Stochastic Structural Dynamics. Lecture-28. Monte Carlo simulation approach-4

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1 Stochastic Structural Dyamics Lecture-8 Mote Carlo simulatio approach-4 Dr C S Maohar Departmet of Civil Egieerig Professor of Structural Egieerig Idia Istitute of Sciece Bagalore 56 Idia maohar@civil.iisc.eret.i

2 Recall Pseudoradom umber geerator Simulatio of radom variables Scalar/vector Gaussia/o-Gaussia Completely specified/partially specified Methods Trasformatio method Accept-Reject method

3 ourier represetatio of a Gaussia radom process Let X t be a zero mea, statioary, Gaussia radom process defied as X t a cos tb si t; Assumptios Here a ~,, b ~,, aa k, bb k ab k k,,,, k k, X t a cos t b si t 3 3

4 X t X t a cos tb si t a cos tb si t a cos t b si ta cos t b si t m RXX cos 4 4

5 ourier represetatio of a Gaussia radom process (cotiued) Cosider the psd fuctio S S XX R S cos XX R S cos XX 5 5

6 Compare this with R XX cos By choosig S two AC-s coicide., we see that the 6 6

7 By discretizig the psd fuctio as show we ca simulate samples of X( t) usig the ourier represetatio.5 cos si ; X t a t b t psd. Area S frequecy rad/s 7 7

8 Example Simulate samples of a zero mea statioary Gaussia radom process with followig properties: I S exp ; I ; 6 X t a cos t b si t i T 5 s; ; ;.53 rad/s max max rad/s. t.49 s 8

9 .4. cotiuous discrete..8 psd w rad/s 9

10 target measamples.5.5 a sample mea std dev target stdev x(t) time s

11 samples.9.8 simulatio target.7.6 PD x

12 x(t) time s

13 mu c u x k u x u u u x x Let y S I yy g b g b u I g g g g g b g b 4 g 4 g g g 4 g g ; 8 rad/s;.6 g g 3

14 Target Simulatio PSD (m/s ) /(rad/s) requecy rad/s 4

15 Accl m/s Time s 5

16 Simulatio ormal PD Probability R exp cos ; ; ; x x exp ; Accl m/s p X x 6

17 samples.5 Target ad Simulated PSD Simulatio5Target PSD requecy (rad/s) 7

18 Probability distributio fuctio of X(t).9.8 Target Simulatio.7.6 Probability Level 8

19 x(t) x(t) x(t) x(t) x(t) 5 Samples of the radom process X(t) Time (s) 9

20 Simulatio of partially specified o - Gaussia radom processes : ataf's trasformatio Let Xt ( ) be a radom process whose first order pdf ad the AC fuctios are available. o further iformatio about the process is available. X t eed ot be statioary. How to simulate samples of X( t)? t &. X t mx t Defie Y t so that Y t Y t X Itroduce a ew radom process Z(t) through the trasformatio Z t P Y t Y t Here PD of, radom variable. Z is a zero mea Gaussia radom process with a ukow covariace fuctio.

21 Z t P Y t Y Y t PY Z t * Y Y, ;, Y t Y t P z P z z z dzdz * XX t, t PY zp Y z z, z;, t, t dzdz Remarks * t t * t, t & t, t RHS is kow ad, is ot kow XX * z, z ;, t, t Steps * t Solve for t, t Simulate Z t Simulate Y ad hece X( t).

22 Probability distributio fuctio of X(t) R exp ; ;.833 ~.5,.5 p x U X Simulatio Target Probability Level

23 samples.35 Target ad Simulated PSD PSD..5.3 Target.5 Simulatio requecy (rad/s) 3

24 x(t) x(t) x(t) x(t) x(t).5 Samples of the radom process X(t)

25 Probability distributio fuctio of X(t).9.8 Probability R exp ; ; ;.768 Simulatio Target x x px x exp ; x x 5

26 Simulatio5samples Target ad Simulated PSD Target PSD - - requecy (rad/s) 6

27 x(t) x(t) x(t) x(t) x(t) 5 Samples of the radom process X(t)

28 Simulatio of vector Gaussia radom process t Let X t ad Y be two joitly statioary, zero mea Gaussia processes with R X t X t R Y t Y t ote: XY XY YY XX R X t Y t XY R X t Y t Y t X t R R X t Y t Y t X t R R R R XX YX R R XY YY YX YX 8

29 RXX SXX exp id; SXX RXX expid RYY SXX exp id; SYY RYY expid RXY SXY exp id; SXY RXY expid RYX SYX exp id; SYX RYX expid S XY XY i * exp exp XY ; S R i d R i d S XY XY YX YX 9

