mengdi zheng 09232014_apma
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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:
Comparing CPU time (cost) to construct the polynomials by 5 methods:
Comparing the minimum polynomial orders that the Stieltjes method starts to fail (Binomial):
10 20 40 80 100
10−4
10−3
10−2
10−1
100
polynomial order i
CP
U ti
me
to e
valu
ate
orth
(i)
NowakStieltjesFischerModified ChebyshevLanczos
C*i2n=100,p=1/2
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0 20 40 60 80 100 120 140 1600
20
40
60
80
100
120
140
160
n (p=1/10) for measure defined in (28)
poly
nom
ial o
rder
i
�=1E−8
�=1E−10
�=1E−13
i = n
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���
����
0 10 20 30 40 50 60 70 80 90 100
10−20
10−15
10−10
10−5
100
polynomial order i
orth
(i)
NowakStieltjesFischerModified ChebyshevLanczos
n=100, p=1/2������������� ��������������������
Multi-element (ME) Gauss quadrature integration theorem (new):
h-convergence of integration of GENZ1 function over Binomial distributions (Lanczos/ME-PCM)
100 10110−6
10−5
10−4
10−3
10−2
Nes
abso
lute
erro
r
c=0.1,w=1
GENZ1d=2m=3bino(120,1/2) �������������
h-convergence of integration of GENZ4 function over Binomial distributions (Lanczos/ME-PCM)
100 10110−13
10−12
10−11
10−10
10−9
Nes
abso
lute
erro
rs
c=0.1,w=1
GENZ4d=2m=3bino(120,1/2)
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Comparing sparse grid and tensor product grid in 8 dimensions by integration of GENZ1 function over Binomial distribution (Lanczos/ME-PCM)
17 153 969 484510−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
r(k)
abso
lute
erro
r
sparse grid
tensor product grid
Genz1sparse 8d�1,...,8�Bino(5,1/2)c1,...,8=0.1w1,...,8=1
ADAPTIVE integration mesh of ME-PCM (idea)
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Example: KdV equation/ homogeneous BC/moment statistics!
Define errorr (for moment statistics)
ADAPTIVE V.s. NON-ADAPTIVE mesh (moment statistics/KdV/Poisson RV/Nowak)!
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.!
2 3 4 5 6
10−5
10−4
10−3
10−2
Number of PCM points on each element
erro
rs
2 el, even grid2 el, uneven grid4 el, even grid4 el, uneven grid5 el, even grid5 el, uneven grid
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����������� ��� ���
2 3 4 5 6
10−5
10−4
10−3
Number of PCM points on each element
erro
rs
2 el, even grid2 el, uneven grid4 el, even grid2 el, uneven grid5 el, even grid5 el, uneven grid
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