<s>
the	O
matrix	O
is	O
also	O
assumed	O
positive-definite	B-Algorithm
.	O
</s>
<s>
Kantorovich	O
in	O
1948	O
proposed	O
calculating	O
the	O
smallest	O
eigenvalue	O
of	O
a	O
symmetric	B-Algorithm
matrix	I-Algorithm
by	O
steepest	B-Algorithm
descent	I-Algorithm
using	O
a	O
direction	O
of	O
a	O
scaled	O
gradient	O
of	O
a	O
Rayleigh	O
quotient	O
in	O
a	O
scalar	O
product	O
,	O
with	O
the	O
step	O
size	O
computed	O
by	O
minimizing	O
the	O
Rayleigh	O
quotient	O
in	O
the	O
linear	O
span	O
of	O
the	O
vectors	O
and	O
,	O
i.e.	O
</s>
<s>
Samokish	O
proposed	O
applying	O
a	O
preconditioner	O
to	O
the	O
residual	B-Algorithm
vector	O
to	O
generate	O
the	O
preconditioned	O
direction	O
and	O
derived	O
asymptotic	O
,	O
as	O
approaches	O
the	O
eigenvector	O
,	O
convergence	B-Architecture
rate	I-Architecture
bounds	O
.	O
</s>
<s>
D'yakonov	O
suggested	O
spectrally	O
equivalent	O
preconditioning	O
and	O
derived	O
non-asymptotic	O
convergence	B-Architecture
rate	I-Architecture
bounds	O
.	O
</s>
<s>
Block	O
locally	O
optimal	O
multi-step	O
steepest	B-Algorithm
descent	I-Algorithm
for	O
eigenvalue	O
problems	O
was	O
described	O
in	O
.	O
</s>
<s>
Local	O
minimization	O
of	O
the	O
Rayleigh	O
quotient	O
on	O
the	O
subspace	O
spanned	O
by	O
the	O
current	O
approximation	O
,	O
the	O
current	O
residual	B-Algorithm
and	O
the	O
previous	O
approximation	O
,	O
as	O
well	O
as	O
its	O
block	O
version	O
,	O
appeared	O
in	O
.	O
</s>
<s>
The	O
costs	O
per	O
iteration	B-Algorithm
and	O
the	O
memory	O
use	O
are	O
competitive	O
with	O
those	O
of	O
the	O
Lanczos	O
method	O
,	O
computing	O
a	O
single	O
extreme	O
eigenpair	O
of	O
a	O
symmetric	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Linear	B-Architecture
convergence	I-Architecture
is	O
theoretically	O
guaranteed	O
and	O
practically	O
observed	O
.	O
</s>
<s>
Accelerated	O
convergence	O
due	O
to	O
direct	O
preconditioning	O
,	O
in	O
contrast	O
to	O
the	O
Lanczos	O
method	O
,	O
including	O
variable	O
and	O
non-symmetric	O
as	O
well	O
as	O
fixed	O
and	O
positive	O
definite	O
preconditioning	O
.	O
</s>
<s>
Allows	O
trivial	O
incorporation	O
of	O
efficient	O
domain	B-Algorithm
decomposition	I-Algorithm
and	O
multigrid	O
techniques	O
via	O
preconditioning	O
.	O
</s>
<s>
Warm	O
starts	O
and	O
computes	O
an	O
approximation	O
to	O
the	O
eigenvector	O
on	O
every	O
iteration	B-Algorithm
.	O
</s>
<s>
Blocking	O
allows	O
utilizing	O
highly	O
efficient	O
matrix-matrix	O
operations	O
,	O
e.g.	O
,	O
BLAS	B-Application
3	O
.	O
</s>
<s>
The	O
block	O
size	O
can	O
be	O
tuned	O
to	O
balance	O
convergence	O
speed	O
vs.	O
computer	O
costs	O
of	O
orthogonalizations	O
