Generic Form of Bayesian Monte Carlo For Models With Partial Monotonicity

Transcription

1 Generc Form of Bayesan Monte Carlo For Models Wth Partal Monotoncty M. Raabalnead a*, C. Sptas b a,b Delft Unversty of Technology, Delft, the Netherlands Abstract: Ths paper presents a generc method for the safety assessments of models wth partal monotoncty. For ths purpose, a Bayesan nterpolaton method s developed and mplemented n the Monte Carlo process. ntegrated approach s the generalzaton of the recently developed technques used n safety assessment of monotonc models and t substantally ncreases the effcency of Monte Carlo method. The formulaton of ths development s provded n ths paper wth an example showng ts ablty to dramatcally mprove effcency of smulaton. Ths s acheved by employng pror nformaton obtaned from monotonc models and outcomes of the precedng smulatons. The theory and numercal algorthms of ths method for mult-dmensonal problems and ther ntegraton wth the probablstc fnte element model of a real-world example are presented. Keywords: Relablty, Bayesan, Dynamc Bounds, Monte Carlo, Gaussan, Beta. 1. INTRODUCTION In ths paper, we ntroduce a Bayesan Monte Carlo Method for Partally Monotonc (BMCPM) models. Applyng the BMCPM sgnfcantly reduces smulaton tme of monotonc models for a desred accuracy levels. Monotonc models are used extensvely n practce as shown n [1-3]. There are already two methods ntroduced for the relablty assessment of monotonc models n [4]. The method of Dynamc Bounds (DB) ncorporates the monotonc nformaton of the structures n the relabalty assessment. Ths method s appled to a complex case study n New Orleans [1]. The method of Improved Dynamc Bounds (IDB) s also developed for the relablty assessment of monotonc models gven the model response order. The applcaton of ths method to flood defence systems s presented n [5]. The Bayesan Monte Carlo method also has been developed to capture model s pror nformaton as presented n [6]. Ths paper presents a sgnfcant mprovement by ntegratng these methods nto one generc form. We present a flexble formulaton that can be appled to fully monotonc, partally monotonc, and nonmonotonc models. Ths novel approach also ntegrates the pror nformaton form the neghborng ponts. These neghborng ponts represent the nformaton of pror smulatons. In other words, we capture the nformaton of pror smulaton as well as the partally monotonc models. Ths premse s based on assumpton that the global uncertanty s related to local uncertantes [4, 7]. These outcomes are nterestng for the nterpolaton schemes. We, however, progress further and use the outcomes for the relablty estmaton of nfrastructures. We use the Monte Carlo method as the bass for our smulatons. Wth ths modelng approach, smulaton tmes can be reduced n predctve tools developed to forecast system relablty estmates. A logcal dependence between neghborng ponts s assumed for each randomly generated pont. The uncertanty of ths assumpton s nvestgated for all calculated data ponts and error lmts are developed. Ths approach s flexble as t permts ncluson of addtonal pror nformaton n the modelng f warranted. Generalze Beta (GB) dstrbuton s used to capture the monotonc pror, and the Gaussan dstrbuton s used for the non-monotonc prors. Regardng the calculaton tme, the presented approach s recommended for the relablty assessment of complex structures where every realzaton counts regardng the overall calculaton efforts. It s assumed that readers are famlar wth the Bayesan technques [8] and applcaton of Bayes Theorem n practce [9].

