_1+s_y N_2$
with multipliers $s_x,s_y$. These stay $\CO(10^{-6})$ during continuation,
and the only effect of the construction is that
the 2D kernel of translational invariance of $\pa_u G$  is removed from
the linearization of the extended system $(\wt G,q)$. See the end of
cmds1.m for further comments.
[In cmds1.m we also monitor the “positions” $(x_0,y_0)$
of $X$ defined as $x_0=\int_X X_1\dd S,\, y_0=\int_X X_2\dd S$. These behave
as expected, namely: very small drifts for continuation without PCs,
but no drifts for continuation with the PCs hspc.]

The BCs sNBC allow motion of $\pa X$ in
the “support–plane” $z=0$, and
are implemented as

$$\text{ {\tt r(idx)=grXz*(X(idx,1).\^{}2+X(idx,2).\^{}2)}($\stackrel!=0$)}$$

in sGhs, where as usual idx are the boundary indices, and
grXz is the $z$–derivative (operator)
on $X$, as before obtained from grX=grad(X,p.tri)
and interpolation to the nodes. This forces the $x,y$–coordinates of
the points on $X$ directly above the $z=0$ layer to also
fulfill $x^2+y^2=1$, i.e., we obtain a “cylindrical socket”
for the hemispheres. To mitigate this effect, in hsinit we
initialize with a somewhat specialized mesh over the unit disk $D$
with higher density towards $\pa D$, which is then mapped to the
unit hemisphere via $z=\sqrt{1-x^2-y^2}$. Nevertheless, after continuation
to larger $V$, which is combined with some mesh–adaptation by area,
we obtain a mismatch between $rA/3$ and $V$,
see Fig. <ref>(a,b), and compare Fig. <ref>(b).

In Fig. <ref>(c,d) we give an example of VPMCF from a perturbation
of hs1r/pt50, here of the form $X|_{t=0}=X+0.4(\cos(\vt)(\max(z)-z
+0.1({\rm rand}-0.5))N$, where $\vt$ is the angle in the $z=0$ plane.
We do not use any BCs
in the rhs vpmcff.m. Instead, we use the correction

N(p.idx,3)=0; N=normalizerow(N);

in getN.m, to let $N$ lie in the $x$–$y$–plane, cf. Remark <ref>(b)
for a similar trick.
Without hsNc, the continuation of hemispheres works as before,
but the solutions of the VPMCF start to (slowly) lift off the $z=0$ plane,
and then (quickly) evolve towards a planar $X$ such that also $V\to 0$.
With hsNc, the solutions flow back to hemispheres,
and $V$ is again conserved up to $0.5\%$, and “after convergence”
(e.g., from $X|_{t=2}$) we can start continuation again, showing

§.§ Spheres

The demo spheres, containing only the (somewhat minimal) files
sG.m, getM.m, sphereinit.m, and
is mostly meant to illustrate volume preserving mean curvature
flow (VPMCF) near  spheres, see Fig. <ref>.
[Additionally, it contains convtest.m for convergence
tests, see Fig. <ref>.]
We refer to sphereinit.m and
cmds1.m for comments (and to geomflow.m and
vpmcff.m from libs/Xcont for the VPMCF)
and here only note:

* The comparison between $rA/3$ and $V$ in (a) shows
a very small error which indicates that the solutions can be safely regarded
as spheres.

* For convex closed initial $X$ (meaning that $X=\pa\Om$ for
a convex domain $\Om\subset\R^3$), the VPMCF converges to a sphere.
See also, e.g., <cit.> for theoretical background.
This also holds for “slightly” non–convex initial $X$ as
in Fig. <ref>(d).
[In detail, $X(0)$ here is obtained as $X(0)=S_{r_0}+(\sin(\vt)(|x|-{r_0})+\xi)N$, where $\xi=0.2({\tt rand}-0.5)$ with rand a
random variable on each node.]

* Our (explicit Euler) implementation of the VPMCF does not
conserve $V$ exactly, but with “reasonable accuracy”, i.e.:
Even for quite “non–spherical” initial $X(0)$, the error
$|1-V_\infty/V_0|{<}0.01$, where $V_\infty=\lim_{t\to\infty}V(t)<V_0$
in all our tests, i.e., $V_\infty$ is always slightly smaller than
[This “volume–accuracy” of geomflow for VPMCF
depends weakly on the Euler–stepsize dt, and more strongly
on the fineness of the discretization of $X$. This can be checked
by changing sw in spheres/cmds1.m for initializing $X$.]