and	O
the	O
Rayleigh-Ritz	B-Algorithm
method	I-Algorithm
on	O
every	O
iteration	B-Algorithm
.	O
</s>
<s>
called	O
the	O
eigenvector	O
residual	B-Algorithm
.	O
</s>
<s>
called	O
the	O
preconditioned	O
residual	B-Algorithm
.	O
</s>
<s>
To	O
dramatically	O
accelerate	O
the	O
convergence	O
of	O
the	O
locally	O
optimal	O
preconditioned	O
steepest	B-Algorithm
ascent	I-Algorithm
(	O
or	O
descent	O
)	O
,	O
one	O
extra	O
vector	O
can	O
be	O
added	O
to	O
the	O
two-term	O
recurrence	O
relation	O
to	O
make	O
it	O
three-term	O
:	O
</s>
<s>
The	O
maximization/minimization	O
of	O
the	O
Rayleigh	O
quotient	O
in	O
a	O
3-dimensional	O
subspace	O
can	O
be	O
performed	O
numerically	O
by	O
the	O
Rayleigh	B-Algorithm
–	I-Algorithm
Ritz	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Adding	O
more	O
vectors	O
,	O
see	O
,	O
e.g.	O
,	O
Richardson	B-Algorithm
extrapolation	I-Algorithm
,	O
does	O
not	O
result	O
in	O
significant	O
acceleration	O
but	O
increases	O
computation	O
costs	O
,	O
so	O
is	O
not	O
generally	O
recommended	O
.	O
</s>
<s>
As	O
the	O
iterations	B-Algorithm
converge	O
,	O
the	O
vectors	O
and	O
become	O
nearly	O
linearly	O
dependent	O
,	O
resulting	O
in	O
a	O
precision	O
loss	O
and	O
making	O
the	O
Rayleigh	B-Algorithm
–	I-Algorithm
Ritz	I-Algorithm
method	I-Algorithm
numerically	O
unstable	O
in	O
the	O
presence	O
of	O
round-off	O
errors	O
.	O
</s>
<s>
Furthermore	O
,	O
orthogonalizing	O
the	O
basis	O
of	O
the	O
three-dimensional	O
subspace	O
may	O
be	O
needed	O
for	O
ill-conditioned	B-Algorithm
eigenvalue	O
problems	O
to	O
improve	O
stability	O
and	O
attainable	O
accuracy	O
.	O
</s>
<s>
This	O
is	O
a	O
single-vector	O
version	O
of	O
the	O
LOBPCG	B-Application
method	O
—	O
one	O
of	O
possible	O
generalization	O
of	O
the	O
preconditioned	O
conjugate	B-Algorithm
gradient	I-Algorithm
linear	O
solvers	O
to	O
the	O
case	O
of	O
symmetric	B-Algorithm
eigenvalue	O
problems	O
.	O
</s>
<s>
Extreme	O
simplicity	O
and	O
high	O
efficiency	O
of	O
the	O
single-vector	O
version	O
of	O
LOBPCG	B-Application
make	O
it	O
attractive	O
for	O
eigenvalue-related	O
applications	O
under	O
severe	O
hardware	O
limitations	O
,	O
ranging	O
from	O
spectral	B-Algorithm
clustering	I-Algorithm
based	O
real-time	O
anomaly	B-Algorithm
detection	I-Algorithm
via	O
graph	O
partitioning	O
on	O
embedded	O
ASIC	O
or	O
FPGA	B-Architecture
to	O
modelling	O
physical	O
phenomena	O
of	O
record	O
computing	O
complexity	O
on	O
exascale	B-General_Concept
TOP500	B-Operating_System
supercomputers	B-Architecture
.	O
</s>
<s>
Subsequent	O
eigenpairs	O
can	O
be	O
computed	O
one-by-one	O
via	O
single-vector	O