2 . MATHEMATICAL FORMULATION.1. A NOVEL BAYESIAN INTERPOLATION METHOD Consder a contnuous functon U that we wsh to estmate at a number of dscrete ponts. We defne the u set of dscrete ponts by a vector u assgned to dscrete ponts (pxels). The elements of observed data ponts are d defned by vector d [ d1,, d n ], and ther locatons are stored n a n-dmensonal vector. Let Pu ( DI, ) be the unvarate probablty densty functon (pdf) for an arbtrary pxel u. The data D and nformatonal context I can be found from the smulatons and model, respectvely. The global uncertanty ( () ( n) [,,, ] (e.g., global standard devaton), where each uncertanty s assocated wth ts ( () ( n) respectve dmenson x [ x, x,, x ] of the lmt state equaton Gx ( ) at any pont x s u. The global uncertanty was frst used n [7, 10] to defne a nusance parameter. Wth margnalzaton, the global uncertanty can be wrtten as Pu ( DI, ) Pu (, DI, ) d. ( By applcaton of Bayes Theorem, we fnd Pu ( DI, ) Pu ( ) P( DI, ) d. () Wth the global uncertanty ( ) defned, we can now estmate the value of LSE at an arbtrary pont x from an nterpolaton functon (model) f usng nformaton about ts neghborng ponts. Let the estmate be u ˆ. In ths model, the value of u s estmated by ts neghbor ponts. There are m 1 neghborng ponts for each arbtrary locaton among these ponts. The model f m s defned usng Equaton (), where ndex m s the order of model (functon). We can estmate value of LSE at the pont x close to the mddle of ts neghborng ponts Equaton (3) [11-13]. m rk rk k0, kr m f ( x ) u L ( x ), (3) m 1 where r = abs( ), u s the LSE responses assgned to pont x, and L k s the -th fundamental polynomal defned as.. GAUSSIAN ERROR ESTIMATE FOR ONE-DIMENSIONAL PROBLEMS x x l Lk( x). kl xk x (4) l We frst defne Pu ( ) n wrte hand sde (RHS) of Equaton () for a one-dmensonal problem. The value of error wth a zero mean could be postve or negatve and ts unknown varance s. Assume the standard devaton of error s proportonal to the shortest dstance from ts neghborng data ponts. We use the Gaussan densty functon for the error (Equaton (5)) of the model as a standard error form [14]. The Gaussan error s defned as 1 1 ( )= exp. e Pe (5)

3 where e s the error and fnd the followng multvarate pdf for pxels s an unknown varance. By makng the change of varable from e to u as: u, we 1 1 ( )= exp Pu. u f u m (6) Followng the approach used n [3, 4] to assocate the global and local uncertantes, and assumng the logcal ndependence between the errors and makng approprate substtutons, we obtan the posteror for u as: Pu ( DI, ) u n fm u d fmu 1 1 exp d. n1 =1 (7).3. BETA ERROR ESTIMATE FOR ONE-DIMENSIONAL PROBLEMS For monotonc model, we frst defne Pu ( ) n wrte hand sde (RHS) of Equaton () for a onedmensonal problem. The value of error wth a zero mean could be postve or negatve and ts unknown varance s. We use the Generalzed Beta (GB) densty functon for the error to assure monotonc constrant of the model. The GB dstrbuton ensures that uˆ f( x) s bounded between u 1 and u 1 and ts densty functon for error s a sutable choce. The GB densty s defned as p1 q1 ( xc) ( d x) Pxcd (, ), pq1 B( p, q)( d c) (8) for c x d and B( pq, ) s the Beta functon. Usng the GB dstrbuton at the nterval of [ u 1 u 1] and assumng u 1 u 1, we have ( xu ) ( u x) Pxu (, ), p1 q u 1 pq1 B( p, q)( u 1 u (9) where u 1 x u 1. The estmate of the pxel value s u = xand the error functon s defned as e = u u 1. (10) Substtuton of Equaton (10) nto Equaton (9) gves p1 q1 ( u u ( u 1u) 1 1 pq1 B( p, q)( u 1 u Pu ( u, u, p, q). (1 Now, followng the steps ndcated n [15, 16] to assocate the global and local uncertantes, and assumng the logcal ndependence between the errors and makng approprate substtutons, we obtan the posteror for u as:

4 n p1 q1 ( d u ( u 1d ) Pu ( DI, ) pq1 1, c B( p, q)( u 1 u, ( ( u u ) ( u u ) d B( p, q )( u u ) p1 q1 1 1 pq1 1 1 where B (.,.) s the Beta functon as ndcted n Equaton (8), p and q are the local Beta parameters obtaned by the followng equatons. p ( u uˆ )( uˆ u u u uˆ uˆ u ), ( u 1 u (13) q uˆ u u u u uˆ u uˆ u u uˆ uˆ u uˆ ( u 1 u, (14).4. INTEGRATED ERROR ESTIMATE FOR MODELS WITH PARTIAL MONOTONICITY In ths secton, we ntegrate the two dstrbutons nto one model to capture pror nformaton of partally monotonc models. Ths s a necessty for analyss of dependent varables. Assume a lmt state equaton Gx ( ), where x s composed of vectors x mon and x nonmon. We use the Generalzed Beta and Gaussan dstrbutons to capture the monotonc and non-monotonc part of the model, respectvely. These were descrbed n Sectons.3 and.. In ths case, we defne a vector of global uncertantes ( () ( n) () [,,, ], where each global uncertanty s assocated wth ts correspondng dmenson () x, where x () ( () ( n) belongs to x [ x, x,, x ] of the lmt state equaton Gx ( ). The exact value of Gx ( ) at any pont x s u. Havng the data ponts at neghborhood of pont x, we can estmate the response model usng functon f accordng to Equaton (3). Ths estmate s uˆ and the error s obtaned by the followng equaton e = u uˆ = u f ( u,, u ). (15) q 1 p where p s the number of requred data ponts for response estmaton and q s the response order. These numbers ( p, q ) depend on dmensons of the problem and the model order. For example, n a lnear estmaton of a two dmensonal problem, three data ponts are requred, and p s equal to 3 whle the model s lnear and q 1. In presence of hgher number of data ponts, data ponts wth the shortest dstance from u are selected snce a closer neghbor s assumed to have a greater nfluence on the estmate than the other neghbors. Equaton () s used for the multdmensonal problems and the frst term n ts RHS s Pu ( ( m) ( m ( n) ( ) Pu (,,,, ), (16) () where n s the problem dmenson, m s the number of monotonc varables, u s the target pxel and s the global uncertanty assocated wth -th dmenson. Followng the process for monotonc model (Equaton (1) and non-monotonc models (Equaton (6)) and assumng ndependent global uncertantes of dfferent dmensons gves

5 Pu ( ( m) ( m ( n) (,,,,, ) m k 1 ( ) ( ) 1 u f ( u,, u ) u u u u (17) exp. B( p, q )( u u ) ( k) ( k) ( k) p ( ) 1 ( k) q 1 k m 1 1 q 1 p ( k) ( k) ( k) ( k) ( k) ( k) p 1 ( ) q n k ( k ) k km1 The second term of the RHS of Equaton () s P( D, I) whch can be obtaned on the bass of Equatons (7) and (. The rest of the process s a straght forward process and the JPDF of u s obtaned. We present an applcaton example of the proposed method n the next secton. 3. APPLICATION TO THE FLOOD WALL IN NEW ORLEANS The method s demonstrated here for the 17 th Street Flood wall. The falure of ths structure was studed before, and we use ths real-world example to compare the outcomes wth prevous studes. Ths recent structural falure problem has been nvestgated by many researchers [1, 17, 18]. Located on the 17 th Street Canal n New Orleans, USA, the 17 th Street Flood Wall was breached durng Hurrcane Katrna when the surge level exceeded 8.0 feet. All data and nformaton about the geometry and materal propertes of the 17 th Street Flood Wall were obtaned from publshed materals on nternet webstes [17, 18]. The 17 th Street Flood Wall can be consdered as a monotonc model, and as shown n [1], the frst three nfluental varables for the 17 th Street Flood Wall suffced to provde the desred level of accuracy usng a fnte element model as depcted n Fgure 1. The same model s used here to descrbe mplementaton of our method for three nfluental varables to nvestgate fal-safe characterstcs of 17 th Street Flood Wall and develop probablty of falure estmates. The canddates for the frst three nfluental varables are shown n Table 3 n [1]. The product moment correlaton ( ) crteron for the frst three nfluental varables s used here to obtan probablty estmates shown n [1]. In ths analyss, parameters for sol number 3, 8, and are ndeed the controllng varables for falure of the flood wall 1 accordng to [1]. Fgure 1. The fnte element model of 17th Street Flood Wall n New Orleans. The controllng varables of the 17 th Street Flood Wall v 1,..., v 3 are shown n [1]. Predcted varable estmate are dependent on these varables, and dfferent estmates would be obtaned wth a dfferent set of varables. For each randomly generated data pont (pxel) n the lmt state equaton, we developed JPDF of the estmate. The ntegraton of ont pdf over the stable or unstable regons determnes locaton of the target pxels (or data ponts). As the smulaton progresses, the accuracy of estmates wll mprove wth ncreasng sze of ensemble populaton (data ponts) that reduces the predctve errors. Fgure shows a comparson between the estmated ont pdf of a two-dmensonal Gaussan problem ( v1, v ) for 5 versus 0 data ponts. 1 The rank correlaton shows that varables 3, 8, and 4 are the most nfluental varables and ths sequence may change when the structure's response becomes nonlnear at the hgh water level, W.L.=+8 ft (.4 m).

