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

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 osition Method Sharif Rahman Sharif Rahman The University The The University The University of University of of Iowa of Iowa 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)

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

INTRODUCTION INTRODUCTION  Dimensional Decomposition (Hoeffding, 1948) Dimensional Decomposition (Hoeffding, 1948) NONLINEAR NONLINEAR SYSTEM

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

PDD METHOD PDD METHOD  Orthonormal (ON) Polynomial Basis Orthonormal (ON) Polynomial Basis

PROPOSED METHOD PROPOSED METHOD  Cl Cl Cl Class assica i cal Ort l O h rthogona ogonal Polynom l P l ynomials i l

PROPOSED METHOD  Fourier-Polynomial Expansions Fourier-Polynomial Expansions

PROPOSED METHOD PROPOSED METHOD  Polynomial Dimensional Decomposition Polynomial Dimensional Decomposition

PROPOSED METHOD  Formulation of Coefficients Formulation of Coefficients

PROPOSED METHOD PROPOSED METHOD  Dimension-Reduction Integration Dimension-Reduction Integration

PROPOSED METHOD PROPOSED METHOD  Dimension-Reduction Integration Dimension-Reduction Integration

PROPOSED METHOD  Computat C C omputationa i i onal Eff l Eff Effort ort Approx Approximat ation No. o. of Funct of Function Evaluat on Evaluations ons Cost Scali ost Scaling Univariate Bivariate … … … S -variate

EXAMPLES EXAMPLES  A Cubi A C A C bi P l bic c Polynom ynomial i l 0.5 0.5 B (-3,+3,3,3) N (0,1) B (-  5,+  5,1,1) 0.4 U (-  0 3 0.3 f i ( x i ) 0.2 0.1  i = 0,  i = 1 0.0 -4 -3 -2 -1 0 1 2 3 4 x x i

EXAMPLES EXAMPLES  Tail T il P il Pro robabili b bili bili i bilities es 10 0 10 0 10 0 10 0 Gaussian-Hermite Uniform-Legendre m = 3, n = 4 m = 3, n = 4 10 -1 10 -1 Monte Carlo Univariate 10 -2 10 -2 F Y ( y ) F Y ( y ) Bivariate Trivariate 10 -3 10 -3 Monte Carlo Univariate 10 -4 10 -4 Bivariate Trivariate 10 -5 10 -5 -200 -150 -100 -50 0 50 100 0 10 20 30 40 50 60 70 80 y y

EXAMPLES EXAMPLES  Tail T il P il Pro robabili b bili bili i bilities es 10 0 10 0 10 0 10 0 Beta-Jacobi Beta-Jacobi (-3  x i  3;  =  = 3) (-  5  x i   5;  =  = 1) 10 -1 10 -1 m = 3, n = 4 m = 3, n = 4 10 -2 10 -2 F Y ( y ) F Y ( y ) 10 -3 10 -3 Monte Carlo Monte Carlo Univariate U i i Univariate 10 -4 Bivariate 10 -4 Bivariate Trivariate Trivariate 10 -5 10 -5 -50 0 -25 2 0 0 2 25 50 0 75 100 100 -120 -80 -40 0 40 80 120 y y

EXAMPLES EXAMPLES  Piezoelectric  Piezoelectric Piezoelectric Transducer Piezoelectric Transducer Transducer Transducer 3 mm Brass cap Ceramic 12.5 mm Electroded Electroded 12.5 mm surfaces 3 mm Brass cap 1.5 mm 11 mm 11 mm 1.5 mm Brass cap Ceramic

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

EXAMPLES EXAMPLES  Marg M M argina i i nal Di l Di Distr strib ib ibut utions o i ons of Frequenc f F requencies i es 10 0 10 0 18 0.18 F 1 f 1 Monte Carlo (5000) F 2 Univariate ( m =3) 10 -1 F 3 Bivariate ( m =3) F 4 0.12 0 12 4 10 -2 F 5 F i (  i ) f i (  i ) F 6 10 -3 f 3 f 4 f 2 f 5 Monte Carlo (5000) ( ) 0.06 f 6 f Univariate ( m =3) 10 -4 Bivariate ( m =3) 0.00 10 -5 0 0 20 20 40 40 60 60 80 80 100 100 120 120 140 140 0 20 40 60 80 100 120 140  i , kHz  i , kHz

EXAMPLES EXAMPLES  Joint J i J i nt Di Di Distr strib ib ibut utions o i ons of Frequenc f F requencies i es

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 p utational effort  Future work: Future work: Time-variant problems, arbitrary Time-variant problems, arbitrary pro probabilit b bilit bility measures, ilit y measures, di di discon sconti ti tinuous nuous & non- & non- smooth responses, smooth responses, etc etc .