44th AIAA Aerospace Sciences Meeting and Exhibit 9 - 12 January 2006, Reno, Nevada

AIAA 2006-896

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Copyright © 2006 by J.A.S. Witteveen and H. Bijl. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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deterministic exact mean Gram−Schmidt PC modes

1

Gram−Schmidt PC: mean Gram−Schmidt PC: variance Gram−Schmidt PC: distribution Askey PC: mean Askey PC: variance Askey PC: distribution

0

10

−2

10

0.5

L1 error

position x

mode 0 mode 2

0

−4

10

mode 3 −6

10 mode 1 −0.5 0

−8

0.2

0.4

time t

0.6

0.8

1

10

0

1

2 3 polynomial order p

4

5

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1 sensor location x=0.5→

position x

0.8

temperature T

Pe=0→

velocity u

0.6

← Pe=1

0.4 Pe=5→

0.2

Pe=10→

sensor location xsl

0 0

0.2

0.4 0.6 position x

0.8

Pe=20 Pe=∞→ 1

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10

0

−2

10

10

−4

10

−6

10

0

L1 error

L1 error

10

GSPC: mean GSPC: var. GSPC: distr. APC: mean APC: var. APC: distr.

−2

−4

10

−6

1 2 polynomial order p

3

10

1

GSPC: mean GSPC: variance GSPC: distribution APC: mean APC: variance APC: distribution 2

3

4 5 # terms (P+1)

6

7

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3

Gram−Schmidt PC: mean Gram−Schmidt PC: variance Gram−Schmidt PC: distribution Askey PC: mean Askey PC: variance Askey PC: distribution

0

10

2.5 2

L1 error

polynomial chaos expansion coefficient K

p

Gram−Schmidt PC Askey PC

1.5 1

−2

10

−4

10

0.5 −6

0 0

1

2 3 polynomial order p

4

10

5

0

1

2 3 polynomial order p

4

5

{ |[!r# 0 */"# BCk0£ 3# V\6!3/"# /Ÿ#+‹/R%r~ { (|i# i# /R;R1 /"|#<.% "># 0%# / ˆS ŒL‰‹ŠL›£ŒŽ‰‹œLPšÏBG•Z’l–mÈ4—§—>P˜–ž—§•7™R˜‰¨ÉŽšGÇRŠ9”•Ê›13‰‹—>ªG›H›B™™ PœGœžŸ¢ ËŸ¢šGœL•l•Ê’ ‰¢–žŸ Ç£”G–œL— ÇRœG•Z™Ž–mP3ª7›Pœ Í —1ΏnËk™ œGŸ¢ËšGœL•Ê‰¢–žŸ Ç£”G–œL—B¬Ìœžj–7Ÿ‹œmŠLšGœžR•Z–žŸ‰ešŽ™ ŒL›®ª ‰e—>›PR‰¢˜

1

0.8

Gram−Schmidt PC: mean Gram−Schmidt PC: variance Gram−Schmidt PC: distribution Askey PC: mean Askey PC: variance Askey PC: distribution

0

10

0.7 −2

0.6

L1 error

probability distribution function F

K

0.9

2

10

exact Gram−Schmidt PC expansion Askey PC expansion

0.5 0.4 0.3

−4

10

p=3−4

0.2

−6

10

p=0

0.1 0 0

10

p=1−2 −8

1

2 3 spring stiffness K

4

5

10

0

1

2 3 polynomial order p

4

5

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GSPC: mean GSPC: var. GSPC: distr. APC: mean APC: var. APC: distr.

0

10

0.8 p=4

0.6 0.4

L1 error

probability distribution function FT

1

p=1

−2

10

−4

10

0.2

0 −0.005

−6

0

0.005

0.01 0.015 0.02 temperature T

0.025

0.03

10

1

2

3

4 5 # terms (P+1)

6

7

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10

Legendre (p=1,4) exact

0.8 p=4

−2

10

0.6

L1 error

probability distribution function FT

1

p=1 0.4

−4

10 0.2

0 −0.005

−6

0

0.005

0.01 0.015 0.02 temperature T

0.025

0.03

10

0

GSPC: mean GSPC: var. GSPC: distr. APC: mean APC: var. APC: distr. 1

2 3 polynomial order p

4

5

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L Poisseuille flow

adiabatic wall 2h

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350

GSPC (p=4) MC (N=1000)

probability distribution function FT

360

temperature Tw

340 mean

330 320

±σ

Tw

310 300 290 280 0

0.02

0.04 0.06 flow direction x

0.08

0.1

1

GSPC (p=1,4) MC (N=1000)

0.8 p=1

0.6

p=4

0.4 0.2 0 320

330

340 350 360 temperature T

370

380

out

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Modeling Arbitrary Uncertainties Using Gram-Schmidt ...

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The training set consisted of about 500hrs of audio. The LDA feature vectors were obtained by first com- puting 13 ... Again, Hybrid Monte Carlo seems to be best among .... 6th European Conference on Speech Communication and Technol-.

Semiautonomous Vehicular Control Using Driver Modeling
Apr 13, 2014 - steering or the actual steering for fully autonomous control. If the autonomous controller is unable to construct a control that renders the vehicle safe over ...... Table I. In the precontroller survey, 54% of subjects admitted re- sp

Acoustic Modeling Using Exponential Families - Semantic Scholar
For general exponential models, there is no analytic solution for maximizing L(θ) and we use gradient based numerical op- timization methods. This requires us ...

Missing feedbacks, asymmetric uncertainties, and the ... - CiteSeerX
albedo, water vapor, and cloud feedbacks in the climate ... feedback big enough to worry about? The answer is a definite ''yes. .... temperature data were deuterium excess-corrected to gen- ..... (2006), Climate-carbon cycle feedback analysis,.

Semiautonomous Vehicular Control Using Driver Modeling
we describe a real-time semiautonomous system that utilizes em- pirical observations of a driver's pose to inform an autonomous controller that corrects a ...

Acoustic Modeling Using Exponential Families - Semantic Scholar
shape parameter), Chi-square, and of course the diagonal and full covariance ..... Again, Hybrid Monte Carlo seems to be best among the Markov Chain .... Carlo methods,” Dept. of Computer Science, University of Toronto,. Tech. Rep.

Length-contraction-magnetic-force between arbitrary currents.pdf ...
sphere A in Figure 2 will become the ellipsoid A', with its small axis ... reference O. We rotate the frame O such that its x axis becomes. parallel to v. The so ...

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Diffusion Equations over Arbitrary Triangulated ... - Semantic Scholar
3, MAY/JUNE 2008. • The authors are with the Department of Mathematics, University of ...... search Fund for the Doctoral Program of Higher Education. (No. .... [54] C.L. Wu, J.S. Deng, W.M. Zhu, and F.L. Chen, “Inpainting Images on Implicit ...

Electromagnetic interaction of arbitrary radial ...
Jul 27, 2009 - The parameter x is a convenient tool to control the quality of the NTB spherical cloak. .... Image Sci. Vis 25, 1623 2008. 15 L. W. Cai and J.

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Accelerating Light Beams along Arbitrary Convex Trajectories
May 25, 2011 - invariant (non-diffracting) yields the Airy beam solution, which carries ..... at z ј 0, coincides with the phase of the analytic expansion of the Ai ...

Arbitrary-precision computation of Chebyshev ...
it is better to use trigonometric relations, while for n

Arbitrary Dual-Band RLC Circuits
between the theoretical and practical results. 2 SINGLE-BAND TO DUAL-BAND CONVERSION. If the reactance of each passive element (non-resistive) of.