Slow 
Exponential 
Growth 
for 
Clenshaw 
Curtis 
Sparse 
Grids 
http://people.sc.fsu.edu/jburkardt/presentations/. 
. 
. 
. 
. 
. 
slow 
growth 
paper.pdf 


John 
Burkardt, 
Clayton 
Webster 


June 
9, 
2015 


Abstract 


Smolyak 
demonstrated 
a 
procedure 
for 
ecient 
multidimensional 
quadrature 
based 
on 
a 
family 
of 
one-dimensional 
quadrature 
rules. 
A 
particular 
family 
of 
nested 
Clenshaw 
Curtis 
rules 
is 
most 
commonly 
used 
in 
this 
context. 
Nestedness 
controls 
the 
point 
growth 
of 
sparse 
grids 
in 
high 
dimension, 
but 
in 
lower 
dimensions, 
it 
can 
actually 
result 
in 
signicant 
ineciencies. 
A 
simple 
change 
to 
the 
Smolyak 
construction 
greatly 
reduces 
the 
point 
count 
when 
relatively 
low 
dimensions 
are 
involved. 
Similar 
benets 
can 
be 
obtained 
when 
the 
Smolyak 
procedure 
is 
based 
on 
other 
families 
of 
quadrature 
rules. 
Numerical 
tables 
show 
that 
the 
modied 
rule 
has 
a 
signicant 
point 
count 
reduction, 
while 
preserving 
the 
precision 
of 
the 
unmodied 
rule. 


Introduction 


Sparse 
grids, 
as 
dened 
by 
Smolyak 
[9], 
are 
powerful 
and 
ecient 
tools 
for 
interpolation, 
quadrature, 
and 
optimization 
of 
functions 
with 
dependence 
on 
a 
multidimensional 
argument. 
By 
strictly 
controlling 
the 
number 
of 
function 
evaluations 
required, 
sparse 
grids 
can 
handle 
problems 
in 
dimensions 
far 
beyond 
the 
reach 
of 
any 
other 
methods 
except 
Monte 
Carlo. 
One 
of 
the 
techniques 
that 
can 
be 
used 
to 
extend 
the 
power 
of 
a 
sparse 
grid 
is 
nestedness; 
that 
is, 
the 
reuse 
of 
data 
from 
lower 
order 
calculations. 
A 
very 
common 
example 
of 
a 
sparse 
grid 
is 
based 
on 
the 
nested 
Clenshaw 
Curtis 
rules. 
Nestedness 
comes 
at 
a 
price, 
which 
for 
the 
Clenshaw 
Curtis 
rules 
means 
that 
each 
successive 
element 
of 
the 
family 
uses 
about 
twice 
as 
many 
points 
as 
the 
previous 
one, 
an 
example 
of 
exponential 
growth. 
For 
various 
reasons, 
this 
exponential 
growth 
does 
not 
dominate 
the 
behavior 
of 
sparse 
grids 
in 
high 
dimension 
and 
low 
to 
moderate 
level. 
However, 
in 
relatively 
low 
dimensions, 
say 
d 
= 
5, 
a 
Clenshaw 
Curtis 
sparse 
grid 
incurs 
obvious, 
and 
as 
it 
turns 
out, 
easily 
avoidable, 
expense 
because 
of 
its 
simple-minded 
reliance 
on 
nested 
rules. 


In 
an 
analisis 
of 
the 
properties 
of 
sparse 
grids 
based 
on 
Clenshaw 
Curtis 
rules, 
Novak 
and 
Ritter 
[6] 
showed 
that 
the 
exactness 
of 
the 
sparse 
grid 
is 
related 
in 
a 
simple 
way 
to 
the 
exactness 
of 
the 
1D 
quadrature 
rules 
used 
to 
construct 
it; 
this 
result 
shows 
that 
there 
is 
actually 
considerable 
freedom 
available 
when 
specifying 
how 
a 
sparse 
grid 
is 
to 
be 
constructed. 
Using 
this 
guideline, 
a 
simple 
modication 
allows 
the 
creation 
of 
Clenshaw 
Curtis 
sparse 
grids 
that 
are 
smaller, 
but 
of 
the 
same 
exactness 
as 
the 
classic 
grids. 


Sparse 
grids 
can 
also 
be 
constructed 
from 
Legendre 
quadrature 
rules. 
Since 
these 
rules 
are 
not 
nested, 
the 
construction 
pattern 
associated 
with 
Clenshaw 
Curtis 
rules 
is 
surely 
inappropriate 
as 
a 
model. 
Again, 
the 
Novak 
and 
Ritter 
guideline 
can 
be 
used 
in 
order 
to 
arrive 
at 
an 
ecient 
sparse 
grid 
with 
known 
exactness. 


