Application of Bayesian Filters to Heat Conduction Problems

Transcription

1 EngOpt 28 - Internatonal Conference on Engneerng Optmzaton Ro de Janero, Brazl, - 5 June 28. Applcaton of Bayesan Flters to Heat Conducton Problems Helco R. B. Orlande, George S. Dulkravch 2 and Marcelo J. Colaço 3 Department of Mechancal Engneerng, COPPE, Federal Unversty of Ro de Janero, Brazl, helco@mecanca.coppe.ufrj.br Department of Mechancal and Materals Engneerng, Florda Internatonal Unversty, Mam, U.S.A., dulkrav@fu.edu Department of Mechancal and Materals Engneerng, Mltary Insttute of Engneerng, Ro de Janero, RJ, Brazl, colaco@asme.org. Abstract In ths paper we present a general descrpton of state estmaton problems wthn the Bayesan framework. State estmaton problems are addressed n whch evoluton and measurement stochastc models are used to predct the dynamc behavor of physcal systems. Specfcally, the applcaton of two Bayesan flters to lnear and non-lnear unsteady heat conducton problems s demonstrated; a) the use of Kalman flter, and b) the use of Partcle Flter wth the samplng mportance resamplng algorthm. 2. Keywords: Kalman flter, Partcle flter, state estmaton 3. Introducton State estmaton problems, also desgnated as nonstatonary nverse problems [], are of great nterest n nnumerable practcal applcatons. In such knds of problems, the avalable measured data s used together wth pror knowledge about the physcal phenomena and the measurng devces, n order to sequentally produce estmates of the desred dynamc varables. Ths s accomplshed n such a manner that the error s mnmzed statstcally [2]. For example, the poston of an arcraft can be estmated through the tme-ntegraton of ts velocty components snce departure. However, t may also be measured wth a GPS system and an altmeter. State estmaton problems deal wth the combnaton of the model predcton (ntegraton of the velocty components that contan errors due to the velocty measurements) and the GPS and altmeter measurements that are also uncertan, n order to obtan more accurate estmatons of the system varables (arcraft poston). State estmaton problems are solved wth the so-called Bayesan flters [,2]. In the Bayesan approach to statstcs, an attempt s made to utlze all avalable nformaton n order to reduce the amount of uncertanty present n an nferental or decson-makng problem. As new nformaton s obtaned, t s combned wth prevous nformaton to form the bass for statstcal procedures. The formal mechansm used to combne the new nformaton wth the prevously avalable nformaton s known as Bayes theorem [,3]. The most wdely known Bayesan flter method s the Kalman flter [,2,4-9]. However, the applcaton of the Kalman flter s lmted to lnear models wth addtve Gaussan noses. Extensons of the Kalman flter were developed n the past for less restrctve cases by usng lnearzaton technques [,3,6,7,8]. Smlarly, Monte Carlo methods have been developed n order to represent the posteror densty n terms of random samples and assocated weghts. Such Monte Carlo methods, usually denoted as partcle flters among other desgnatons found n the lterature, do not requre the restrctve hypotheses of the Kalman flter. Hence, partcle flters can be appled to non-lnear models wth non-gaussan errors [,4,8-7]. In ths paper we apply the Kalman flter and the partcle flter to heat conducton problems. These Bayesan flters are used to predct the temperature n a medum where the heat conducton model and temperature measurements contan errors. Lnear and nonlnear heat conducton problems are examned, as well as Gaussan and non-gaussan nose. Before focusng on the heat conducton applcatons of nterest, the state estmaton problem s defned and the Kalman and partcle flters are descrbed. 4. State Estmaton Problem In order to defne the state estmaton problem, consder a model for the evoluton of the vector x n the form xk = fk( xk, v k ) (.a) n where the subscrpt k =, 2,, denotes a tme nstant t k n a dynamc problem. The vector x R x s called the state vector and contans the varables to be dynamcally estmated. Ths vector advances n accordance wth the state evoluton model gven by n equaton (.a), where f s, n the general case, a non-lnear functon of the state varables x and of the state nose vector v R v. n Consder also that measurements z z k R are avalable at t k, k =, 2,. The measurements are related to the state varables x through the general, possbly non-lnear, functon h n the form zk = hk( xk, n k) (.b) n where n R n s the measurement nose. Equaton (.b) s referred to as the observaton (measurement) model. The state estmaton problem ams at obtanng nformaton about x k based on the state evoluton model (.a) and on the measurements z: k = { z, =, K, k} gven by the observaton model (.b) [-7]. The evoluton-observaton model gven by equatons (.a,b) are based on the followng assumptons [,4]: () The sequence xk for k =, 2,, s a Markovan process, that s, π( xk x, x, K, xk ) = π( xk xk ) (2.a) () The sequence zk for k =, 2,, s a Markovan process wth respect to the hstory of x k, that s, π( zk x, x, K, xk) = π( zk xk) (2.b)