* During continuation, the position of X is not fixed, i.e.,
we have the threefold (approximate)
translational symmetry in $x,y,z$, and hence always a three-dimensional
(approximate) kernel. This could be removed by suitable trannslational
PCs (see the demo halfspheres). However, the approximate
kernel is not a problem for the continuation here (since
in the Newton loops the right hand sides are orthogonal to this kernel),
and since we are mostly interested in the VPMCF we refrain from
these PCs here to keep the demo as sleek and simple as possible.

§.§ Hemispheres

In the demo hemispheres we continue  in volume $V$ hemispheres $X$
sitting on the $z=0$–plane, i.e.,

$\pa_r X_3=0$ where $r=\sqrt{x^2+y^2}$,

and test VPMCF for pertubations of such hemispheres, see Fig. <ref>.
Additionally we use the PCs

∫_X N_1S=∫_X N_2S=0,   X=X_0+uN_0.

Files in pde2path/demos/geomtut/hemispheres.

cmds1.m   continuation in $V$, and VPMCF flow test.

hsinit.m, sGhs.m   init and rhs

qf2.m, qjac2.m   PC hspc, and derivative

getN.m   mod of getN, correction at $z=0$

diskpdeo2.m   mod of (–)default  diskpdeo2.m to have
a finer mesh at $r=1$.

Compared to the spheres in <ref> this requires a few more files,
listed in Table <ref>. The PCs hspc (and $u$–derivatives)
are implemented in qf2, qjac2, and the rhs $G(u)=H-H_0$ in
sGhs is augmented to $\wt G(u)=H-H_0+s_x N_1+s_y N_2$
with multipliers $s_x,s_y$. These stay $\CO(10^{-6})$ during continuation,
and the only effect of the construction is that
the now 2D kernel of translational invariance of $\pa_u G$  is removed from
the linearization of the extended system $(\wt G,q)$. See the end of
cmds1.m for further comments.
[In cmds1.m we also monitor the “positions” $(x_0,y_0)$
of $X$ defined as $x_0=\int_X X_1\dd S,\, y_0=\int_X X_2\dd S$. These behave
as expected, namely: small drifts for continuation without PCs,
but no drifts for continuation with the PCs hspc.]

The BCs sNBC allow movement
of $\pa X$ in the “support–plane” $z=0$, and
are implemented as

$$\text{ {\tt r(idx)=grXz*(X(idx,1).\^{}2+X(idx,2).\^{}2)}($\stackrel!=0$)}$$

in sGhs, where as usual idx are the boundary indizes, and
grXz is the $z$–derivative (operator)
on $X$, as before obtained from grX=grad(X,p.tri)
and interpolation to the nodes. This forces the $x,y$–coordinates of
the points on $X$ directly above the $z=0$ layer to be essentially
equal to those in the $z=0$ layer, i.e., we obtain a 'cylindrical base'
for the hemispheres. To mitigate this effect, in hsinit we
initialize with a somewhat specialized mesh over the unit disk $D$
with higher density towards $\pa D$, which is then mapped to the
unit hemisphere via $z=\sqrt{1-x^2-y^2}$. Nevertheless, after continuation
to larger $V$, which is combined with some mesh–adaptation by area,
we obtain a slight mismatch between $rA/3$ and $V$,
see Fig. <ref>(a,b), and compare Fig. <ref>(b).

In Fig. <ref>(c,d) we give an example of VPMCF from a perturbation
of hs1r/pt50, here of the form $X|_{t=0}=X+0.4(\cos(\vt)(\max(z)-z
+0.1({\rm rand}-0.5))N$, where $\vt$ is the angle in the $z=0$ plane.
We do not use any BCs
in the rhs vpmcff.m. Instead, we use the correction

N(p.idx,3)=0; N=normalizerow(N)

in getN.m.
Without hsNc, the continuation of hemispheres
essentially works as before,
but the solutions of the VPMCF start to (slowly) lift off the $z=0$ plane,
and then (quickly) evolve towards a planar $X$ such that also $V\to 0$.
With hsNc, the solutions flow back to hemispheres,
and $V$ is again conserved up to $0.5\%$, and “after convergence”
(e.g., from $X(2)$) we can start continuation again, showing

§.§ Spherical caps via 2D finite elements

For the sake of completeness and possible generalization,
we show how the spherical caps with DBCs
can be treated in a classical FEM setting.
Let $\Om\subset\R^2$ be a bounded domain and
$X_0$ be a surface with parametrization $\phi_0:\Omega\ra\R^3$,
and as before define a new surface via
$X=X_0+u\,N_0$, $u:\Om\to\R$.  Then CMC reads

=0,  with $u|_{\pa \Om}=0$.