The 
Patterson 
family 
of 
quadrature 
rules 
represents 
an 
interesting 
mix 
of 
the 
features 
of 
the 
Clenshaw 
Curtis 
and 
Legendre 
families 
of 
rules. 
The 
family 
is 
nested 
(although 
in 
a 
dierent 
way 
from 
the 
Clenshaw 
Curtis 
family) 
and 
of 
increased 
accuracy 
(although 
less 
than 
the 
Legendre 
family). 
Again, 
Novak 
and 
Ritter 
can 
be 
used 
to 
specify 
a 
construction 
plan 
of 
guaranteed 
exactness 
and 
demonstrable 
eciency. 


As 
suggested 
earlier, 
the 
eciency 
to 
be 
gained 
by 
this 
approach 
is 
most 
prominent 
when 
the 
sparse 
grid 
is 
generated 
in 
relatively 
low 
dimensions. 
However, 
many 
sparse 
grid 
techniques 
that 
are 
applied 
to 
high 


1 



dimensions 
actually 
select 
a 
small 
subspace 
for 
preferential 
treatment, 
or 
apply 
anisotropic 
weights 
that 
have 
a 
similar 
eect; 
such 
cases 
may 
also 
be 
be 
regarded 
as 
low 
dimensional, 
and 
hence 
possibly 
aected 
by 
the 
improvements 
considered 
here. 


In 
order 
to 
focus 
on 
the 
main 
point 
of 
the 
argument, 
a 
number 
of 
simplifying 
assumptions 
will 
be 
made: 


• 
sparse 
grids 
are 
being 
constructed 
in 
order 
to 
estimate 
the 
integral 
I(f) 
of 
a 
function 
of 
a 
multidimensiona
l 
argument; 
• 
the 
multidimensional 
integration 
region 
is 
the 
unit 
hypercube 
[..1, 
+1]m; 
• 
Q 
represents 
an 
indexed 
family 
of 
quadrature 
rules 
for 
the 
interval 
[..1, 
+1], 
with 
typical 
element 
Qi; 
• 
the 
same 
family 
Q 
will 
be 
used 
to 
select 
each 
component 
of 
the 
sparse 
grid; 
The 
outline 
of 
the 
remainder 
of 
this 
paper 
is 
as 
follows: 
Section 
2 
presents 
some 
background 
material 
on 
the 
Clenshaw 
Curtis 
family 
of 
1D 
quadrature 
rules, 
the 
Smolyak 
construction 
procedure, 
the 
notion 
of 
exactness 
for 
quadrature, 
and 
the 
Novak 
and 
Ritter 
exactness 
constraint. 
Section 
3 
presents 
the 
classic 
construction 
of 
sparse 
grids 
from 
the 
Clenshaw 
Curtis 
family, 
and 
then 
reconstructs 
such 
sparse 
grids 
using 
Novak 
and 
Ritter. 
Sections 
4 
and 
5 
carry 
out 
similar 
operations 
for 
sparse 
grids 
based 
on 
Legendre 
and 
Patterson 
families. 
Section 
6 
presents 
some 
simple 
numerical 
tests 
indicating 
that 
the 
modied 
approach 
produces 
sparse 
grids 
that 
outperform 
the 
classic 
variety. 


Construction 
of 
Sparse 
Grids 
for 
Quadrature 


. 
+1

A 
version 
of 
the 
univariate 
quadrature 
problem 
seeks 
to 
estimate 
the 
integral 
I(f)= 
f(x) 
dx.A

..1 


quadrature 
rule 
Q() 
for 
this 
problem 
is 
a 
set 
of 
n 
points 
x 
and 
weights 
w 
which 
produce 
the 
integral 
estimate: 


n

I(f) 
˜ 
Q(f)= 
wif(xi). 
i=1 


Such 
a 
quadrature 
rule 
is 
said 
to 
have 
exactness 
of 
degree 
d 
if 
the 
integral 
estimate 
is 
exact 
whenever 
the 
integrand 
f 
is 
a 
polynomial 
of 
degree 
d 
or 
less. 
A 
common 
strategy 
for 
quadrature 
involves 
assembling 
an 
indexed 
family 
Q 
of 
quadrature 
rules 
of 
increasing 
exactness. 
By 
applying 
rules 
of 
increasing 
index 
to 
a 
given 
problem, 
a 
reasonable 
estimate 
of 
the 
quadrature 
error 
may 
be 
made. 


A 
version 
of 
the 
multivariate 
quadrature 
problem 
may 
be 
posed 
in 
the 
same 
way 
for 
a 
function 
of 
a 
variable 
x 
. 
[..1, 
+1]m 
. 
Quadrature 
rules 
for 
this 
problem 
may 
be 
constructed 
by 
making 
m 
selections 
from 
Q 
and 
forming 
the 
product 
rule. 
A 
signicant 
drawback 
of 
this 
approach 
arises 
because, 
if 
the 
1D 
rule 
of 


m

exactness 
d 
requires 
n 
points, 
then 
the 
product 
rule 
of 
corresponding 
exactness 
requires 
npoints, 
a 
fact 
which 
rules 
out 
the 
product 
approach 
except 
for 
low 
dimensions 
or 
degrees 
of 
exactness. 