2 () The sequence x k depends on the past observatons only through ts own hstory, that s, π( k k, : k ) = π( k k ) where π ( ab ) denotes the condtonal probablty of a when b s gven. x x z x x (2.c) In addton, for the evoluton-observaton model gven by equatons (.a,b) t s assumed that for j the nose vectors v and v j, as well as n and n j, are mutually ndependent and also mutually ndependent of the ntal state x. The vectors v and n j are also mutually ndependent for all and j []. Dfferent problems can be consdered wth the above evoluton-observaton model, namely []: () The predcton problem, concerned wth the determnaton of π ( xk z : k ) ; () The flterng problem, concerned wth the determnaton of π ( xk z : k) ; () The fxed-lag smoothng problem, concerned the determnaton of π ( xk z : k+ p), where p s the fxed lag; (v) The whole-doman smoothng problem, concerned wth the determnaton of π ( xk z : K), where z: K = { z, =, K, K} s the complete sequence of measurements. Ths paper deals only wth the flterng problem. By assumng that π( x z) = π( x ) s avalable, the posteror probablty densty π ( xk z : k) s then obtaned wth Bayesan flters n two steps [-7]: predcton and update, as llustrated n fgure. The Kalman flter and the partcle flter used n ths work are dscussed below. π(x ) π(x x ) Predcton π(x ) Update π(z x ) π(x z ) π(x 2 x ) Predcton π(x 2 z ) Update π(z 2 x 2 ) π(x 2 z :2 ) M Fgure. Predcton and update steps for the Bayesan flter [] 5. The Kalman Flter For the applcaton of the Kalman flter t s assumed that the evoluton and observaton models gven by equatons (.a,b) are lnear. Also, t s assumed that the noses n such models are Gaussan, wth known means and covarances, and that they are addtve. Therefore, the posteror densty π ( xk z : k) at t k, k =, 2, s Gaussan and the Kalman flter results n the optmal soluton to the state estmaton problem, that s, the posteror densty s calculated exactly [,2,4-9]. Wth the foregong hypotheses, the evoluton and observaton models can be wrtten respectvely as: xk = Fkxk + v k (3.a) zk = Hkxk + n k (3.b) where F and H are known matrces for the lnear evolutons of the state x and of the observaton z, respectvely. By assumng that the noses v and n have zero means and covarance matrces Q and R, respectvely, the predcton and update steps of the Kalman flter are gven by [,2,4-9]:

3 Predcton: Update: xk = Fkx k (4.a) Pk = FkPk Fk T + Q k (4.b) T ( ) T Kk = PkHk HkPkHk + R k (5.a) k x = xk + Kk( zk Hkx k) (5.b) ( ) Pk = I-KkHk P k (5.c) The matrx K s called Kalman s gan matrx. Notce above that after predctng the state varable x and ts covarance matrx P wth equatons (4.a,b), a posteror estmates for such quanttes are obtaned n the update step wth the utlzaton of the measurements z. For other cases for whch the hypotheses of lnear Gaussan evoluton-observaton models are not vald, the use of the Kalman flter does not result n optmal solutons because the posteror densty s not analytc. The applcaton of Monte Carlo technques then appears as the most general and robust approach to non-lnear and/or non-gaussan dstrbutons [,4,8-7]. Ths s the case despte the avalablty of the so-called extended Kalman flter and ts varatons, whch generally nvolves a lnearzaton of the problem. A Monte Carlo flter s descrbed below. 6. The Partcle Flter The Partcle Flter Method [,4,8-7] s a Monte Carlo technque for the soluton of the state estmaton problem. The partcle flter s also known as the bootstrap flter, condensaton algorthm, nteractng partcle approxmatons and survval of the fttest [8]. The key dea s to represent the requred posteror densty functon by a set of random samples (partcles) wth assocated weghts, and to compute the estmates based on these samples and weghts. As the number of samples becomes very large, ths Monte Carlo characterzaton becomes an equvalent representaton of the posteror probablty functon, and the soluton approaches the optmal Bayesan estmate. We present below the so-called Sequental Importance Samplng (SIS) algorthm for the partcle flter, whch ncludes a resamplng step at each nstant, as descrbed n detal n references [8,9]. The SIS algorthm makes use of an mportance densty, whch s a densty proposed to represent another one that cannot be exactly computed, that s, the sought posteror densty n the present case. Then, samples are drawn from the mportance densty nstead of the actual densty. Let { x:, = k, K, N } be the partcles wth assocated weghts { w k, =, K, N} and x: k = { xj, j =, K, k} be the set of all states N up to t k, where N s the number of partcles. The weghts are normalzed so that wk =. Then, the posteror densty at t k can be = dscretely approxmated by: N π( x: k z: k) wkδ( x: k x : k) (6.a) = where δ(.) s the Drac delta functon. By takng hypotheses (2.a-c) nto account, the posteror densty (6.a) can be wrtten as [9]: N π( xk z: k) wkδ( xk x k) (6.b) = A common problem wth the SIS partcle flter s the degeneracy phenomenon, where after a few states all but one partcle wll have neglgble weght [,4,8-7]. Ths degeneracy mples that a large computatonal effort s devoted to updatng partcles whose contrbuton to the approxmaton of the posteror densty functon s almost zero. Ths problem can be overcome by ncreasng the number of partcles, or more effcently by approprately selectng the mportance densty as the pror densty π ( xk x k ). In addton, the use of the resamplng technque s recommended to avod the degeneracy of the partcles. Resamplng nvolves a mappng of the random measure { * x k, wk} nto a random measure { x k, N } wth unform weghts. It can be performed f the number of effectve partcles wth large weghts falls below a certan threshold number. Alternatvely, resamplng can also be appled ndstnctvely at every nstant t k, as n the Samplng Importance Resamplng (SIR) algorthm descrbed n [8,9]. Such algorthm can be summarzed n the followng steps, as appled to the system evoluton from t k- to t k [8,9]: Step. For =,, N draw new partcles x k from the pror densty π ( xk x k ) and then use the lkelhood densty to calculate the correspondent weghts w k = π ( zk x k). Step 2. Calculate the total weght w = T w. k w k T w N = w and then normalze the partcle weghts, that s, for =,, N let = k