30 RXY SXY expid S i XY XY XY si ; RXY XY ixy cos isi d XY cos XY sid i XY si XY cosd R XY is real valued XY The above is true if XY cosd & XY XY XY XY 3

31 R sid cos XY XY XY & XY XY XY XY R cos si d XY XY XY 3

32 SXX lim XT XT T T * S lim YY YT YT T T * S lim XY XT YT T T * S lim YX YT XT T T S * S S XX YX S S XY YY 3

33 ourier represetatios X t a cos tb si t; Y t c cos td si t; k Assumptios a ~,, b ~,, aa k, bb k, ab k,,,, X t a cos t b si t k k k k k X X k k k X t X t X cos 33

34 Y t c cos td si t; k Assumptios k k k k k c ~,, d ~,, Y Y c c k, d d k, c d, k,,, k k k Y t c cos t d si t cos Y t Y t Y 34

35 X t a cos tb si t; Y t c cos td si t; k X t Y t acostbsitck cosk tdk sik t k k k k k k a cos t b si tc cos t d si t ac k k k k k k cos tcos t a d cos tsi t k k k bc si tcos t bd si tsi t k k k k 35

36 X t Y t k ac cos tcos t ad cos tsi t bc si tcos t bd si tsi t Take k k k k k k k k ac ; ad ; bc ; bd k ac k k ad k k bc k k bd k cos cos cos si R t t t t XY ac ad k bc si tcos t si tsi t k bd k urthermore, assume & ac bd ad bc cos si R XY ac ad 36

37 R R R cos XX X cos YY Y cos si 3 XY ac ad Cosider S S XX XX R XX SXX expid SXX cosd SXX cosd 37

38 R R cos XX X XX S YY YY SXX cos 4 SXX Select X so that RXX R XX Similarly defie S ad SYY select Y so that RYY R YY. 38

39 RXY SXY expid cos si XY ixy i d XY cos XY sid Cosider XY XY XY XY & 39

40 R cosd XY XY XY sid XY XY cos si Compare this with R cos si XY ac ad If we select ac R XY XY R XY & ad XY 4

41 Summary X t a cos tb si t; Y t c cos td si t; k k k k k k a ~,, b ~,, X X a a k, b b k, a b, k,,, k k k c ~,, d ~,, Y Y cc k, dd k, cd k,,,, k k k ac ; ad ; bc ; bd k ac k k ad k k bc k k bd k & ac bd ad bc 4

42 Summary (cotiued) X ac bc a b X bc ac ~, c ac bc Y d bc ac Y S S ; ; XX YY X Y ac XY & XY bc 4

43 Summary (cotiued) R R R XX X cos YY Y cos cos si XY ac ad 43

44 otes X t Y t Y t X t R R Y t X t R R R R cos si XY ac ad R YX R XY YX YX cos si XY ac ad XY XY 44

45 Exercise : Simulatio of spatially varyig earthquake groud acceleratio Cosider X t ad Y( t) to be two radom processes represetig groud acceleratios i the horizotal directio at two statios separated by a distace. X t ad Y( t) ca be take to be joitly statioary, Gaussia ad zero mea radom processes. The auto-psd fuctios of X t ad Y( t)may be take to be of the form S I H H d XY 45

46 S I H H with i g f H & H i i g g f f The above psd is fashioed after the Kaai-Tajimi psd fuctio i which a additoal filter H is itroduced to esure that the groud displacemet is well behaved at low frequecies. The coherecy fuctio ca be take to be give by d XY XY exp dxy exp i V Develop a code to simulate samples of Xt ( ) ad Yt ( ). 46

47 The coherecy fuctio ca be take to be give by d XY XY exp dxy exp i V Develop a code to simulate samples of Xt ( ) ad Yt ( ). Assume f g 5.6 rad/s, =.6, I= (m/s/s),.8 rads/, =.5, d m, =. /m, XY ad V =5 m/s. rom the esemble of time histories geerated estimate the psd ad cross psd fuctios ad compare them with the target values. 47

48 Geeralizatio X t a cos tb si t; Y t c cos td si t; k k Z t e cos t f si t; m m k k k k k m m m m m a ~,, b ~,, X X aa k, bb k, ab k,,,, k k k c ~,, d ~,, Y Y cc k k, d d k, c d, k,,, k k e ~,, f ~,, Z Z e e k, f f k, e f, k,,, k k k 48

49 Geeralizatio (cotiued) ac ; ad ; bc ; bd k ac k k ad k k bc k k bd k ae ; a f ; be ; b f k ae k k af k k be k k bf k ce ; c f ; de ; d f k ce k k cf k k de k k df k & ac bd ad bc & ae bf af be & ce df cf de 49