The 
sparse 
grid 
construction 
of 
Smolyak 
[9] 
showed 
how 
simpler 
product 
rules 
could 
be 
combined 
in 
a 
way 
that 
achieved 
the 
exactness 
of 
a 
given 
product 
rule. 
Of 
course, 
if 
enough 
simple 
product 
rules 
are 
involved, 
the 
total 
number 
of 
points 
can 
grow 
arbitrarily. 
Thus, 
a 
key 
idea 
in 
the 
implementation 
of 
Smolyak's 
procedure 
was 
to 
prefer 
quadrature 
families 
Q 
that 
were 
nested, 
which 
greatly 
reduces 
the 
point 
count 
of 
the 
sparse 
grid. 


Smolyak's 
formula 
produces 
a 
sequence 
of 
m-dimensional 
sparse 
grids 
with 
an 
index 
. 
=0, 
1, 
2, 
::. 
often 
called 
the 
level: 
. 


m 
- 
1 


A(`, 
m)= 
(..1)`..ji| 
(Qi1 

· 


. 
Qim 
)

. 
..jij
0`..jijm..1 


It 
is 
natural 
to 
expect 
that, 
for 
a 
given 
spatial 
dimension 
m, 
the 
sequence 
of 
sparse 
grids 
A(`, 
m);. 
= 
0, 
1, 
2, 
::. 
produce 
integral 
estimates 
of 
increasing 
exactness. 
Novak 
and 
Ritter 
were 
able 
to 
show 
that, 
if 
based 
on 
the 
classic 
Clenshaw-Curtis 
family, 
the 
sparse 
grid 
of 
level 
. 
would 
have 
exactness 
2. 
+ 
1. 
It 
will 
be 
seen 
shortly 
that 
this 
theorem 
suggests 
a 
more 
ecient 
way 
to 
employ 
the 
Clenshaw 
Curtis 
family; 
it 
can 
also 
be 
extended 
to 
other 
families 
of 
1D 
rules. 
At 
the 
moment, 
it 
is 
enough 
to 
note 
that 
this 
theorem 


2 



Table 
1: 
Exactness 
and 
CCE/CCL/CCS 
1D 
rule 
sizes 


i 
= 
index 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
ei 
(required) 
1 
3 
5 
7 
9 
11 
13 
15 
17 
19 
21 
ni 
(CCE) 
1 
3 
5 
9 
17 
33 
65 
125 
257 
513 
1025 
ni 
(CCL) 
1 
3 
5 
7 
9 
11 
13 
15 
17 
19 
21 
ni 
(CCS) 
1 
3 
5 
9 
9 
17 
17 
17 
17 
33 
33 


demonstrates 
that, 
at 
least 
for 
smooth 
integrands, 
a 
sequence 
of 
sparse 
grids 
can 
be 
used 
to 
produce 
integral 
estimates 
of 
rapidly 
improving 
accuracy. 


Using 
the 
Clenshaw-Curtis 
Quadrature 
Family 


Clenshaw 
and 
Curtis 
[1] 
presented 
a 
family 
of 
easily-computed 
1D 
quadrature 
rules 
with 
positive 
weight 
and 
nested 
abscissas. 
Their 
purposes 
required 
the 
evaluation 
of 
a 
sequence 
of 
estimates, 
and 
so 
nesting 
was 
a 
valuable 
way 
of 
reducing 
the 
number 
of 
times 
the 
integrand 
was 
evaluated. 
The 
rules 
were 
dened 
on 
the 
interval 
[..1, 
+1]. 
The 
rst 
rule 
(i 
= 
0) 
has 
size 
n0 
= 
1, 
but 
all 
subsequent 
rules 
have 
size 
ni 
=2i 
+ 
1, 
thus 
exhibiting 
exponential 
growth. 
This 
1D 
quadrature 
family 
will 
be 
denoted 
by 
CCE. 
Because 
of 
symmetry 
and 
the 
fact 
that 
the 
rules 
are 
all 
of 
odd 
size, 
the 
exactness 
e 
of 
each 
rule 
is 
the 
same 
as 
its 
size, 
that 
is 
e 
= 
n. 


Novak 
and 
Ritter 
showed 
that 
the 
exactness 
of 
sparse 
grids 
constructed 
from 
a 
family 
Q 
was 
related 
to 
the 
exactness 
of 
the 
members 
of 
that 
family. 
In 
particular, 
they 
showed 
that, 
if 
every 
1D 
rule 
Qi 
had 
exactness 
at 
least 
2i 
+ 
1, 
then 
the 
sparse 
grid 
of 
level 
. 
is 
guaranteed 
to 
have 
exactness 
at 
least 
2. 
+ 
1. 