4 Step 3. Resample the partcles as follows: Step 3.. Construct the cumulatve sum of weghts (CSW) by computng c = c + wk for =,, N, wth c =. Step 3.2. Let = and draw a startng pont u from the unform dstrbuton U[, N ]. Step 3.3. For j =,, N Move along the CSW by makng uj = u + N ( j ). Whle uj > c make = +. j Assgn sample x k = x k. j Assgn weght wk = N. Although the resamplng step reduces the effects of the degeneracy problem, t may lead to a loss of dversty and the resultant sample can contan many repeated partcles. Ths problem, known as sample mpovershment, can be severe n the case of small process nose. In ths stuaton, all partcles collapse to a sngle partcle wthn few nstants t k [,8,9]. Another drawback of the partcle flter s related to the large computatonal cost due to the Monte Carlo method, whch may lmt ts applcaton to complcated physcal problems. 7. Results and Dscussons In ths sesson we apply the Bayesan flters descrbed above to the estmaton of the transent temperature feld n heat conductng meda. Lnear and non-lnear heat conducton problems are addressed, as well as dfferent models for the noses. The problems under study are descrbed below and the results obtaned wth the Kalman and partcle flters are dscussed. 7.. Lnear Heat Conducton Problem Consder heat conducton n a sem-nfnte one-dmensonal medum, ntally at the unform temperature T*. The temperature at boundary x = s kept at T = o C. Physcal propertes are constant and there s no heat generaton n the medum. The formulaton for ths problem s gven by: 2 T T = 2 α t x n x >, for t > (7.a) T = at x=, for t > (7.b) T = T* for t=, n x > (7.c) The analytcal soluton for ths problem s gven by [8]: x T( x, t) = T * erf 4αt (8) The dscretzaton of equaton (7.a) by usng explct fnte-dfferences results n: k+ k k k T = rt + ( 2 r) T + rt+ (9) where the superscrpt k denotes the tme step, the subscrpt denotes the fnte-dfference node and α t r = 2 ( x) () For the soluton of problem (7.a-c) wth fnte-dfferences we have to mpose a boundary condton at some fcttous surface at x=l. We assume L suffcently large for the tme range of nterest, so that the fnte doman behaves as a sem-nfnte medum. Temperature at the surface x=l was assumed to be T*. The fnte-dfference soluton for problem (7.a-c) can be wrtten as: k k T = FT + S () where ( 2 r) r T r ( 2 r) r Τ = M F = O O O M S = T N r ( 2 r) r r ( 2 r) rt * (2.a-c) In the equatons above N s the number of nternal nodes n the fnte-dfference dscretzaton. Matrx F s NxN and vectors T and S s Nx. Equaton () s n approprate form for the applcaton of the Kalman flter (see equaton 3.a). In the present example t s assumed that state and measured varables are the transent temperatures nsde the medum at the equdstant fnte dfference nodes. Therefore, matrx H for the observaton model (see equaton 3.b) s the dentty matrx. The medum s consdered to be concrete, wth thermal dffusvty α = 4.9x -7 m 2 /s. The standard devaton for the measurement errors s consdered constant and equal to 2 o C. The effects of the standard devaton of the errors n the state model are examned below. Fgures 2.a and 2.b present the exact temperatures and the measured temperatures n the regon, respectvely. The fnal tme s taken as 25 seconds and measurements are supposed avalable n the regon every second. The thckness of the medum s consdered to be L =. m and the regon s dscretzed wth N = 5 nternal nodes.

5 Temperature Temperature T, C 4 2 T, C x, m Fgure 2.a temperatures x, m 5 Fgure 2.b temperatures contanng Gaussan errors wth standard devaton of 2 o C Fgures 3.a,b present a comparson of exact, measured and predcted temperatures at postons x =.2 m and x =. m, respectvely. The predcted temperatures were obtaned wth the Kalman flter. The 99% confdence ntervals for the predcted temperatures are also presented n ths fgure. The results presented n fgures 3.a-d were obtaned wth a standard devaton for the evoluton model errors of o C. These fgures clearly show a great mprovement n the predcted temperatures as compared to the measured temperatures at dfferent postons nsde the medum. Note that predcted temperatures are much closer to the exact ones than the measurements. In fact, f the model errors are reduced, the predcted temperatures tend to follow the model more closely. Ths s exemplfed n fgures 4.a,b, where the standard devaton of the evoluton model errors was reduced to.5 o C. On the other hand, f such errors are large, the predctons of the Kalman flter tend to follow the measurements nstead of the model. Such a fact s clear from the analyss of fgures 5.a,b, whch present the results for a standard devaton for the evoluton model errors of 2 o C. The applcaton of the partcle flter wth the samplng mportance resamplng algorthm descrbed above s presented n fgures 6.a,b. These fgures present the exact, measured and predcted temperatures at postons x =.2 m and x =. m, respectvely. Ffty partcles were used n ths case. A comparson of fgures 3.a,b and 6.a,b show that the partcle flter, smlarly to the Kalman flter, provded accurate predctons for the temperature n the regon. However, as expected, the computatonal cost of the partcle flter was substantally larger than that of the Kalman flter. For ths example, the predctons obtaned wth the Kalman flter took.5 seconds and those obtaned wth the partcle flter took 6.9 seconds on a.8 GHz Centrno Duo computer wth Mbyte of RAM memory. We note that for ths case the computatonal cost of the partcle flter could be reduced to 2.3 seconds by decreasng the number of partcles to, wthout sgnfcant loss of accuracy n the predcted temperatures T(x=.2 m), C 2 T(x=. m), C (a) (b) Fgure 3 Temperature at (a) x =.2 m and (b) x =. m standard devaton for the evoluton model errors of o C Kalman flter

