Sparse Grid Quadrature and Interpolation
Note: We have encountered potentially high dimensional quadrature at several points:!
⇡(q|�obs
) =⇡(�
obs
|q)⇡0(q)RRp ⇡(�obs|q)⇡0(q)dq
Bayes’ Relation:!
QoI:! y(t, x) = E[uK(t, x,Q)] =
Z
�u
K(t, x, q)⇢Q(q)dq
Discrete Projection:! uk(t, x) =1
�khu, ki⇢ =
1
�k
Z
�u(t, x, q) k(q)⇢Q(q)dq
Stochastic Quadrature Methods: Monte Carlo!
hu, ki⇢ =1
R
RX
r=1
u(t, x, qr) k(qr)⇢Q(q
r) + "R
Notes:!• Errors satisfy!• Optimal for sufficiently large p.!
E["R] = 0 and "R = O‘ ( 1pR) for large R
Deterministic Quadrature Methods
1-D Quadrature Relations:!
I(1)f =
Z
�1
f(q)⇢Q(q)dq ⇡RX
r=1
f(qr)wr = Q(1)f
Gaussian Quadrature:!
I(1)f =1
2
Z 1
�1f(q)dq ⇡ 1
2
RX
r=1
f(qr)wr,
I(1)f =1p2⇡
Z
Rf(q)e�q2/2dq ⇡
RX
r=1
f(qr)wr
r Nodes qr Weights wr
1 0 2
2 ± 1p3
1
3 0 89
±q
35
59
4 ±p
15+2p30p
3549
6(18+p30)
±p
15�2p30p
3549
6(18�p30)
1-D Quadrature Relations
Nested Quadrature Techniques: Consider on [0,1]!
Q(1)` f =
RX̀
r=1
f (qr` )wr`
Trapezoid Rule:!
Q(1)` f =
1
2h`
"f(0) + f(1) + 2
R`�1X
r=1
f(qr` )
#
where!
h` =1
2`�1, R` = 2`�1+1 , qr` = rh` =
r
2`�1, wr =
1
2h`,1
h`, · · · , 1
h`,
1
2h`
�
0 0.25 0.5 0.75 10
1
2
3
4
5
6
Level
Clenshaw--Curtis:!
qr` =
1
2
1� cos
⇡(r � 1)
R` � 1
�, r = 1, · · · , R`
0 0.25 0.5 0.75 10
1
2
3
4
5
6
Level
Tensor Product Formulation
l1
Levell2
0
0.5
1
0
0.5
1
0
0.5
1
0 0.5 10
0.5
1
0 0.5 1 0 0.5 1 0 0.5 1
Level
Integrals:!
I(p)f =
Z
�f(q)⇢Q(q)dq
Tensor Product Quadrature:!
Q(p)` f =
⇣Q(1)
`1⌦ · · ·⌦Q(1)
`p
⌘f
⌘R`1X
r1=1
· · ·R`pX
rp=1
f�q r11 , · · · , q rp
p
�wr1
`1· · ·wrp
`p
where!
R =pY
i=1
R`i
Errors:!���I(p)f �Q(p)` f
��� = O‘ (R�↵/p` )
for functions f in the space!
C↵([0, 1]p) =
(f : [0, 1]p ! R
����� max
|k0|↵
�����@|k0|f
@q k11 · · · @q kp
p
�����1
< 1)
Sparse Grid Construction Motivation:!
R
0 1
1 x y
2 x
2xy y
2
3 x
3x
2y xy
2y
3
4 x
4x
3y x
2y
2xy
3y
4
Difference Relations: Define!
�(1)` f =
⇣Q(1)
` �Q(1)`�1
⌘f
where!
Q(1)` f =
RX̀
r=1
f(qr` )wr`
1-D Nodal Points!
⇥(1)` =
n
q 1` , · · · , q
R``
o
Example: Trapezoid rule!
` = 2: ⇥(1)2 = {0, 1
2 , 1} and w = [ 14 ,12 ,
14 ]
` = 1 : ⇥(1)1 = {0, 1} and w = [ 12 ,
12 ]
Thus!
�(1)2 f = �1
4f(0) +
1
2f(1/2)� 1
4f(1)
Note: Weights can be negative!
Sparse Grid Construction Sparse Grid Quadrature Rule:!
Q(p)` f =
X
|`0|`+p�1
⇣�(1)
`1⌦ · · ·⌦�(1)
`p
⌘f
where `0 = (`1, · · · , `p) 2 Npis a multi-index with |`0| =
Ppi=1 `i.
Note: Tensor product formula can be expressed as!