This 
condition 
is 
easily 
veried 
for 
the 
CCE 
family; 
Novak 
and 
Ritter 
require 
that 
the 
exactness 
sequence 
for 
the 
1D 
family 
be 
at 
least 
2i 
+ 
1 
but 
it 
has 
alredy 
been 
observed 
that 
ei 
= 
ni 
=2i 
+1 
= 
2i 
+ 
1 
(the 
case 
i 
= 
0 
is 
treated 
seperately), 
and 
hence 
the 
desired 
exactness 
of 
sparse 
grids 
constructed 
from 
the 
CCE 
family 
follows. 


Since 
eciency 
is 
a 
vital 
concern 
for 
high 
dimensional 
problems, 
it 
is 
natural 
to 
wonder 
whether 
there 
is 
an 
unnecessary 
cost 
associated 
with 
the 
fact 
that 
the 
CCE 
family 
exponentially 
exceeds 
the 
linear 
Novak 
and 
Ritter 
requirement. 
This 
excess 
is 
most 
obvious 
if 
one 
imagines 
using 
the 
Smolyak 
procedure 
to 
construct 
a 
sequence 
of 
1D 
\sparse 
grids", 
which 
simply 
produces 
the 
elements 
of 
the 
quadrature 
family. 


The 
nesting 
of 
the 
CCE 
family, 
which 
is 
so 
benecial 
in 
higher 
dimensions, 
is 
forcing 
the 
doubling 
in 
size 
that 
is 
so 
troubling 
in 
lower 
dimensions. 
It 
is 
worth 
asking 
if 
there 
are 
simple 
alternatives 
to 
the 
CCE 
family 
that 
will 
moderate 
the 
doubling 
eect. 


An 
obvious 
choice 
would 
be 
to 
abandon 
nesting. 
The 
CCE 
family 
selected 
a 
particular 
sequence 
of 
sizes 
to 
guarantee 
nesting. 
Instead, 
it 
is 
possible 
to 
choose 
a 
sequence 
of 
sizes 
n 
that 
guarantee 
the 
Novak 
and 
Ritter 
condition, 
namely 
ni 
=2i 
+1;i 
=0, 
1, 
2, 
:::. 
This 
family 
will 
be 
denoted 
by 
CCL. 
Because 
these 
rules 
have 
odd 
size, 
the 
exactness 
condition 
is 
still 
met, 
although 
the 
perfect 
nesting 
of 
the 
CCE 
family 
has 
been 
abandoned. 


A 
second 
choice 
is 
to 
preserve 
nesting, 
but 
to 
change 
the 
way 
the 
individual 
quadrature 
rules 
are 
indexed. 
This 
rule 
will 
be 
denoted 
CCS, 
for 
the 
\slow 
growth” 
option. 
The 
informal 
denition 
of 
this 
family 
is 
that 
the 
i-th 
member 
of 
the 
family 
is 
the 
smallest 
member 
of 
the 
CCE 
family 
that 
has 
exactness 
at 
least 
2i 
+ 
1. 
Another 
way 
of 
viewing 
the 
CCS 
family 
is 
to 
imagine 
that 
the 
elements 
in 
the 
sequence 
are 
the 
same 
as 
those 
in 
the 
CCE 
sequence, 
but 
that 
any 
element, 
once 
it 
occurs 
in 
the 
sequence 
at 
position 
i, 
may 
be 
repeated 
at 
one 
or 
more 
subsequent 
positions, 
if 
it 
satises 
the 
corresponding 
exactness 
requirements. 


Table 
1 
compares 
the 
sizes 
of 
the 
1D 
rules 
in 
the 
CCE, 
CCL 
and 
CCS 
families: 


It 
is 
clear 
that 
the 
three 
families 
dier 
little 
for 
low 
index 
i, 
but 
that 
the 
two 
newly-dened 
families 
avoid 
the 
exponential 
explosion 
aicting 
the 
higher-index 
elements 
of 
the 
CCE 
family. 


The 
dierences 
are 
suggested 
by 
Table 
2, 
which 
counts 
the 
number 
of 
points 
in 
the 
corresponding 
sparse 
grids 
for 
several 
moderate 
dimensions. 
The 
improvement 
oered 
by 
the 
new 
families 
is 
strongest 
in 
the 


3 



sparse 
grids 
of 
lowest 
dimension. 
Table 
2: 
CCE/CCL/CCS 
sparse 
grid 
point 
counts 
in 
2D, 
6D, 
10D 


. 
= 
level 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
Dimension 
2 
n. 
(CCE) 
1 
5 
13 
29 
65 
145 
321 
705 
1,537 
3,329 
7,169 
n. 
(CCL) 
1 
5 
13 
29 
57 
105 
177 
281 
425 
611 
855 
n. 
(CCS) 
1 
5 
13 
29 
49 
81 
129 
161 
225 
257 
385 
Dimension 
6 
n. 
(CCE) 
1 
13 
85 
389 
1,457 
4,865 
15,121 
44,689 
127,105 
350,657 
943,553 
n. 
(CCL) 
1 
13 
85 
389 
1,433 
4,533 
12,961 
33,817 
82,153 
188,039 
408,995 
n. 
(CCS) 
1 
13 
85 
389 
1,409 
4,289 
11,473 
27,697 
61,345 
126,401 
244,289 
Dimension 
10 
n. 
(CCE) 
1 
21 
221 
1,581 
8,801 
41,265 
171,425 
652,065 
2,320,385 
7,836,545 
25,370,753 
n. 
(CCL) 
1 
21 
221 
1,581 
8,761 
40,425 
162,385 
584,665 
? 
? 
? 
n. 
(CCS) 
1 
21 
221 
1,581 
8,721 
39,665 
155,105 
536,705 
1,677,665 
4,810,625 
12,803,073 


