<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
the	O
quasi-Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
is	O
a	O
method	O
for	O
numerical	B-Algorithm
integration	I-Algorithm
and	O
solving	O
some	O
other	O
problems	O
using	O
low-discrepancy	O
sequences	O
(	O
also	O
called	O
quasi-random	O
sequences	O
or	O
sub-random	O
sequences	O
)	O
.	O
</s>
<s>
This	O
is	O
in	O
contrast	O
to	O
the	O
regular	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
or	O
Monte	B-Algorithm
Carlo	I-Algorithm
integration	I-Algorithm
,	O
which	O
are	O
based	O
on	O
sequences	O
of	O
pseudorandom	B-Error_Name
numbers	I-Error_Name
.	O
</s>
<s>
Monte	O
Carlo	O
and	O
quasi-Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
are	O
stated	O
in	O
a	O
similar	O
way	O
.	O
</s>
<s>
Quasi-Monte	O
Carlo	O
uses	O
a	O
low-discrepancy	O
sequence	O
such	O
as	O
the	O
Halton	O
sequence	O
,	O
the	O
Sobol	O
sequence	O
,	O
or	O
the	O
Faure	O
sequence	O
,	O
whereas	O
Monte	O
Carlo	O
uses	O
a	O
pseudorandom	B-Error_Name
sequence	O
.	O
</s>
<s>
Quasi-Monte	O
Carlo	O
has	O
a	O
rate	O
of	O
convergence	O
close	O
to	O
O( 	O
1/N	O
)	O
,	O
whereas	O
the	O
rate	O
for	O
the	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
is	O
O( N−	O
0.5	O
)	O
.	O
</s>
<s>
The	O
Quasi-Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
recently	O
became	O
popular	O
in	O
the	O
area	O
of	O
mathematical	O
finance	O
or	O
computational	O
finance	O
.	O
</s>
<s>
Hence	O
,	O
the	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
and	O
the	O
quasi-Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
are	O
beneficial	O
in	O
these	O
situations	O
.	O
</s>
<s>
The	O
approximation	O
error	O
of	O
the	O
quasi-Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
is	O
bounded	O
by	O
a	O
term	O
proportional	O
to	O
the	O
discrepancy	O
of	O
the	O
set	O
x1	O
,	O
...	O
,	O
xN	O
.	O
</s>
<s>
The	O
inequality	O
can	O
be	O
used	O
to	O
show	O
that	O
the	O
error	O
of	O
the	O
approximation	O
by	O
the	O
quasi-Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
is	O
,	O
whereas	O
the	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
has	O
a	O
probabilistic	O
error	O
of	O
.	O
</s>
<s>
For	O
one-dimensional	O
integration	O
,	O
quadrature	O
methods	O
such	O
as	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
,	O
Simpson	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
,	O
or	O
Newton	B-Algorithm
–	I-Algorithm
Cotes	I-Algorithm
formulas	I-Algorithm
are	O
known	O
to	O
be	O
efficient	O
if	O
the	O
function	O
is	O
smooth	O
.	O
</s>
<s>
Hence	O
,	O
a	O
method	O
that	O
can	O
overcome	O
this	O
curse	B-Algorithm
of	I-Algorithm
dimensionality	I-Algorithm
should	O
be	O
used	O
for	O
multidimensional	O
integrations	O
.	O
</s>
<s>
The	O
standard	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
is	O
frequently	O
used	O
when	O
the	O
quadrature	O
methods	O
are	O
difficult	O
or	O
expensive	O
to	O
implement	O
.	O
</s>
<s>
Monte	O
Carlo	O
and	O
quasi-Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
are	O
accurate	O
and	O
relatively	O
fast	O
when	O
the	O
dimension	O
is	O
high	O
,	O
up	O
to	O
300	O
or	O
higher	O
.	O
</s>
<s>
Morokoff	O
and	O
Caflisch	O
studied	O
the	O
performance	O
of	O
Monte	O
Carlo	O
and	O
quasi-Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
for	O
integration	O
.	O
</s>
<s>
In	O
the	O
paper	O
,	O
Halton	O
,	O
Sobol	O
,	O
and	O
Faure	O
sequences	O
for	O
quasi-Monte	O
Carlo	O
are	O
compared	O
with	O
the	O
standard	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
using	O
pseudorandom	B-Error_Name
sequences	O
.	O
</s>
<s>
They	O
found	O
that	O
the	O
Halton	O
sequence	O
performs	O
best	O
for	O
dimensions	O
up	O
to	O
around	O
6	O
;	O
the	O
Sobol	O
sequence	O
performs	O
best	O
for	O
higher	O
dimensions	O
;	O
and	O
the	O
Faure	O
sequence	O
,	O
while	O
outperformed	O
by	O
the	O
other	O
two	O
,	O
still	O
performs	O
better	O
than	O
a	O
pseudorandom	B-Error_Name
sequence	O
.	O
</s>
<s>
Still	O
,	O
in	O
the	O
examples	O
studied	O
by	O
Morokoff	O
and	O
Caflisch	O
,	O
the	O
quasi-Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
did	O
yield	O
a	O
more	O
accurate	O
result	O
than	O
the	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
with	O
the	O
same	O
number	O
of	O
points	O
.	O
</s>
<s>
Morokoff	O
and	O
Caflisch	O
remark	O
that	O
the	O
advantage	O
of	O
the	O
quasi-Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
is	O
greater	O
if	O
the	O
integrand	O
is	O
smooth	O
,	O
and	O
the	O
number	O
of	O
dimensions	O
s	O
of	O
the	O
integral	O
is	O
small	O
.	O
</s>
<s>
In	O
order	O
to	O
overcome	O
some	O
of	O
these	O
difficulties	O
,	O
we	O
can	O
use	O
a	O
randomized	O
quasi-Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Since	O
the	O
low	O
discrepancy	O
sequence	O
are	O
not	O
random	O
,	O
but	O
deterministic	O
,	O
quasi-Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
can	O
be	O
seen	O
as	O
a	O
deterministic	O
algorithm	O
or	O
derandomized	O
algorithm	O
.	O
</s>
<s>
The	O
resulting	O
method	O
is	O
called	O
the	O
randomized	O
quasi-Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
and	O
can	O
be	O
also	O
viewed	O
as	O
a	O
variance	O
reduction	O
technique	O
for	O
the	O
standard	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
.	O
</s>
