Stochastic Computing by Stochastic Computing by a New Polynomial a - - PowerPoint PPT Presentation

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Stochastic Computing by Stochastic Computing by a New Polynomial a - - PowerPoint PPT Presentation

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th AIAA Conference, Palm Springs, CA, May 50 th 50 AIAA Conference, Palm Springs, CA, May 2009 2009 Stochastic Computing by Stochastic Computing by a New Polynomial a New Polynomial Dimensional Decom Dimensional Decomposition Method p

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SLIDE 1

50 50th

th AIAA Conference, Palm Springs, CA, May

AIAA Conference, Palm Springs, CA, May 2009 2009

Stochastic Computing by Stochastic Computing by a New Polynomial a New Polynomial Dimensional Decom Dimensional Decomposition Method

  • sition Method

Sharif Rahman Sharif Rahman The The University University of

  • f Iowa

Iowa

p

The The University University of

  • f Iowa

Iowa Iowa City, IA 52245 Iowa City, IA 52245

Work supported by Work supported by NSF (DMI-0355487 and CMMI-0653279) NSF (DMI-0355487 and CMMI-0653279)

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SLIDE 2

OUTLINE OUTLINE

 Introduction Introduction  Polynomial Dimensional Polynomial Dimensional Decomposition (PDD) Method Decomposition (PDD) Method  Examples Examples  Conclusions & Conclusions & Future Work Future Work

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SLIDE 3

INTRODUCTION INTRODUCTION

 Dimensional Decomposition (Hoeffding, 1948) Dimensional Decomposition (Hoeffding, 1948)

NONLINEAR NONLINEAR SYSTEM

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SLIDE 4

INTRODUCTION INTRODUCTION

 Existing Component Functions (Xu/Rahman) Existing Component Functions (Xu/Rahman)

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SLIDE 5

PDD METHOD PDD METHOD

 Orthonormal (ON) Polynomial Basis Orthonormal (ON) Polynomial Basis

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SLIDE 6

PROPOSED METHOD PROPOSED METHOD

Cl Cl i l O h l P l i l  Cl Class assica cal Ort rthogona

  • gonal Polynom

ynomials

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SLIDE 7

PROPOSED METHOD

 Fourier-Polynomial Expansions Fourier-Polynomial Expansions

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SLIDE 8

PROPOSED METHOD PROPOSED METHOD

 Polynomial Dimensional Decomposition Polynomial Dimensional Decomposition

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SLIDE 9

PROPOSED METHOD

 Formulation of Coefficients Formulation of Coefficients

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SLIDE 10

PROPOSED METHOD PROPOSED METHOD

 Dimension-Reduction Integration Dimension-Reduction Integration

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SLIDE 11

PROPOSED METHOD PROPOSED METHOD

 Dimension-Reduction Integration Dimension-Reduction Integration

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SLIDE 12

PROPOSED METHOD C i C i l Eff  Computat

  • mputationa
  • nal Eff

Effort

  • rt

Approx Approximat ation No.

  • . of Funct
  • f Function Evaluat
  • n Evaluations
  • ns

Cost Scali

  • st Scaling

Univariate Bivariate

… … …

S-variate

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SLIDE 13

EXAMPLES EXAMPLES

A C A C bi P l i l  A Cubi bic c Polynom ynomial

0.5 0 3 0.4 0.5

N(0,1) U(- B(-3,+3,3,3) B(-5,+5,1,1)

0.2 0.3

fi(xi)

  • 4
  • 3
  • 2
  • 1

1 2 3 4

x

0.0 0.1 i = 0, i = 1

xi

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SLIDE 14

EXAMPLES EXAMPLES

T il P b bili bili i  Tail il Pro robabili bilities es

10 0 10 0 10 -1 10 0

Gaussian-Hermite m = 3, n = 4 Monte Carlo

10 -1 10 0

Uniform-Legendre m = 3, n = 4

10 -3 10 -2

FY(y) Univariate Bivariate Trivariate

10 -3 10 -2

FY(y) Monte Carlo

10 -5 10 -4 10 -5 10 -4

Univariate Bivariate Trivariate

  • 200
  • 150
  • 100
  • 50

50 100

y

10 20 30 40 50 60 70 80

y

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SLIDE 15

EXAMPLES EXAMPLES

T il P b bili bili i  Tail il Pro robabili bilities es

10 0 10 0 10 -1 10 0

Beta-Jacobi (-3  xi  3;

 =  = 3)

m = 3, n = 4

10 -1 10 0

Beta-Jacobi (-5  xi  5;  =  = 1) m = 3, n = 4

10 -3 10 -2

FY(y) Monte Carlo

10 -3 10 -2

FY(y) Monte Carlo U i i

10 -5 10 -4

Univariate Bivariate Trivariate

2 2 100 10 -5 10 -4

Univariate Bivariate Trivariate

  • 120
  • 80
  • 40

40 80 120

y

  • 50
  • 25

25 50 75 100

y

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SLIDE 16

EXAMPLES EXAMPLES

 Piezoelectric Piezoelectric Transducer Transducer  Piezoelectric Piezoelectric Transducer Transducer

Brass cap 3 mm 12.5 mm Electroded Ceramic Brass cap 12.5 mm 3 mm Electroded surfaces 11 mm 11 mm 1.5 mm 1.5 mm Brass cap Ceramic

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SLIDE 17

EXAMPLES EXAMPLES

A F A F M d Sh (M I )  A Few ew Mode e Sh Shapes apes (M (Mean ean Input nput)

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SLIDE 18

EXAMPLES EXAMPLES

M i M i l Di ib i f F i  Marg argina nal Di Distr strib ibut utions o

  • ns of Frequenc

requencies es

0 18 10 0 0 12 0.18 f1 Monte Carlo (5000) Univariate (m=3) Bivariate (m=3) 10 -1 10 F1 F2 F3 F4 0.06 0.12

fi(i)

f2 f3 f4 f5 f 10 -3 10 -2

Fi(i)

4

Monte Carlo (5000) F5 F6 20 40 60 80 100 120 140 0.00 f6 10 -5 10 -4 ( ) Univariate (m=3) Bivariate (m=3) 20 40 60 80 100 120 140

i, kHz 20 40 60 80 100 120 140 i, kHz

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SLIDE 19

EXAMPLES EXAMPLES

J i J i Di ib i f F i  Joint nt Di Distr strib ibut utions o

  • ns of Frequenc

requencies es

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SLIDE 20

CONCLUSIONS/FUTURE WORK CONCLUSIONS/FUTURE WORK

 A new polynomial dimensional decomposition A new polynomial dimensional decomposition method was developed method was developed

  • Decomposition with increasing dimensions
  • Fourier-polynomial expansions
  • Innovative dimension-reduction integration

 No sample points required; yet, generates No sample points required; yet, generates convergent approximations convergent approximations  Accurate with modest com Accurate with modest computational effort utational effort p  Future work: Future work: Time-variant problems, arbitrary Time-variant problems, arbitrary b bilit ilit di ti & pro probabilit bility measures, y measures, di discon sconti tinuous nuous & non- non- smooth responses, smooth responses, etc etc.