4 
Using 
the 
Legendre 
Quadrature 
Family 


Gaussian 
quadrature 
rules 
oer 
increased 
exactness, 
at 
the 
same 
time 
constraining 
the 
choice 
of 
abscissa 
location. 
An 
n-point 
Gauss-Legendre 
rule, 
for 
instance, 
dened 
on 
the 
interval 
[..1, 
1], 
will 
have 
exactness 
ei 
=2ni 
- 
1. 
A 
quadrature 
family 
built 
from 
the 
Legendre 
rule 
will 
benet 
from 
the 
increased 
accuracy, 
but 
has 
the 
disadvantage 
that 
there 
is 
almost 
no 
oportunity 
for 
nesting, 
aside 
from 
the 
occurrence 
of 
the 
abscissa 
x 
= 
0 
in 
every 
rule 
of 
odd 
size. 


If 
it 
is 
desired 
to 
construct 
sparse 
grids 
using 
the 
Gauss-Legendre 
family, 
it 
is 
reasonable 
to 
imitate 
the 
procedure 
used 
for 
the 
classic 
Clenshaw-Curtis 
example, 
which 
can 
be 
understood 
as 
choosing 
the 
next 
rule 
by 
adding 
a 
new 
abscissa 
within 
each 
subinterval 
of 
the 
current 
rule. 
This 
will 
again 
amount 
to 
exponential 
growth, 
so 
the 
family 
will 
be 
denoted 
by 
GLE. 
Since 
the 
rules 
are 
open, 
the 
growth 
pattern 
diers 
from 
the 
CCE 
family, 
producing 
rules 
of 
size 
ni 
=1, 
3, 
7, 
15, 
31, 
63, 
:::, 
2i+1 
- 
1. 


Since 
there 
is 
little 
nesting 
advantage 
available 
in 
this 
case, 
it 
makes 
more 
sense 
to 
be 
guided 
by 
the 
Novak 
and 
Ritter 
criterion. 
Taking 
it 
strictly, 
the 
GLL 
family 
can 
be 
constructed, 
with 
a 
linearly 
growing 
size 
sequence 
of 
ni 
=1, 
2, 
3, 
4, 
:::, 
i 
+ 
1, 
which 
just 
satises 
the 
exactness 
constraint. 


However, 
it 
is 
tempting 
to 
consider 
forming 
a 
family 
only 
using 
odd 
rules; 
this 
takes 
whatever 
advantage 
is 
to 
be 
had 
from 
the 
repeated 
abscissa 
x 
= 
0, 
at 
the 
expense 
of 
often 
using 
slightly 
more 
powerful 
rules. 
This 
family 
will 
be 
denote 
GLO. 


Table 
3 
compares 
the 
point 
counts 
for 
the 
three 
Gauss-Legendre-based 
quadrature 
families 
in 
moderate 
dimensions. 
It 
is 
hardly 
surprising 
that 
the 
GLE 
family 
grows 
so 
rapidly, 
but 
it 
is 
truly 
disconcerting 
to 
see 
that 
GLO 
sparse 
grids 
use 
far 
fewer 
points 
than 
GLL, 
and 
do 
so 
by 
using 
bigger 
1D 
rules! 
It 
is 
a 
testament 
to 
the 
somewhat 
unpredictable 
power 
of 
nesting, 
in 
this 
case 
applied 
to 
a 
single 
repeated 
abscissa! 


5 
Using 
the 
Patterson 
Quadrature 
Family 


Patterson 
[7] 
produced 
a 
family 
of 
quadrature 
rules 
that 
are 
completely 
nested 
like 
the 
CCE 
family, 
with 
an 
exponential 
growth 
in 
size 
like 
the 
GLE 
family, 
and 
an 
exactness 
that 
may 
be 
estimated 
as 
ei 
˜ 
3 
ni, 
about 


2 


halfway 
between 
the 
exactnesses 
of 
Clenshaw-Curtis 
and 
Gauss-Legendre 
rules. 


Let 
GPE 
denote 
the 
quadrature 
family 
formed 
by 
the 
Patterson 
rules, 
with 
sizes 
1, 
3, 
7, 
15, 
31, 
:::. 
Let 
GPS 
denote 
the 
\slow 
growth” 
quadrature 
family 
formed 
by 
applying 
the 
Novak 
and 
Ritter 
constraint 
to 
the 
Patterson 
rules. 