6 Results on Fgure show that the accuracy of predcted estmates s mprovng wth the progresson of MC smulatons. Takng advantage of these characterstcs saves enormous computatonal tme n the MC smulatons. Pu ( v, v, I, 5 data ponts) b) Pu ( v1, v3, I, 0 data ponts) a) 1 3 ( v Fgure. A comparson between the estmated ont pdf of a two dmensonal Gaussan problem 1, v) versus 0 data ponts. for 5 The number of smulatons requred for the Bayesan Monte Carlo method for the 17 th applcaton for water level +8 ft s shown n Street I-Wall Table 1. Results are provded for the Bayesan Monte Carlo for a monotonc model, Bayesan Monte Carlo for a non-monotonc model, classcal Monte Carlo (MC) and Dynamc Bounds (DB) methods, showng that only a fracton of MC smulaton s requred when the Generalzed Beta dstrbuton s used to capture monotoncty. The method of Dynamc Bounds s brefly descrbed here snce t has been fully descrbed n prevous publcatons (see References). It s used n the comparsons shown here wth a monotonc model. The proposed method n ths paper s flexble and can be used for totally monotonc, non-monotonc, and partally monotonc models. Ths novel approach, therefore, s not subected to the lmtatons of use of the DB method. An uncertanty model assocaton s used to relate the local uncertanty to the global uncertanty n the form of, where corresponds to the cubc root of the dstance of x wth ts closest neghbour. The cubc root relaton for the thrd order model response s used [5].

7 Table 1. The calculated probabltes of falure for the 17th Street I Wall structure obtaned wth the BMC for monotonc models, BMC for non monotonc models, MC and DB methods usng product moment correlaton [1] for three most nfluental varables. Number of Method W.L. (ft) p ˆ f ( pˆ f ) smulatons BMC for monotonc model (G. Beta error estmaton) BMC for non-monotonc model (Gaussan error estmaton) +8 (.4 m) (.4 m) Dynamc Bounds method +8 (.4 m) Monte Carlo method +8 (.4 m) CONCLUSIONS A technque s developed n ths paper to sgnfcantly ncrease the effcency of Monte Carlo method for partally monotonc, totally monotonc and non-monotonc models. Ths s n great mportance for complex fnte element applcatons or other tme consumng processes whch are employed for modelng lnear and nonlnear behavor of structures. Ths technque ntegrates advantages of three recently developed relablty methods (DB [19], IDB [4] and BMC [0]) nto a generc form for the Monte Carlo famly. The theoretcal formulaton and numercal mplementaton detals of a novel Bayesan nterpolaton method developed for ths purpose are provded. The proposed Bayesan model ntegrates the Gaussan and Generalzed Beta densty functons. Our proposed method enoys ncluson of a new concept n Bayesan formulaton that relates the global and local uncertantes [10]. As a result, an unbased estmate for the Monte Carlo method has been obtaned. In other words, ths novel technque preserves fundamental propertes of the classcal MC method, and greatly mproves the computatonal effcency by usng dfferent types of pror nformaton. Ths newly developed method s appled to nvestgate load-response characterstcs of the 17 th Street Flood Wall. Results of prevous numercal realzatons (smulatons), termed here as the pror nformaton, are used n the current smulaton wth the Bayes theory and the Bayesan Monte Carlo method. The pror nformaton s obtaned from prevously completed Monte Carlo smulatons. A partal or total monotoncty of a model s consdered as another source of pror nformaton ntegrated to the smulaton by use of the Generalzed Beta dstrbuton. As a result, separaton of data ponts n ths manner avods unnecessary smulatons n the Monte Carlo method and substantally reduces computatonal burdens. REFERENCES 1. Raabalnead, M., et al., Applcaton of Dynamc Bounds n the safety assessment of flood defences, a case study: 17th Street Flood Wall, New Orleans. Georsk Journal., June ().. Raabalnead, M., P. van Gelder, and J.k. Vrlng, Probablstc Fnte Elements wth Dynamc Lmt Bounds: A Case Study: 17th Street Flood Wall, New Orleans, n 6th Internatonal Conference on Case Hstores n Geotechncal Engneerng008, Mssour Unversty of Scence and Technology. 3. Raabalnead, M., P. van Gelder, and J.K. Vrlng, Improved Dynamc Lmt Bounds n Monte Carlo Smulatons, n 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamcs, and Materals Conference008, Amercan Insttute of Aeronautcs and Astronautcs.