This is a quasilinear elliptic equation for $u:\Om\to\R$,
and after solving feq2 we can again update $X_0=X_0+uN_0$ and repeat.
To discretize feq2 we now use the FEM
in $\Om$. The main issue is how to compute $H$ (and $A$ and $V$ and
similar quantities) without the ,
and Table <ref> lists the pertinent files.

Overview of files in geomtut/spcap2.

cmds1   Main script; initialization, continuation, plotting.

spcapinit   Initialization, rather standard, except for p.X=[x,y,0*x] for the initialization of $X$.

oosetfemops   Setting mass matrix $M$, and first order differentiation
matrices which are used to compute $H$ in sG (via getmeancurv).

sG, qV   rhs and volume constraint.

getN   compute normal vector.

getA, getV   compute $A(X)$ and $V(X)$, overload of Xcont functions.

getmeancurv   $H$ from h1, see also  get1ff, get2ff for
1st and 2nd fundamental forms.

e2rs   ElementToRefineSelector function, based on triangle areas.

cmcbra, pplot   like Xcont/cmcbra and Xcont/pplot,
but overloaded since p.tri not present here.

getGupde   overload of library function to deal with larger bandwidth.

To implement $H(X)$, $X=X_0+uN$, we here directly use the definition

H=1/2 h_11g_22-2h_12g_12+h_22g_11/g_11g_22-g_12^2

of the mean curvature
based on the fundamental forms of $X$. This is brute force
and in particular neither confirms to a weak (FEM) formulation
nor allows simple Jacobians, see Remark <ref>.

a) In getN and get1ff we compute the
nodal values for $X_x$ and $X_y$ (via weighted averages of
the adjacent triangles), and the corresponding
nodal values of $N$ and $g_{ij}$; this is needed here as
$X_x$ and $X_y$ appear nonlinearly in $N$, and similarly $g_{ij}$
appear nonlinearly in $H$. On the other hand, the second derivatives
$h_{ij}$ only appear linearly in $H$, and hence we
take the second derivatives in get2ff using the element
differentiation matrices ${\tt Kx}$ and ${\tt Ky}$. Thus, ${\tt H}$
in r=-2*H+p.mat.M*(H0*ones(p.nu,1)) in sG is (an approximation of)
the element wise mean curvature, and hence ${\tt H0}$ must also be multiplied
by the mass matrix p.mat.M.

b) Since $G$ as implemented uses products
of differentiation matrices, the associated Jacobian $\pa_u G$ has
more bandwidth than before, i.e., the sparsity pattern $S$ of $\pa_u G$
is that of $M^2$, rather than that of $M$, and thus we use
${\tt S=M^2>0}$ in getGupde.

caption=spcapinit, oosetfemops, sG, getmeancurv and getV from spcap1. See sources for, e.g.,
get1ff, get2ff, and the Element2RefineSelector function e2rs used for mesh adaptation.

Despite the caveats in Remark <ref>, we can now set up a
simple script (Listing <ref>) and produce a continuation
diagram and sample plots fully analogous to <ref>.
For mesh–refinement, to select
triangles to refine we use the areas on $X$, see e2rs,
but otherwise the adaptive mesh refinement works as usual
in the legacy setting of  via oomeshada.

linerange=3-13,caption=Short script cmds1.m from geomtut/spcap.,label=spl2/cmds1.m

As already said, the main advantage of this setup is simplicity in the
sense that
the function getmeancurv (based on get1ff and get2ff)
is a direct translation of the differential geometric definition.
Moreover,  we can work with fixed preassembled differentiation
matrices (independent of $X$) as long as the mesh (in $\Om$) is fixed.
The main disadvantage is that this
implementation of h1 is not a weak form but mixes
FEM and FD differentiation matrices.