4 



Table 
3: 
GLE/GLL/GLO 
sparse 
grid 
point 
counts 
in 
2D, 
6D, 
10D 


. 
= 
level 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
Dimension 
2 
n. 
(GLE) 
1 
5 
21 
73 
221 
609 
1,573 
3,881 
9,261 
21,553 
49,205 
n. 
(GLL) 
1 
5 
13 
29 
53 
89 
137 
201 
281 
381 
501 
n. 
(GLO) 
1 
5 
9 
17 
33 
45 
81 
97 
161 
181 
281 
Dimension 
6 
n. 
(GLE) 
1 
13 
109 
713 
3,953 
19,397 
86,517 
357,153 
1,382,361 
5,065,693 
17,709,469 
n. 
(GLL) 
1 
13 
85 
389 
1,433 
4,541 
12,841 
33,193 
79,729 
180,077 
385,901 
n. 
(GLO) 
1 
13 
73 
257 
737 
1,925 
4,509 
9,837 
20,445 
40,025 
75,917 
Dimension 
10 
n. 
(GLE) 
1 
21 
261 
2,441 
18,881 
126,925 
764,365 
4,208,385 
21,493,065 
102,935,845 
466,201,781 
n. 
(GLL) 
1 
21 
221 
1,581 
8,761 
40,405 
162,025 
581,385 
1,904,465 
5,778,965 
? 
n. 
(GLO) 
1 
21 
201 
1,201 
5,281 
19,165 
61,285 
177,525 
474,885 
1,192,425 
2,835,589 


Table 
4: 
GPE/GPS 
sparse 
grid 
point 
counts 
in 
2D, 
6D, 
10D 


. 
= 
level 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
Dimension 
2 
n. 
(GPE) 
1 
5 
17 
49 
129 
321 
769 
1,793 
4,097 
9,217 
20,481 
n. 
(GPS) 
1 
5 
9 
17 
33 
33 
65 
97 
97 
161 
161 
Dimension 
6 
n. 
(GPE) 
1 
13 
97 
545 
2,561 
10,625 
40,193 
141,569 
471,041 
1,496,065 
4,571,137 
n. 
(GPS) 
1 
13 
73 
257 
737 
1,889 
4,161 
8,481 
16,929 
30,689 
53,729 
Dimension 
10 
n. 
(GPE) 
1 
21 
241 
2,001 
13,441 
77,505 
397,825 
1,862,145 
8,085,505 
32,978,945 
127,574,017 
n. 
(GPS) 
1 
21 
201 
1,201 
5,281 
19,105 
60,225 
169,185 
434,145 
1,041,185 
2,347,809 


Table 
4 
compares 
the 
point 
counts 
for 
the 
two 
Gauss-Patterson-based 
quadrature 
famlies. 
In 
this 
case, 
the 
advantages 
of 
using 
the 
GPS 
rule 
are 
already 
evident 
at 
level 
2, 
and 
strongly 
persist 
through 
dimension 


10. 
Thus, 
the 
Novak 
and 
Ritter 
guidelines 
have 
suggested 
improved 
versions 
of 
the 
CCE, 
GLE 
and 
GPE 
families 
for 
use 
in 
sparse 
grid 
construction. 
Since 
these 
families 
all 
represent 
approaches 
to 
the 
quadrature 
problem, 
table 
5 
compares 
the 
point 
counts. 


Numerical 
Examples 


Genz[2] 
prescribed 
a 
number 
of 
test 
integands 
for 
multidimensional 
quadrature. 
Given 
two 
sparse 
grid 
families, 
a 
comparison 
can 
be 
made 
by 
applying 
successive 
elements 
of 
each 
family 
to 
the 
test 
integrand, 
and 
observing 
the 
relationship 
between 
the 
number 
of 
function 
evaluations 
required 
and 
the 
quadrature 
error. 


Figures 
[1], 
[2] 
and 
[3] 
display 
the 
results 
of 
such 
comparisons 
between 
the 
CCE 
and 
CCS, 
GLO 
and 
GLS, 
and 
GPE 
and 
GPS 
families, 
for 
quadrature 
in 
a 
space 
of 
dimension 
6. 
For 
the 
GLO 
versus 
GLS 
comparison, 
there 
is 
a 
consistent 
advantage 
for 
the 
slow 
growth 
family. 
The 
advantage 
is 
not 
so 
clear 
for 
the 
GPE 
versus 
GPS 
comparison. 


Figure 
[4] 
comparse 
the 
CCS, 
GLO 
and 
GPS 
families. 