6 T(x=.2 m), C 3 2 T(x=. m), C (a) (b) Fgure 4 Temperature at (a) x =.2 m and (b) x =. m standard devaton for the evoluton model errors of.5 o C Kalman flter T(x=.2 m), C 3 2 T(x=. m), C (a) (b) Fgure 5 Temperature at (a) x =.2 m and (b) x =. m standard devaton for the evoluton model errors of 5 o C Kalman flter 5 4 Partcle Flter wth Re-samplng - Number of Partcles = Partcle Flter wth Re-samplng - Number of Partcles = T(x=.2 m), C 2 T(x=. m), C (a) (b) Fgure 6 Temperature at (a) x =.2 m and (b) x =. m standard devaton for the evoluton model errors of o C Partcle flter wth resamplng Fgures 7.a,b present results smlar to those of fgures 6.a,b, but wthout usng the resamplng technque. These fgures show that the predctons obtaned wthout resamplng were qute uncertan, that s, wth large confdence ntervals. On the other hand, sample mpovershment can be observed when resamplng was used, whch s characterzed by the very narrow confdence ntervals for the predcted temperatures shown n fgures 6.a,b.

7 We now examne a case nvolvng unformly dstrbuted errors n the evoluton model, nstead of Gaussan errors. For such case, the applcaton of the Kalman flter does not result n optmal solutons. Consequently, only the partcle flter was consdered for the predcton of the temperatures n the regon. The results obtaned wth errors n the evoluton model unformly dstrbuted n the nterval [-,] o C are presented n fgures 8.a,b for x =.2 m and x =. m, respectvely. These fgures show that the partcle flter was not affected by the dstrbuton of the errors and results smlar to those obtaned for the Gaussan dstrbuton (see fgures 6.a,b) were obtaned. 5 4 Partcle Flter - Number of Partcles = 5 - CPU Tme = sec 6 5 Partcle Flter - Number of Partcles = 5 - CPU Tme = sec 3 4 T(x=.2 m), C 2 T(x=. m), C (a) (b) Fgure 7 Temperature at (a) x =.2 m and (b) x =. m standard devaton for the evoluton model errors of o C Partcle flter wthout resamplng 5 4 Partcle Flter wth Re-samplng - Number of Partcles = Partcle Flter wth Re-samplng - Number of Partcles = T(x=.2 m), C 2 T(x=. m), C (a) (b) Fgure 8 Temperature at (a) x =.2 m and (b) x =. m evoluton model errors havng unform dstrbuton n [-,] o C Partcle flter wth resamplng 7.2. Non-lnear Heat Conducton Problem Consder heat conducton n a one-dmensonal medum wth thckness L, ntally at the unform temperature T*. The boundary at x = s kept nsulated and a constant heat flux q* s mposed at x = L. Thermophyscal propertes are temperature dependent and there s no heat generaton n the medum. The formulaton for ths problem s gven by: T T CT ( ) = kt ( ) t x x n < x < L, for t > (3.a) T = x at x=, for t > (3.b) T kt ( ) = q* x at x=l, for t > (3.c) T = T * for t=, n x > (3.d) We examne here a physcal problem smlar to those examned n references [9,2] nvolvng the heatng of a graphte sample wth an oxy-acetylene torch. The temperature dependence of the thermal conductvty, k(t), and volumetrc heat capacty, C(T), of the graphte, measured for dfferent temperatures, were curve-ftted (see fgure 9) wth exponentals of the form:

8 ( ) 3 CT = A+ Ae T A (4.a) 2 ( ) 3 k T = B + B e T B (4.b) 2 wth the adjusted parameters A, A 2, A 3, B, B 2 and B 3 gven n table. Table. Parameters of the curve-ftted thermophyscal propertes Parameter Mean A (Jm -3 ºC - ) 5,68,6 A 2 (Jm -3 ºC - ) 4,83,57 A 3 (ºC) 547. B (Wm - ºC - ) B 2 (Wm - ºC - ) 83.5 B 3 (ºC) 277. Capacdade Volumetrc Térmca Heat Capacty Volumetrca (Jm (J/m -3o C 3. C) - ) 6x 6 5x 6 4x 6 3x 6 2x 6 x 6 Thermal Condutvdade conductvty térmca Heat Capacdade capacty térmca Temperature Temperatura ( o ( C) Fgure 9. Temperature-dependent thermophyscal propertes Thermal Condutvdade Conductvty térmca (W/m. C) (Wm -o C - ) The sample was supposed to be ntally at the room temperature of T* = 2 o C and the mposed heat flux was q* = 5 W/m 2. The thckness of the slab was taken as L =. m. Fnte volumes were used for the soluton of ths nonlnear heat conducton problem, wth the regon dscretzed wth 5 equal volumes. The fnal tme was supposed to be 9 seconds and the tme step was taken as second. The errors n the state evoluton and observaton models were supposed to be addtve, Gaussan, uncorrelated, wth zero mean and constant standard devatons. In order to examne a very strct case, the standard devaton for the state evoluton model was set to 5 o C and for the observaton model as o C. Fgures.a and.b present the exact and measured temperatures, respectvely, for such non lnear heat conducton problem. It s apparent from the analyss of these fgures that extremely large measurement errors were consdered for ths case. State evoluton and measurement models were used for the predcton of the temperature varaton n the regon. Due to the nonlnear character of the problem under consderaton, only the partcle flter, n the form of the samplng mportance resamplng algorthm descrbed above, was used for the predctons. Ffty partcles were used n such flterng algorthm. Fgures.a-d present a comparson of exact, measured and predcted temperatures at the postons x =.2 m,. m,.2 m and.4 m, respectvely. An analyss of these fgures reveals the robustness of the partcle flter. Despte the very large errors n the evoluton and observaton models, the predcted average temperatures are n excellent agreement wth the exact values. Furthermore, the 99% confdence ntervals for the predcted temperatures are sgnfcantly smaller than the dsperson of the measurements. Dfferently from the case analyzed above nvolvng lnear heat conducton, sample mpovershment s not observed n fgures.a-d. Ths s probably due to the large nose n the state model used for the nonlnear case. The computatonal tme for ths case was 6.3 seconds.

9 Temperature T, C x, m Fgure.a temperatures Temperature T, C x, m Fgure.b temperatures contanng Gaussan errors wth standard devaton

10 6 Partcle Flter wth Re-samplng - Number of Partcles = T(x=.2 m), C Fgure.a Temperature at x =.2 m for the nonlnear heat conducton problem Partcle flter 6 Partcle Flter wth Re-samplng - Number of Partcles = T(x=. m), C Fgure.b Temperature at x =. m for the nonlnear heat conducton problem Partcle flter

11 7 Partcle Flter wth Re-samplng - Number of Partcles = T(x=.2 m), C Fgure.c Temperature at x =.2 m for the nonlnear heat conducton problem Partcle flter 6 Partcle Flter wth Re-samplng - Number of Partcles = T(x=.4 m), C Fgure.d Temperature at x =.4 m for the nonlnear heat conducton problem Partcle flter 8. Conclusons Ths paper dealt wth the applcaton of the Kalman flter and of the partcle flter to the estmaton of the transent temperature feld n lnear and non-lnear heat conducton problems. The partcle flter was coded n the form of the samplng mportance resamplng algorthm. State estmaton problems wth lnear evoluton-observaton models, contanng addtve Gaussan noses, were solved wth these Bayesan flter technques. For lnear-gaussan models, the Kalman flter and the partcle flter provded results of smlar accuracy. However, the computatonal cost of the partcle flter was sgnfcantly larger than that of the Kalman flter. On the other hand, n non-lnear and/or non-gaussan models the basc hypotheses requred for the applcaton of the Kalman flter are not vald. The partcle flter appears n the lterature as an accurate estmaton technque of general use, ncludng for non-lnear and/or non- Gaussan models. In ths paper, the partcle flter was successfully appled to a non-lnear heat conducton problem. Such Monte Carlo technque provded accurate estmaton results, even for a strct test-case nvolvng very large errors n the evoluton and observaton models.