LOBPCG	B-Application
supplemented	O
with	O
an	O
orthogonal	O
deflation	O
or	O
simultaneously	O
as	O
a	O
block	O
.	O
</s>
<s>
Iterating	O
several	O
approximate	O
eigenvectors	O
together	O
in	O
a	O
block	O
in	O
a	O
locally	O
optimal	O
fashion	O
in	O
the	O
block	O
version	O
of	O
the	O
LOBPCG	B-Application
.	O
</s>
<s>
allows	O
fast	O
,	O
accurate	O
,	O
and	O
robust	O
computation	O
of	O
eigenvectors	O
,	O
including	O
those	O
corresponding	O
to	O
nearly-multiple	O
eigenvalues	O
where	O
the	O
single-vector	O
LOBPCG	B-Application
suffers	O
from	O
slow	O
convergence	O
.	O
</s>
<s>
The	O
block	O
size	O
can	O
be	O
tuned	O
to	O
balance	O
numerical	O
stability	O
vs.	O
convergence	O
speed	O
vs.	O
computer	O
costs	O
of	O
orthogonalizations	O
and	O
the	O
Rayleigh-Ritz	B-Algorithm
method	I-Algorithm
on	O
every	O
iteration	B-Algorithm
.	O
</s>
<s>
The	O
block	O
approach	O
in	O
LOBPCG	B-Application
replaces	O
single-vectors	O
and	O
with	O
block-vectors	O
,	O
i.e.	O
</s>
<s>
All	O
columns	O
are	O
iterated	B-Algorithm
simultaneously	O
,	O
and	O
the	O
next	O
matrix	O
of	O
approximate	O
eigenvectors	O
is	O
determined	O
by	O
the	O
Rayleigh	B-Algorithm
–	I-Algorithm
Ritz	I-Algorithm
method	I-Algorithm
on	O
the	O
subspace	O
spanned	O
by	O
all	O
columns	O
of	O
matrices	O
and	O
.	O
</s>
<s>
Each	O
column	O
of	O
is	O
computed	O
simply	O
as	O
the	O
preconditioned	O
residual	B-Algorithm
for	O
every	O
column	O
of	O
The	O
matrix	O
is	O
determined	O
such	O
that	O
the	O
subspaces	O
spanned	O
by	O
the	O
columns	O
of	O
and	O
of	O
are	O
the	O
same	O
.	O
</s>
<s>
The	O
outcome	O
of	O
the	O
Rayleigh	B-Algorithm
–	I-Algorithm
Ritz	I-Algorithm
method	I-Algorithm
is	O
determined	O
by	O
the	O
subspace	O
spanned	O
by	O
all	O
columns	O
of	O
matrices	O
and	O
,	O
where	O
a	O
basis	O
of	O
the	O
subspace	O
can	O
theoretically	O
be	O
arbitrary	O
.	O
</s>
<s>
However	O
,	O
in	O
inexact	O
computer	O
arithmetic	O
the	O
Rayleigh	B-Algorithm
–	I-Algorithm
Ritz	I-Algorithm
method	I-Algorithm
becomes	O
numerically	O
unstable	O
if	O
some	O
of	O
the	O
basis	O
vectors	O
are	O
approximately	O
linearly	O
dependent	O
.	O
</s>
<s>
Numerical	O
instabilities	O
typically	O
occur	O
,	O
e.g.	O
,	O
if	O
some	O
of	O
the	O
eigenvectors	O
in	O
the	O
iterative	B-Algorithm
block	O
already	O
reach	O
attainable	O
accuracy	O
for	O
a	O
given	O
computer	O
precision	O
and	O
are	O
especially	O
prominent	O
in	O
low	O
precision	O
,	O
e.g.	O
,	O
single	O
precision	O
.	O
</s>
<s>
The	O
art	O
of	O
multiple	O
different	O
implementation	O
of	O
LOBPCG	B-Application
is	O
to	O
ensure	O
numerical	O
stability	O
of	O
the	O
Rayleigh	B-Algorithm
–	I-Algorithm