5 



Table 
5: 
CCS/GLO/GPS 
point 
counts 
in 
2D, 
6D, 
10D 


. 
= 
level 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
Dimension 
2 
n. 
(CCS) 
1 
5 
13 
29 
49 
81 
129 
161 
225 
257 
385 
n. 
(GLO) 
1 
5 
9 
17 
29 
41 
65 
81 
121 
141 
201 
n. 
(GPS) 
1 
5 
9 
17 
33 
33 
65 
97 
97 
161 
161 
Dimension 
6 
n. 
(CCS) 
1 
13 
85 
389 
1,409 
4,289 
11,473 
27,697 
61,345 
126,401 
244,289 
n. 
(GLO) 
1 
13 
73 
257 
737 
1,925 
4,509 
9,837 
20,445 
40,025 
75,917 
n. 
(GPS) 
1 
13 
73 
257 
737 
1,889 
4,161 
8,481 
16,929 
30,689 
53,729 
Dimension 
10 
n. 
(CCS) 
1 
21 
221 
1,581 
8,721 
39,665 
155,105 
536,705 
1,677,665 
4,810,625 
12,803,073 
n. 
(GLO) 
1 
21 
201 
1,201 
5,281 
19,165 
61,285 
177,525 
474,885 
1,192,425 
2,835,589 
n. 
(GPS) 
1 
21 
201 
1,201 
5,281 
19,105 
60,225 
169,185 
434,145 
1,041,185 
2,347,809 



Figure 
1: 
CCE 
(green) 
vs 
CCS 
(red) 
on 
Genz 
Functions 
Error 
decay 
for 
Clenshaw-Curtis 
Exponential 
(CCE) 
and 
Clenshaw-Curtis 
Slow 
(CCS) 
sparse 
grids 
on 
Genz 
product 
peak, 
corner 
peak, 
and 
gaussian 
integrands, 
Dimension 
= 
6. 



Figure 
2: 
GLO 
(green) 
vs 
GLS 
(red) 
on 
Genz 
Functions 
Error 
decay 
for 
Gauss-Legendre 
Odd 
(GLO) 
and 
Gauss-Legendre 
Slow 
(GLS) 
sparse 
grids 
on 
Genz 
product 
peak, 
corner 
peak, 
and 
gaussian 
integrands, 
Dimension 
= 
6. 


6 



Figure 
3: 
GPE 
(green) 
vs 
GPS 
(red) 
on 
Genz 
Functions 
Error 
decay 
for 
Gauss-Patterson 
Exponential 
(GPE) 
and 
Gauss-Patterson 
Slow 
(GPS) 
sparse 
grids 
on 
Genz 
product 
peak, 
corner 
peak, 
and 
gaussian 
integrands, 
Dimension 
= 
6. 



Figure 
4: 
CCS 
(red) 
vs 
GLO 
(green) 
vs 
GPS 
(black) 
on 
Genz 
Functions 
Error 
decay 
comparison 
for 
CCS/GLO/GPS 
families, 
Dimension 
6 


7 



Conclusion 


Smolyak's 
sparse 
grid 
procedure 
is 
widely 
used 
for 
computations 
in 
moderate 
and 
high 
dimension, 
and 
was 
devised 
to 
avoid 
the 
exponential 
growth 
in 
point 
count 
incurred 
by 
a 
straight-forward 
product 
rule 
approach. 
It 
is 
thus 
somewhat 
surprising 
that, 
in 
the 
most 
common 
example 
of 
a 
sparse 
grid, 
the 
1D 
quadrature 
family 
is 
designed 
to 
exhibit 
exponential 
point 
growth. 
It 
turns 
out 
that 
this 
peculiarity 
is 
rarely 
an 
issue, 
since 
the 
1D 
rules 
of 
very 
large 
order 
play 
very 
little 
role 
in 
practical 
high 
dimensional 
problems. 


However, 
the 
bigger 
1D 
rules 
do 
come 
into 
play 
in 
lower 
dimensional 
sparse 
grids. 
Since 
such 
sparse 
grids 
are 
also 
of 
importance, 
the 
methods 
discussed 
here 
oer 
an 
approach 
that 
can 
detect 
unnecessary 
growth 
in 
the 
1D 
quadrature 
rules, 
and 
indicate 
how 
to 
adjust 
the 
sequence 
of 
rules 
according 
to 
the 
Novak 
and 
Ritter 
criterion. 
No 
new 
quadrature 
rules 
need 
to 
be 
devised; 
rather, 
some 
rules 
can 
be 
re-used 
for 
several 
steps 
of 
the 
sparse 
grid 
construction 
process. 


The 
tables 
suggest 
that 
the 
benet 
of 
this 
approach 
varies 
widely 
depending 
on 
the 
peticular 
family, 
the 
spacial 
dimension, 
and 
the 
range 
of 
sparse 
grid 
levels 
of 
interest. 
The 
numerical 
results 
conrm 
that 
the 
adjusted 
rules 
perform 
with 
the 
desired 
accuracy. 