Q(p)` f =
X
max `0`
⇣�(1)
`1⌦ · · ·⌦�(1)
`p
⌘f
where max `0 ⌘ max{`1, · · · , `p}.
Sparse Grid Nodal Set:!
⇥(p)` =
[
|`0|`+p�1
⇥(1)`1
⇥ · · ·⇥⇥(1)`p
.
Sparse Grid Construction Example:!Consider Clenshaw-Curtis with ⇥(1)
1 = { 12} and ⇥(1)
2 = {0, 12 , 1}. For p = 2,
`0 = (`1, `2) so |`0| = `1 + `2. For ` = 4, the sparse grid nodal set is
⇥(2)4 =
⇣⇥(1)
1 ⇥⇥(1)1
⌘, (`1 = 1, `2 = 1)
[⇣⇥(1)
1 ⇥⇥(1)2
⌘[⇣⇥(1)
2 ⇥⇥(1)1
⌘
[⇣⇥(1)
1 ⇥⇥(1)3
⌘[⇣⇥(1)
2 ⇥⇥(1)2
⌘[⇣⇥(1)
1 ⇥⇥(1)3
⌘
[⇣⇥(1)
1 ⇥⇥(1)4
⌘[⇣⇥(1)
2 ⇥⇥(1)3
⌘[⇣⇥(1)
3 ⇥⇥(1)2
⌘[⇣⇥(1)
4 ⇥⇥(1)1
⌘
l1
Levell2
0
0.5
1
0
0.5
1
0
0.5
1
0 0.5 10
0.5
1
0 0.5 1 0 0.5 1 0 0.5 1
Level
X
2
l1
X
X
X
X
X
X X X
X
1 2 3 4
3
4
2
1
l
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
Sparse Grid: 29 points!Tensored Grid: 81 points!
Sparse Grid Construction Error Analysis:!
kIf �A(q, p)fk = O‘⇣R�↵
log(R)
(p�1)(↵+1)⌘
Grid Sizes:!
p R` Sparse Grid R Tensored Grid R = (R`)p
2 5 13 25
9 29 81
5 5 61 3125
9 241 59,049
10 5 221 9,765,625
9 1581 > 3⇥ 109
50 5 5101 > 8⇥ 1034
9 171,901 > 5⇥ 1047
100 5 20,201 > 7⇥ 1069
9 1,353,801 > 2⇥ 1095
Anisotropic Sparse Grids Multi-Index Set:!
I(`) =(`0 2 Np
��� `0 · a =pX
i=1
ai`i `+ p� 1
)
where a 2 Rp+ is a vector of weights.
7
2
l1
X
X
X
X
X
X X X
X
X
X
X
X
X
X
X
7
6
5
4
3
2
1
1 2 3 4 5 6
l
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
Interpolating Polynomials 1-D Interpolation: Seek polynomials that satisfy!
uM1(qm) = um = u(qm) , m = 1, · · · ,M1
Lagrange Interpolation: Employ representation!
uM1(q) =M1X
m=1
umLm(q)
where Lm(q) are Lagrange interpolating polynomials defined by
Lm(q) =M1Y
j=0j 6=m
q � q j
qm � q j
=(q � q 1) · · · (q � qm�1)(q � qm+1) · · · (q � qM1)
(qm � q 1) · · · (qm � qm�1)(qm � qm+1) · · · (qm � qM1)
Note: By construction, the Lagrange polynomials satisfy!
Lm(q n) = �mn , 1 m,n M1
1-D Interpolating Polynomials Example: Consider the Runge function!f(q) =
1
1 + 25q2with points!
qj = �1 + (j � 1)2
M1, j = 1, · · · ,M1 + 1 qj = � cos
⇡(j � 1)
M1 � 1
, j = 1, · · · ,M1
−1 −0.5 0 0.5 1−0.2
0
0.2
0.4
0.6
0.8
1
1.2
q
f(q)
−1 −0.5 0 0.5 10
0.2
0.4
0.6
0.8
1
q
f(q)
−1 −0.5 0 0.5 1−80
−60
−40
−20
0
20
q
f(q)
−1 −0.5 0 0.5 10
0.2
0.4
0.6
0.8
1
q
f(q)
Multi-Dimensional Interpolation Tensor Products and Sparse Grid: Interpolation formulae analogous to quadrature rules.!
Sparse Grid Software: !• MATLAB: Sparse Grid Interpolation Toolbox – Be careful of Clenshaw-Curtis,
which are actually Newton-Cotes points – versus Chebychev!• Dakota!
Uses: !• Spectral collocation methods to propagate input uncertainties.!
• Construction of surrogate models.!