# Towards a general purpose machine translation system for Sranantongo

Just Zwennicker⋆† David Stap
Language Technology Lab
University of Amsterdam

###### Abstract

Machine translation for Sranantongo (Sranan, srn), a low-resource Creole
language spoken predominantly in Surinam, is virgin territory. In this study
we create a general purpose machine translation system for srn. In order to
facilitate this research, we introduce the SRNcorpus, a collection of parallel
Dutch (nl) to srn and monolingual srn data. We experiment with a wide range of
proven machine translation methods. Our results demonstrate a strong baseline
machine translation system for srn.

Towards a general purpose machine translation system for Sranantongo

Just Zwennicker⋆† David Stap Language Technology Lab University of Amsterdam

##  1 Introduction

The official language in Surinam is Dutch, however the language that you will
most likely hear on the streets of Paramaribo, its capital city, is Sranan. It
is a lingua franca that finds its origins in the 17th century when the
transatlantic slave trade brought people of different cultures and languages
together. This Creole language broke down the language barrier between the
different social groups.

There are approximately 600k speakers of Sranan worldwide Vissio and Zakharov
(2021), most of which live in Surinam, and in the Netherlands, their former
colonizer. Around 1975, when Surinam became independent, a large group of
people emigrated from Surinam to the Netherlands. That so called first
generation is mostly fluent in Sranan, but this is often not the case for the
second and third generation. A machine translation (MT) system for nl/srn
could assist with (re)learning the language Lent et al. (2022) and facilitate
familiarizing themselves with their cultural background.

Although Sranan is the second largest language spoken in Surinam there are
relatively few written sources available. This could in part be explained by
the stigmatization of the language, as is the case with many Creole languages
Lent et al. (2022).

##  2 Method, setup and results

SRNcorpus We have collected data from various domains to create the SRNcorpus.
It consists of parallel nl-srn as well as monolingual srn data. The largest
data source contains religious data, in particular the Jehova Witness bible
translations (Željko Agic and Vulic, 2020)111Unfortunately the JW300 became
unavailable during our research, reportedly due to copyright issues. This
limited the options for our multilingual setup, as we were not able to secure
data for other language pairs containing Sranan. Furthermore we scraped around
3k non-religious, parallel sentences from an online Sranan dictionary222SIL
parallel sentences. For numerous words listed in the dictionary, it contains
an example sentence showing how it could be used, along with the Dutch
translation of that sentence. This website also contained links to various
(children) stories333SIL stories in Sranan, yielding over 6.5k monolingual
sentences. Lastly the SRNcorpus contains 5 smaller parallel sources from
various domains. See Table 1 for a detailed description of its contents.

While religious data is over-represented, our goal is to create a general
purpose translation system. We therefore create non-religious validation and
test sets, by sampling two times 256 parallel sentences from the non-religious
sources within SRNcorpus to create test and validation sets.
Other data In addition to SRNcorpus, we used the following parallel data for
our multilingual and transfer learning experiments: Wikimatrix Schwenk et al.
(2021) (nl-English (en) 3.3M; nl-Portugese (pt) 1.2M), TED2020 Reimers and
Gurevych (2020) (nl-en 317K; nl-pt 260K) and Europarl Koehn (2005) (nl-en 2M;
nl-pt 1.9M). We used pt since srn inherited features from this language Seuren
(1981), and similarity between parent and child languages can stimulate
transfer Johnson et al. (2017).

source | type | language(s) | Domain | #sentences
---|---|---|---|---
JW300 | parallel | nl-srn | religious | 307,866
SIL | parallel | nl-srn | general | 2,927
Z-Library-1 | parallel | nl-srn | stories | 351
Z-Library-2 | parallel | nl-srn | stories | 163
Z-Library-3 | parallel | nl-srn | stories | 399
Naks Sranan fb | parallel | nl-srn | general | 62
Dutch DOJ | parallel | nl-srn | legal | 220
SIL | monolingual | srn | stories | 6,572
total: |  |  |  | 318,560
Table 1: Contents of SRNcorpus. The Summer Institute of Linguistics (SIL)
sentences are scraped from an online Sranan dictionary. Z-library 1, 2 and 3
are authored by Tori di switi fu leisi / H.C. Tiendalli, Lafu tori / J. Redan
and Dri Anansi tori / H.C. Tiendalli respectively. Naks Sranan is the facebook
page of a cultural organisation and contains member introductions. Finally,
the Dutch DOJ data contains warrants for arrest.