Ritz	I-Algorithm
method	I-Algorithm
at	O
minimal	O
computing	O
costs	O
by	O
choosing	O
a	O
good	O
basis	O
of	O
the	O
subspace	O
.	O
</s>
<s>
The	O
arguably	O
most	O
stable	O
approach	O
of	O
making	O
the	O
basis	O
vectors	O
orthogonal	O
,	O
e.g.	O
,	O
by	O
the	O
Gram	B-Algorithm
–	I-Algorithm
Schmidt	I-Algorithm
process	I-Algorithm
,	O
is	O
also	O
the	O
most	O
computational	O
expensive	O
.	O
</s>
<s>
For	O
example	O
,	O
LOBPCG	B-Application
implementations	O
,	O
utilize	O
unstable	O
but	O
efficient	O
Cholesky	O
decomposition	O
of	O
the	O
normal	O
matrix	O
,	O
which	O
is	O
performed	O
only	O
on	O
individual	O
matrices	O
and	O
,	O
rather	O
than	O
on	O
the	O
whole	O
subspace	O
.	O
</s>
<s>
The	O
constantly	O
increasing	O
amount	O
of	O
computer	O
memory	O
allows	O
typical	O
block	O
sizes	O
nowadays	O
in	O
the	O
range	O
,	O
where	O
the	O
percentage	O
of	O
compute	O
time	O
spend	O
on	O
orthogonalizations	O
and	O
the	O
Rayleigh-Ritz	B-Algorithm
method	I-Algorithm
starts	O
dominating	O
.	O
</s>
<s>
Block	O
methods	O
for	O
eigenvalue	O
problems	O
that	O
iterate	B-Algorithm
subspaces	O
commonly	O
have	O
some	O
of	O
the	O
iterative	B-Algorithm
eigenvectors	O
converged	O
faster	O
than	O
others	O
that	O
motivates	O
locking	O
the	O
already	O
converged	O
eigenvectors	O
,	O
i.e.	O
,	O
removing	O
them	O
from	O
the	O
iterative	B-Algorithm
loop	O
,	O
in	O
order	O
to	O
eliminate	O
unnecessary	O
computations	O
and	O
improve	O
numerical	O
stability	O
.	O
</s>
<s>
The	O
fact	O
that	O
the	O
eigenvectors	O
of	O
symmetric	B-Algorithm
eigenvalue	O
problems	O
are	O
pair-wise	O
orthogonal	O
suggest	O
keeping	O
all	O
iterative	B-Algorithm
vectors	O
orthogonal	O
to	O
the	O
locked	O
vectors	O
.	O
</s>
<s>
For	O
example	O
,	O
LOBPCG	B-Application
implementations	O
,	O
follow	O
,	O
separating	O
hard	O
locking	O
,	O
i.e.	O
</s>
<s>
a	O
deflation	O
by	O
restriction	O
,	O
where	O
the	O
locked	O
eigenvectors	O
serve	O
as	O
a	O
code	O
input	O
and	O
do	O
not	O
change	O
,	O
from	O
soft	O
locking	O
,	O
where	O
the	O
locked	O
vectors	O
do	O
not	O
participate	O
in	O
the	O
typically	O
most	O
expensive	O
iterative	B-Algorithm
step	O
of	O
computing	O
the	O
residuals	B-Algorithm
,	O
however	O
,	O
fully	O
participate	O
in	O
the	O
Rayleigh	O
—	O
Ritz	O
method	O
and	O
thus	O
are	O
allowed	O
to	O
be	O
changed	O
by	O
the	O
Rayleigh	O
—	O
Ritz	O
method	O
.	O
</s>
<s>
LOBPCG	B-Application
includes	O
all	O
columns	O
of	O
matrices	O
and	O
into	O
the	O
Rayleigh	B-Algorithm
–	I-Algorithm
Ritz	I-Algorithm
method	I-Algorithm
resulting	O
in	O
an	O
up	O
to	O
-by	O
-	O
eigenvalue	O
problem	O
needed	O
to	O
solve	O
and	O
up	O