50 Simulatio of a multi - parameter radom process Let f x, t be a radom process evolvig i x ad t. Let f x, t &,,, f x t f x t R Rff, S ff, expidd 4 f x, t A cos tcos xb k A B C D, ff k cos tsi x k + C si tcos x D si tsi x k k,,, zero mea Gaussia radom variables. k k k k k 5

51 A, B, C, D zero mea Gaussia radom variables. k k k k k, A A k rs Ak r ks B B k rs Bk r ks C C k rs Ck r ks D D k rs Dk r ks A B A C A D, k, r, s, k rs k rs k rs BkCrs Bk Drs, k, r, s, C D, k, rs,, k rs 5

52 f xt, f x, t [ A cos tcos xb cos tsi x r s k + C si tcos x D si tsi x] k k [ A cos t cos x B cos t si x C si tcos xd si tsi x] Thisca be reduced to the form R k k rs r s rs r s rs r s rs r s, x cos cos ff k k with S, k k 4 5

53 Markov process approach for simulatio of o - Gaussia radom processes Let Xt ( ) be a statioary Gaussia radom process defied o x, x. x Let p be the first order desity fuctio ad the psd fuctio X (for the purpose of illustratio) be S XX Here is the measquare value of ( ). Cosider the SDE ; dx t Xdt D X db t The psd of respose i the steady state is obtaiable. The steady state solutio to the goverig PK equatio is obtaiable. Demad that these solutios match with the target psd ad pdf. Determie drift ad diffusio coefficiets so that this becomes possible. Xt l r 53

54 Multiply the above equatio by X t ad take the esemble average: dx t Xdt D X db t X t dx t X t X t dt X t DX t db t dr d Aexp Select. exp R R Remark DX t A has o ifluece o psd of Xt ( ). 54

55 dx t Xdt D X db t p xp x D x p x t x x Steady state d d xp x D x p x dx dx d xp x D x p x dx ote : Sice p x is specified, the above equatio eeds to be viewed as a equatio for the ukow D x. 55

56 d xp x D x p x dx x D x up u du x with D x up u du. Summary To geerate samples of Xt ( ) with pdf px ( ) ad psd S p x dx t Xdt D X db t x p x l, obtai steady state solutios of the SDE x l 56

57 VARIACE REDUCTIO 57

58 g( x) ai i g Xi i ( ) Pf px x dx I g x px x dx Ig X a Var I g( X i i i i i i a Ig X P a Select a is a ubiased estimator Var i i i i ( ( ) ( ) I g X P g X P g X P 58

59 Var I g( X P P i i P a Var i Var ap( ) Select i is Miimized subject to a. i Lai P( P) ai i i L ap k ( P) ; k,,, a k a k P ( P ) i 59

60 6 5/3/ 6 ( ) ( ) ( ) ( ) ( ) Var ( ) ( ) i i k i a P P P P P P a P P P P P P P P

61 Illustratio P P Coefficiet of variatio ζ σ P m P P ζ P P P (for small P ) Suppose.& P 5 7 umber of samples eeded. Similarly, for., P 9 umber of samples eeded. 5 6

62 Remarks P( P) () Variace of estimator = is idepedet of size of basic radom variable vector X. () If this variace is large, the utility of estimator becomes questioable. (3) It appears that, i order to reduce the variace of the estimator we eed to icrease sample size. (4) Questio: Ca we reduce the variace of the estimator without icreasig? Variace reductio techiques. 6

63 roblofvariacereduo:hithouticreasigsamplesize P ( ) I g x px x dx This is re-writte as Ig( x) p X x P PhV x dx h x V x where h is a valid pdf ad satisfies the coditio p X emv ctiowtoreducew? x h x. V Var Θ 63

64 P ( x) h x dx where ( ) I g x px x ( x). h P h X V V x h x ote: at this stage the fuctio h is yet udefied ad eeds to be suitably selected. V V x Expectatio defied with respect to the pdf h. 64

65 Let J V where V i i are draw from h. i V x i We have show that J is a ubiased estimator for which miimzes the samplig variace with the lowest samplig variace beig Var V ( ) Var J. Var V ( ) I g V px V h V V P P 65

66 v v We ow select h v such that Var ( V) is miimized. Clearly if we select h V Igv px v P it follows Var V ( ). This would mea that eve with oe sample we will get the exact estimate of P. The pdf h V V is called the ideal importace samplig desity fuctio (ispdf). 66

Expectation and Variance of a random variable

Expectation and Variance of a random variable Chapter 11 Expectatio ad Variace of a radom variable The aim of this lecture is to defie ad itroduce mathematical Expectatio ad variace of a fuctio of discrete & cotiuous radom variables ad the distributio