8 4. Raabalnead, M., Relablty Methods for Fnte Element Models. I ed009, Amsterdam, the Netherlands: IOS Press. 5. Raabalnead, M., Z. Demrblek, and T. Mahd, Determnaton of Falure Probabltes of Flood Defence Systems wth Improved Dynamc Bounds Method. submtted for publcaton, Plaxs, fnte element code for sol and rock analyses. 00. D - Verson Raabalnead, M., P. van Gelder, and N. van Erp, Applcaton of Bayesan Interpolaton n Monte Carlo Smulaton, n Safety, Relablty and Rsk Analyss (ESREL)008, Taylor and Francs Group, London, UK. p Jaynes, E. and G. Bretthorst, Probablty theory: the logc of scence003: Cambrdge Unv Pr. 9. Sva, D. and J. Skllng, Data analyss: a Bayesan tutoral006: Oxford Unversty Press, USA. 10. Raabalnead, M., Relablty Methods for Fnte Element Models, n Hydraulc Engneerng009, TUDelft: Delft, the Netherlands. p Boanov, B., H. Hakopan, and A. Saak a n, Splne functons and multvarate nterpolatons1993: Kluwer Academc Pub. 1. De Boor, C. and A. Ron, On multvarate polynomal nterpolaton. Constructve Approxmaton, (3): p Trebel, H., Interpolaton theory, functon spaces, dfferental operators Jaynes, E.T., Probablty theory, the logc of scence Raabalnead, M. and Z. Demrblek, Bayesan Monte Carlo method for monotonc models applyng the Generalzed Beta dstrbuton. Engneerng Falure Analyss, (4): p Raabalnead, M. and T. Mahd, The nclusve and smplfed forms of Bayesan nterpolaton for general and monotonc models usng Gaussan and Generalzed Beta dstrbutons wth applcaton to Monte Carlo smulatons. Natural Hazards, (: p USACE-e, Orleans and Southeast Lousana Hurrcane Protecton System, Volume V The Performance Levees and Floodwalls 006, U.S. Army Corps of Engneers, Report of the Interagency Performance Evaluaton Task Force 18. ILIT, I.L.I.T., Investgaton of the Performance of the New Orleans Flood Protecton Systems n Hurrcane Katrna 006, Berkely Unversty. 19. Raabalnead, M., et al., Applcaton of Dynamc Bounds n the safety assessment of flood defences, a case study: 17th Street Flood Wall, New Orleans. Georsk Journal., (4). 0. Raabalnead, M., Bayesan Monte Carlo method. Relablty Engneerng and System Safety, (10): p