adaptive multi element integration of functions on discrete measures in multi-dimensions

1. Adaptive ME-PCM for SPDEs driven by discrete RVs !
Mengdi Zheng, Applied Mathematics, Brown University!
5 methods to generate orthogonal polynomials
for discrete measures:!
1. (Nowak’s method) S. Oladyshkin, W.
Nowak, Data-driven uncertainty quantification
using the arbitrary polynomial chaos
expansion, Reliability Engineering & System
Safety, 106 (2012), pp. 179–190. !
2. (Fischer’s method) H. J. Fischer, On
generating orthogonal polynomials for
discrete mea- sures, Z. Anal. Anwendungen,
17 (1998), pp. 183–205. !
3. (Stieltjes method) W. Gautschi, On
generating orthogonal polynomials, SIAM J.
Sci. Stat. Comp., 3 (1982), no.3, pp. 289–
317. !
4. (Modified Chebyshev method) the same
paper as above!
5. (Lanczos method) D. Boley, G. H. Golub, A
Comparing othogonality of 5 methods to
construct polynomials:
0 10 20 30 40 50 60 70 80 90 100
0
10
−5
10
−10
10
−15
10
−20
Comparing CPU time (cost) to construct the
polynomials by 5 methods:
0
10
−1
10
−2
10
−3
10
−4
Nowak
Stieltjes
Fischer
Modified Chebyshev
Lanczos
C*i2

2. n=100,p=1/2
10 20 40 80 100
10
polynomial order i
CPU time to evaluate orth(i)
Comparing the minimum polynomial orders that
the Stieltjes method starts to fail (Binomial):
160
140
120
100
80
60
40
20
0
0 20 40 60 80 100 120 140 160
n (p=1/10) for measure defined in (28)
polynomial order i
=1E−8
=1E−10
=1E−13
i = n

3. 10
polynomial order i
orth(i)
Nowak
Stieltjes
Fischer
Modified Chebyshev
Lanczos
n=100, p
=

4. 1
/2
Multi-element (ME) Gauss quadrature
integration theorem (new):
h-convergence of integration of GENZ1 function
over Binomial distributions (Lanczos/ME-PCM)
0
−2
10
−3
10
−4
10
−5
10
−6
10
1
10
10
N
es
absolute error
c=0.1,w=1
GENZ1
d=2
m=3
bino(120,1/2)
h-convergence of integration of GENZ4 function
over Binomial distributions (Lanczos/ME-PCM)
0
−9
10
−10
10
−11
10
−12
10
−13
10
1
10
10
N
es
absolute errors
c=0.1,w=1
GENZ4
d=2
m=3
bino(120,1/2)
Comparing sparse grid and tensor product grid in
8 dimensions by integration of GENZ1 function
over Binomial distribution (Lanczos/ME-PCM)
17 153 969 4845
−3
10
−4
10
−5
10
−6
10
−7
10
−8
10
−9
10
−10
10
r(k)
absolute error
sparse grid
tensor product grid
Genz1
sparse 8d
Bino(5,1/2)
1,...,8c
=0.1
1,...,8
w
1,...,8
=1
ADAPTIVE integration mesh of ME-PCM (idea)

7. Example: KdV equation/ homogeneous BC/
moment statistics!
Define errorr (for moment statistics)
ADAPTIVE V.s. NON-ADAPTIVE mesh
(moment statistics/KdV/Poisson RV/Nowak)!
−3
10
−4
10
−5
2 el, even grid
2 el, uneven grid
4 el, even grid
2 el, uneven grid
5 el, even grid
5 el, uneven grid

8. 4
2 3 4 5 6
−2
10
10
−3
10
−4
10
−5
Number of PCM points on each element
2 el, even grid
2 el, uneven grid
4 el, even grid
4 el, uneven grid
5 el, even grid
5 el, uneven grid

10. 2 3 4 5 6
10
Number of PCM points on each element
errors
errors
Details of this work please see: M. Zheng, X.
Wan, and G.E. Karniadakis, Adaptive-multi-element
polynomial chaos with discrete
measure: Algorithms and application to
SPDEs, Submitted to Applied Numerical
Mathematics, 2013.!
FA 9550-09-1-0613