Domain temperature sampling Temperature sampling is often used to overcome
size differences between language pairs by oversampling lower resource pairs.
In contrast, we applied temperature sampling with the goal of reducing domain
imbalance. We distinguish between religious vs non-religious data within
SRNcorpus.
Experimental setup First we experiment with bilingual models (+
backtranslation). Then we investigate transfer learning, where we first train
a parent model and then finetune on a child model. Finally we train
multilingual models by sharing all parameters between languages and prepending
a target token Johnson et al. (2017). For all experiments we used the
Transformer base architecture Vaswani et al. (2017). We implemented all our
models using JoeyNMT Kreutzer et al. (2019). For evaluation we calculated BLEU
scores using sacrebleu Post (2018).
Results See Table 2 for an overview of our results.

type | $T$ | BPE | nl-srn | srn-nl
---|---|---|---|---
bl | - | 1000 | 22.06 | 15.04
SRN | 3 | 3000 | - | 28.00
SRN | 2 | 4000 | 35.48 | -
SRN | 2 | 3000 | 36.78 | -
SRN | 2 | 2000 | 36.86 | -
SRN | 2 | 1000 | 37.48 | 27.47
SRN | 2 | 500 | 36.60 | -
SRN | 1 | 1000 | 32.92 | 24.93
SRN | 3 | 1000 | 36.99 | 28.47
bt | 2 | 1000 | 38.88 | -
bt⋆ | 2 | 1000 | 37.29 | -
bt | 3 | 1000 | - | 27.41
bt⋆ | 3 | 1000 | - | 27.87
tl | 2 | 32000 | 37.33 | -
tl | 2 | 16000 | 37.49 | -
tl | 2 | 8000 | 38.85 | -
tl | 3 | 32000 | - | 32.02
tl | 3 | 16000 | - | 30.78
tl | 3 | 8000 | - | 29.70
ml | 2 | 32000 | 30.04 | -
ml | 2 | 16000 | 29.88 | 23.02
ml | 2 | 8000 | 29.00 | -
ml | 3 | 32000 | - | 21.71
Table 2: BLEU scores reported on the non-religious test set of the SRNcorpus.
$T$ is domain sampling temperature, BPE is the number of merge ops. bl is
baseline trained using only religious JW300 data. SRN are bilingual models
trained on SRNcorpus. bt are backtranslation experiments ($\star$ indicates
inclusion of synthetic data). tl are transfer learning experiments. ml are
multilingual experiments.

Bilingual We found that for both nl-srn and srn-nl, using SRNcorpus instead of
JW300 increased BLEU scores by 10+ points. 1000 BPE merge operations produced
best results, which is in line with (Gowda and May, 2020) who studied the
effect of the number of BPE merge operations in relation to the parallel
corpus size on translation performance. For nl-srn best results were obtained
with $T=2$, whereas for srn-nl $T=3$ worked best.
Backtranslation For our backtranslation experiments we used the best
performing bilingual srn-nl model (BPE=1000; $T=3$) to translate the
monolingual data from SRNcorpus into nl. We then trained models in both
directions using this synthetic generated data on top of our SRNcorpus. For
BPE and $T$ we used the best values according to the bilingual models. We
found a +1.4 BLEU increase for nl-srn, scoring highest overall for this
direction, while our srn-nl model decreased slightly. In addition we
experimented with applying domain temperature sampling to the synthetic
parallel data as well. We found this to hurt performance for all translation
directions.
Transfer Learning Since Sranan is an English based Creole language, the
language pair nl-en is a natural choice as a parent model. After training the
parent model until convergence, we initialized our child models with the
resulting parameters. We increased the number of merge ops to accommodate for
the increased vocabulary. For nl-srn, BPE=8k resulted in best translation
quality, almost on par with the backtranslation experiment. For srn-nl, best
results were achieved with BPE=32k, resulting in the highest BLEU score for
this direction.
Multilingual For our multilingual models we used our SRNcorpus plus other data
as described in the previous Section. We report scores for nl-srn (which are
obtained by training a one-to-many model on nl-en, nl-pt and nl-srn) and srn-
nl (which are obtained by training a many-to-one model on the reverse
directions). Note that we applied language temperature sampling ($T=5$) to
oversample nl-srn and nl-pt. We found that resulting models perform
substantially worse compared to our other models (except the baseline).

##  3 Conclusion

In this study we have put NMT for Sranan on the map. We introduced the
SRNcorpus and used it for various experiments in search for a performant
general purpose machine translation for nl→srn and srn→nl. Our results
demonstrate a strong baseline machine translation system for Sranantongo,
which future work can build on.

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