to	O
dot	O
products	O
to	O
compute	O
at	O
every	O
iteration	B-Algorithm
,	O
where	O
denotes	O
the	O
block	O
size	O
—	O
the	O
number	O
of	O
columns	O
.	O
</s>
<s>
The	O
original	O
LOBPCG	B-Application
paper	O
describes	O
a	O
modification	O
,	O
called	O
LOBPCG	B-Application
II	O
,	O
to	O
address	O
such	O
a	O
problem	O
running	O
the	O
single-vector	O
version	O
of	O
the	O
LOBPCG	B-Application
method	O
for	O
each	O
desired	O
eigenpair	O
with	O
the	O
Rayleigh-Ritz	B-Algorithm
procedure	O
solving	O
of	O
3-by-3	O
projected	O
eigenvalue	O
problems	O
.	O
</s>
<s>
The	O
global	O
Rayleigh-Ritz	B-Algorithm
procedure	O
for	O
all	O
eigenpairs	O
is	O
on	O
every	O
iteration	B-Algorithm
but	O
only	O
on	O
the	O
columns	O
of	O
the	O
matrix	O
,	O
thus	O
reducing	O
the	O
number	O
of	O
the	O
necessary	O
dot	O
products	O
to	O
from	O
and	O
the	O
size	O
of	O
the	O
global	O
projected	O
eigenvalue	O
problem	O
to	O
-by	O
-	O
from	O
-by	O
-	O
on	O
every	O
iteration	B-Algorithm
.	O
</s>
<s>
Reference	O
goes	O
further	O
applying	O
the	O
LOBPCG	B-Application
algorithm	O
to	O
each	O
approximate	O
eigenvector	O
separately	O
,	O
i.e.	O
,	O
running	O
the	O
unblocked	O
version	O
of	O
the	O
LOBPCG	B-Application
method	O
for	O
each	O
desired	O
eigenpair	O
for	O
a	O
fixed	O
number	O
of	O
iterations	B-Algorithm
.	O
</s>
<s>
The	O
Rayleigh-Ritz	B-Algorithm
procedures	O
in	O
these	O
runs	O
only	O
need	O
to	O
solve	O
a	O
set	O
of	O
3	O
×	O
3	O
projected	O
eigenvalue	O
problems	O
.	O
</s>
<s>
The	O
global	O
Rayleigh-Ritz	B-Algorithm
procedure	O
for	O
all	O
desired	O
eigenpairs	O
is	O
only	O
applied	O
periodically	O
at	O
the	O
end	O
of	O
a	O
fixed	O
number	O
of	O
unblocked	O
LOBPCG	B-Application
iterations	B-Algorithm
.	O
</s>
<s>
Such	O
modifications	O
may	O
be	O
less	O
robust	O
compared	O
to	O
the	O
original	O
LOBPCG	B-Application
.	O
</s>
<s>
Individually	O
running	O
branches	O
of	O
the	O
single-vector	O
LOBPCG	B-Application
may	O
not	O
follow	O
continuous	O
iterative	B-Algorithm
paths	O
flipping	O
instead	O
and	O
creating	O
duplicated	O
approximations	O
to	O
the	O
same	O
eigenvector	O
.	O
</s>
<s>
The	O
single-vector	O
LOBPCG	B-Application
may	O
be	O
unsuitable	O
for	O
clustered	O
eigenvalues	O
,	O
but	O
separate	O
small-block	O
LOBPCG	B-Application
runs	O
require	O
determining	O
their	O
block	O
sizes	O
automatically	O
during	O
the	O
process	O
of	O
iterations	B-Algorithm
since	O
the	O
number	O
of	O
the	O
clusters	O
of	O
eigenvalues	O
and	O
their	O
sizes	O
may	O
be	O
unknown	O
a	O
priori	O
.	O
</s>
<s>
LOBPCG	B-Application
by	O
construction	O
is	O
guaranteed	O
to	O
minimize	O
the	O
Rayleigh	O
quotient	O