Some 
publicly 
available 
software 
for 
sparse 
grid 
computations 
includes 
versions 
of 
the 
slow-growth 
strateg
y 
described 
here. 
In 
particular, 
Petras 
[8] 
presents 
a 
C 
program 
called 
smolpak, 
which 
implements 
slow-growth 
sparse 
grid 
construction 
for 
the 
nested 
Kronrod-Patterson 
family. 
Heiss 
and 
Winschel 
[4] 
describe 
a 
Matlab 
package 
called 
nwspgr 
that 
includes 
slow-growth 
sparse 
grid 
construction 
for 
quadrature 
over 
the 
unit 
hypercube, 
using 
either 
Gauss-Legendre 
or 
nested 
Kronrod-Patterson 
families; 
the 
package 
also 
includes 
slow-growth 
options 
for 
the 
Hermite 
weight 
over 
Rn, 
using 
both 
the 
standard 
Gauss-Hermite 
or 
a 
nested 
Patterson-type 
family 
developed 
by 
Genz 
and 
Keister 
[3]. 
Recently, 
a 
C++ 
package 
known 
as 
tasmanian 
[10] 
has 
been 
developed 
at 
Oak 
Ridge 
National 
Laboratory, 
supporting 
the 
construction 
of 
sparse 
grids 
using 
the 
slow-growth 
option 
for 
several 
quadrature 
families. 


The 
original 
Smolyak 
construction 
was 
a 
fundamental 
breakthrough 
in 
ecient 
analysis 
of 
high-dimensional 
problems. 
This 
discussion 
shows 
that, 
particularly 
in 
lower 
dimensions, 
the 
standard 
procedure 
can 
be 
usefull
y 
modied 
to 
produce 
sparse 
grids 
of 
equivalent 
exactness 
at 
a 
much 
reduced 
cost 
in 
terms 
of 
function 
evaluations. 
These 
results, 
discussed 
only 
in 
the 
isotropic 
case, 
also 
have 
implications 
for 
computations 
in 
higher 
dimensions, 
since 
the 
anisotropic 
approach 
to 
such 
problems 
essentially 
transforms 
them 
to 
a 
kind 
of 
lower 
dimensional 
problem 
in 
which 
the 
same 
eciency 
improvements 
could 
be 
expected. 


References 


[1] 
Charles 
Clenshaw, 
Alan 
Curtis, 
A 
Method 
for 
Numerical 
Integration 
on 
an 
Automatic 
Computer
, 
Numerische 
Mathematik, 
Volume 
2, 
Number 
1, 
December 
1960, 
pages 
197-205. 
[2] 
Alan 
Genz, 
A 
package 
for 
testing 
multiple 
integration 
subroutines, 
in 
Numerical 
Integration: 
Recent 
Developments, 
Software 
and 
Applications, 
edited 
by 
Patrick 
Keast, 
Graeme 
Fairweather, 
Reidel, 
1987, 
pages 
337-340. 
[3] 
Alan 
Genz, 
Bradley 
Keister, 
Fully 
symmetric 
interpolatory 
rules 
for 
multiple 
integrals 
over 
innite 
regions 
with 
Gaussian 
weight, 
Journal 
of 
Computational 
and 
Applied 
Mathematics, 
Volume 
71, 
1996, 
pages 
299-309. 
[4] 
Florian 
Heiss, 
Viktor 
Winschel, 
Likelihood 
approximation 
by 
numerical 
integration 
on 
sparse 
grids, 
Journal 
of 
Econometrics, 
Volume 
144, 
Number 
1, 
May 
2008, 
pages 
62-80. 
[5] 
Erich 
Novak, 
Klaus 
Ritter, 
High 
dimensional 
integration 
of 
smooth 
functions 
over 
cubes, 
Numerisch
e 
Mathematik, 
Volume 
75, 
Number 
1, 
November 
1996, 
pages 
79-97. 
[6] 
Erich 
Novak, 
Klaus 
Ritter, 
Richard 
Schmitt, 
Achim 
Steinbauer, 
Simple 
cubature 
formulas 
with 
high 
polynomial 
exactness, 
Constructive 
Approximation, 
Volume 
15, 
Number 
4, 
December 
1999, 
pages 
499-522. 
8 



[7] 
Thomas 
Patterson, 
The 
optimal 
addition 
of 
points 
to 
quadrature 
formulae, 
Mathematics 
of 
Computation
, 
Volume 
22, 
Number 
104, 
October 
1968, 
pages 
847-856. 
[8] 
Knut 
Petras, 
Smolyak 
cubature 
of 
given 
polynomial 
degree 
with 
few 
nodes 
for 
increasing 
dimension, 
Numerische 
Mathematik, 
Volume 
93, 
Number 
4, 
February 
2003, 
pages 
729-753. 
[9] 
Sergey 
Smolyak, 
Quadrature 
and 
interpolation 
formulas 
for 
tensor 
products 
of 
certain 
classes 
of 
functions, 
Doklady 
Akademii 
Nauk 
SSSR, 
Volume 
4, 
1963, 
pages 
240-243. 
[10] 
Miroslav 
Stoyanov, 
User 
Manual: 
TASMANIAN 
Sparse 
Grids, 
ORNL 
Report, 
August 
2013, 
Oak 
Ridge 
National 
Laboratory. 
9