12 9. Acknowledgements The vsts of Prof. Helco Orlande and Prof. Marcelo Colaço to Florda Internatonal Unversty, n January-February 28, were sponsored by CAPES and CNPq, agences for the fosterng of scence from the Brazlan government. The hosptalty of Prof. George S. Dulkravch and hs famly, as well as of the Department of Mechancal and Materals Engneerng of FIU, s greatly apprecated. Prof. Helco Orlande would lke to acknowledge the course on Statstcal and Computatonal Inverse Problems gven at the Federal Unversty of Ro de Janero by Prof. Jar Kapo and Prof. Vlle Kolehmanen, from the Unversty of Kuopo n Fnland. The authors are grateful for the fnancal support provded for ths work by the US Ar Force Offce of Scentfc Research under grant FA montored by Dr. Todd E. Combs, Dr. Farba Fahroo and Dr. Donald Hearn and by the US Army Research Offce/Materals Dvson under the contract number W9NF montored by Dr. Wllam M. Mullns. The vews and conclusons contaned heren are those of the authors and should not be nterpreted as necessarly representng the offcal polces or endorsements, ether expressed or mpled, of the US Ar Force Offce of Scentfc Research, the US Army Research Offce or the U.S. Government. The U.S. Government s authorzed to reproduce and dstrbute reprnts for government purposes notwthstandng any copyrght notaton thereon.. References. Kapo, J. and Somersalo, E., 24, Statstcal and Computatonal Inverse Problems, Appled Mathematcal Scences 6, Sprnger-Verlag. 2. Maybeck, P., 979, Stochastc models, estmaton and control, Academc Press, New York. 3. Wnkler, R., 23, An Introducton to Bayesan Inference and Decson, Probablstc Publshng, Gansvlle. 4. Kapo, J., Duncan S., Seppanen, A., Somersalo, E., Voutlanen, A., 25, State Estmaton for Process Imagng, Chapter n Handbook of Process Imagng for Automatc Control, edtors: Davd Scott and Hugh McCann, CRC Press. 5. Kalman, R., 96, A New Approach to Lnear Flterng and Predcton Problems, ASME J. Basc Engneerng, vol. 82, pp Sorenson, H., 97, Least-squares estmaton: from Gauss to Kalman, IEEE Spectrum, vol. 7, pp Welch, G. and Bshop, G., 26, An Introducton to the Kalman Flter, UNC-Chapel Hll, TR Arulampalam, S., Maskell, S., Gordon, N., Clapp, T., 2, A Tutoral on Partcle Flters for on-lne Non-lnear/Non- Gaussan Bayesan Trackng, IEEE Trans. Sgnal Processng, vol. 5, pp Rstc, B., Arulampalam, S., Gordon, N., 24, Beyond the Kalman Flter, Artech House, Boston.. Doucet, A., Godsll, S., Andreu, C., 2, On sequental Monte Carlo samplng methods for Bayesan flterng, Statstcs and Computng, vol., pp Lu, J and Chen, R., 998, Sequental Monte Carlo methods for dynamcal systems, J. Amercan Statstcal Assocaton, vol. 93, pp Andreu, C., Doucet, A., Robert, C., 24, Computatonal advances for and from Bayesan analyss, Statstcal Scence, vol. 9, pp Johansen, A. Doucet, A., 28, A note on auxlary partcle flters, Statstcs and Probablty Letters, to appear. 4. Carpenter, J., Clfford, P., Fearnhead, P, 999, An mproved partcle flter for non-lnear problems, IEEE Proc. Part F: Radar and Sonar Navgaton, vol. 46, pp Del Moral, P., Doucet, A., Jasra, A., 27, Sequental Monte Carlo for Bayesan Computaton, Bayesan Statstcs, vol. 8, pp Del Moral, P., Doucet, A., Jasra, A., 26, Sequental Monte Carlo samplers, J. R. Statstcal Socety, vol. 68, pp Andreu, C., Doucet, Sumeetpal, S., Tadc, V., 24, Partcle methods for charge detecton, system dentfcaton and control, Proceedngs of IEEE, vol. 92, pp Ozsk, M., 993, Heat Conducton, Wley, New York. 9. Mota, C. A. A., Mkhalov, M. D., Orlande, H. R. B. and Cotta, R. M., 24, Identfcaton of Heat Flux Imposed by an Oxyacetylene Torch, th AIAA/ISSMO Multdscplnary Analyss and Optmzaton Conference, Albany, New York, August 29 September Mota, C. A. A., Orlande, H. R. B., Wellele, O., Kolehmanen, V. and Kapo, J., 27, Inverse Problem of Smultaneous Identfcaton of Thermophyscal Propertes and Boundary Heat Flux, 9th COBEM -Internatonal Congress of Mechancal Engneerng, November 5-9, 27, Brasla, Brazl.