not	O
slower	O
than	O
the	O
block	O
steepest	O
gradient	B-Algorithm
descent	I-Algorithm
,	O
which	O
has	O
a	O
comprehensive	O
convergence	O
theory	O
.	O
</s>
<s>
Thus	O
,	O
the	O
gradient	B-Algorithm
descent	I-Algorithm
may	O
slow	O
down	O
in	O
a	O
vicinity	O
of	O
any	O
eigenvector	O
,	O
however	O
,	O
it	O
is	O
guaranteed	O
to	O
either	O
converge	O
to	O
the	O
eigenvector	O
with	O
a	O
linear	B-Architecture
convergence	I-Architecture
rate	O
or	O
,	O
if	O
this	O
eigenvector	O
is	O
a	O
saddle	O
point	O
,	O
the	O
iterative	B-Algorithm
Rayleigh	O
quotient	O
is	O
more	O
likely	O
to	O
drop	O
down	O
below	O
the	O
corresponding	O
eigenvalue	O
and	O
start	O
converging	O
linearly	O
to	O
the	O
next	O
eigenvalue	O
below	O
.	O
</s>
<s>
The	O
worst	O
value	O
of	O
the	O
linear	B-Architecture
convergence	I-Architecture
rate	O
has	O
been	O
determined	O
and	O
depends	O
on	O
the	O
relative	O
gap	O
between	O
the	O
eigenvalue	O
and	O
the	O
rest	O
of	O
the	O
matrix	O
spectrum	O
and	O
the	O
quality	O
of	O
the	O
preconditioner	O
,	O
if	O
present	O
.	O
</s>
<s>
The	O
iterative	B-Algorithm
solution	O
by	O
LOBPCG	B-Application
may	O
be	O
sensitive	O
to	O
the	O
initial	O
eigenvectors	O
approximations	O
,	O
e.g.	O
,	O
taking	O
longer	O
to	O
converge	O
slowing	O
down	O
as	O
passing	O
intermediate	O
eigenpairs	O
.	O
</s>
<s>
A	O
good	O
quality	O
random	O
Gaussian	B-Application
function	O
with	O
the	O
zero	O
mean	O
is	O
commonly	O
the	O
default	O
in	O
LOBPCG	B-Application
to	O
generate	O
the	O
initial	O
approximations	O
.	O
</s>
<s>
In	O
contrast	O
to	O
the	O
Lanczos	O
method	O
,	O
LOBPCG	B-Application
rarely	O
exhibits	O
asymptotic	O
superlinear	B-Architecture
convergence	I-Architecture
in	O
practice	O
.	O
</s>
<s>
LOBPCG	B-Application
can	O
be	O
trivially	O
adapted	O
for	O
computing	O
several	O
largest	O
singular	O
values	O
and	O
the	O
corresponding	O
singular	O
vectors	O
(	O
partial	O
SVD	O
)	O
,	O
e.g.	O
,	O
for	O
iterative	B-Algorithm
computation	O
of	O
PCA	O
,	O
for	O
a	O
data	O
matrix	O
with	O
zero	O
mean	O
,	O
without	O
explicitly	O
computing	O
the	O
covariance	O
matrix	O
,	O
i.e.	O
</s>
<s>
PCA	O
needs	O
the	O
largest	O
eigenvalues	O
of	O
the	O
covariance	O
matrix	O
,	O
while	O
LOBPCG	B-Application
is	O
typically	O
implemented	O
to	O
calculate	O
the	O
smallest	O
ones	O
.	O
</s>
<s>
A	O
simple	O
work-around	O
is	O
to	O
negate	O
the	O
function	O
,	O
substituting	O
for	O
and	O
thus	O
reversing	O
the	O
order	O
of	O
the	O
eigenvalues	O
,	O
since	O
LOBPCG	B-Application
does	O
not	O
care	O
if	O
the	O
matrix	O
of	O
the	O
eigenvalue	O
problem	O
is	O
positive	O
definite	O
or	O
not	O
.	O
</s>
<s>
LOBPCG	B-Application
's	O
inventor	O
,	O
Andrew	O
Knyazev	O
,	O
published	O
a	O
reference	O
implementation	O
called	O
Block	O
Locally	O
Optimal	O
Preconditioned	O
Eigenvalue	O
Xolvers	O
(	O
BLOPEX	O
)	O
with	O
interfaces	O
to	O
PETSc	B-Language
,	O
hypre	B-Library
,	O
and	O
Parallel	O
Hierarchical	O
Adaptive	O
MultiLevel	O
method	O
(	O
PHAML	O
)	O
.	O
</s>
<s>
,	O
Julia	B-Application
,	O
MAGMA	O
,	O
Pytorch	B-Algorithm
,	O
Rust	B-Application
,	O
OpenMP	B-Application
and	O
OpenACC	B-Operating_System
,	O
CuPy	B-Application
(	O
A	O
NumPy-compatible	O
array	O
library	O
accelerated	O
by	O
CUDA	B-Architecture
)	O
,	O
</s>
<s>
Google	B-Application
JAX	I-Application
,	O
</s>
<s>
LOBPCG	B-Application
is	O
implemented	O
,	O
but	O
not	O
included	O
,	O
in	O
TensorFlow	B-Language
.	O
</s>
<s>
Software	O
packages	O
scikit-learn	B-Application
and	O
Megaman	O
use	O
LOBPCG	B-Application
to	O
scale	O
spectral	B-Algorithm
clustering	I-Algorithm
and	O
manifold	O
learning	O
via	O
Laplacian	O
eigenmaps	O
to	O
large	O
data	O
sets	O
.	O
</s>
<s>
NVIDIA	O
has	O
implemented	O
LOBPCG	B-Application
in	O
its	O
nvGRAPH	O
library	O
introduced	O
in	O
CUDA	B-Architecture
8	O
.	O
</s>
<s>
Sphynx	O
,	O
a	O
hybrid	O
distributed	O
-	O
and	O
shared-memory-enabled	O
parallel	O
graph	O
partitioner	O
-	O
the	O
first	O
graph	O
partitioning	O
tool	O
that	O
works	O
on	O
GPUs	O
on	O
distributed-memory	O
settings	O
-	O
uses	O
spectral	B-Algorithm
clustering	I-Algorithm
for	O
graph	O
partitioning	O
,	O
computing	O
eigenvectors	O
on	O
the	O
Laplacian	B-Algorithm
matrix	I-Algorithm
of	O
the	O
graph	O
using	O
LOBPCG	B-Application
from	O
the	O
Anasazi	O
package	O
.	O
</s>
<s>
LOBPCG	B-Application
is	O
implemented	O
in	O
ABINIT	B-Application
(	O
including	O
CUDA	B-Architecture
version	O
)	O
and	O
Octopus	B-Application
.	O
</s>
<s>
It	O
has	O
been	O
used	O
for	O
multi-billion	O
size	O
matrices	O
by	O
Gordon	O
Bell	O
Prize	O
finalists	O
,	O
on	O
the	O
Earth	B-Device
Simulator	I-Device
supercomputer	B-Architecture
in	O
Japan	O
.	O
</s>
<s>
Hubbard	B-Algorithm
model	I-Algorithm
for	O
strongly-correlated	O
electron	O
systems	O
to	O
understand	O
the	O
mechanism	O
behind	O
the	O
superconductivity	O
uses	O
LOBPCG	B-Application
to	O
calculate	O
the	O
ground	O
state	O
of	O
the	O
Hamiltonian	O
on	O
the	O
K	B-Device
computer	I-Device
and	O
multi-GPU	O
systems	O
.	O
</s>
<s>
versions	O
of	O
LOBPCG	B-Application
for	O
Kohn-Sham	O
equations	O
and	O
density	O
functional	O
theory	O
(	O
DFT	O
)	O
using	O
the	O
plane	O
wave	O
basis	O
.	O
</s>
<s>
LOBPCG	B-Application
from	O
BLOPEX	O
is	O
used	O
for	O
preconditioner	O
setup	O
in	O
Multilevel	O
Balancing	B-Algorithm
Domain	I-Algorithm
Decomposition	I-Algorithm
by	I-Algorithm
Constraints	I-Algorithm
(	O
BDDC	B-Algorithm
)	O
solver	O
library	O
BDDCML	O
,	O
which	O
is	O
incorporated	O
into	O
OpenFTL	O
(	O
Open	O
Finite	B-Application
element	I-Application
Template	O
Library	O
)	O
and	O
Flow123d	O
simulator	O
of	O
underground	O
water	O
flow	O
,	O
solute	O
and	O
heat	O
transport	O
in	O
fractured	O
porous	O
media	O
.	O
</s>
<s>
LOBPCG	B-Application
has	O
been	O
implemented	O
in	O
LS-DYNA	B-Algorithm
.	O
</s>
<s>
LOBPCG	B-Application
is	O
one	O
of	O
core	O
eigenvalue	O
solvers	O
in	O
PYFEMax	O
and	O
high	O
performance	O
multiphysics	O
finite	B-Application
element	I-Application
software	O
Netgen/NGSolve	O
.	O
</s>
<s>
LOBPCG	B-Application
from	O
hypre	B-Library
is	O
incorporated	O
into	O
open	B-License
source	I-License
lightweight	O
scalable	O
C++	B-Language
library	O
for	O
finite	B-Application
element	I-Application
methods	I-Application
MFEM	B-Language
,	O
which	O
is	O
used	O
in	O
many	O
projects	O
,	O
including	O
BLAST	B-Application
,	O
XBraid	O
,	O
VisIt	O
,	O
xSDK	O
,	O
the	O
FASTMath	O
institute	O
in	O
SciDAC	O
,	O
and	O
the	O
co-design	O
Center	O
for	O
Efficient	O
Exascale	B-General_Concept
Discretizations	O
(	O
CEED	O
)	O
in	O
the	O
Exascale	B-General_Concept
computing	I-General_Concept
Project	O
.	O
</s>
<s>
Iterative	B-Algorithm
LOBPCG-based	O
approximate	O
low-pass	B-Algorithm
filter	I-Algorithm
can	O
be	O
used	O
for	O
denoising	O
;	O
see	O
,	O
e.g.	O
,	O
to	O
accelerate	O
total	B-Algorithm
variation	I-Algorithm
denoising	I-Algorithm
.	O
</s>
<s>
Image	B-Algorithm
segmentation	I-Algorithm
via	O
spectral	B-Algorithm
clustering	I-Algorithm
performs	O
a	O
low-dimension	O
embedding	O
using	O
an	O
affinity	B-Algorithm
matrix	O
between	O
pixels	O
,	O
followed	O
by	O
clustering	O
of	O
the	O
components	O
of	O
the	O
eigenvectors	O
in	O
the	O
low	O
dimensional	O
space	O
,	O
e.g.	O
,	O
using	O
the	O
graph	B-Algorithm
Laplacian	I-Algorithm
for	O
the	O
bilateral	B-Algorithm
filter	I-Algorithm
.	O
</s>
<s>
Image	B-Algorithm
segmentation	I-Algorithm
via	O
spectral	O
graph	O
partitioning	O
by	O
LOBPCG	B-Application
with	O
multigrid	O
preconditioning	O
has	O
been	O
first	O
proposed	O
in	O
and	O
actually	O
tested	O
in	O
and	O
.	O
</s>
<s>
The	O
latter	O
approach	O
has	O
been	O
later	O
implemented	O
in	O
Python	O
scikit-learn	B-Application
that	O
uses	O
LOBPCG	B-Application
from	O
SciPy	B-Application
with	O
algebraic	O
multigrid	O
preconditioning	O
for	O
solving	O
the	O
eigenvalue	O
problem	O
for	O
the	O
graph	B-Algorithm
Laplacian	I-Algorithm
.	O
</s>
