2.3. Quantitative Properties of Finite Difference Schemes. Reading: Tannehill et al. Sections and


 Russell Flynn
 10 months ago
 Views:
Transcription
1 .3. Quattatve Propertes of Fte Dfferece Schemes.3.. Cosstecy, Covergece ad Stablty of F.D. schemes Readg: Taehll et al. Sectos ad Three mportat propertes of F.D. schemes: Cosstecy A F.D. represetato of a PDE s cosstet f the dfferece betwee PDE ad FDE,.e., the trucato error, vashes as the grd terval ad tme step sze approach zero,.e., whe lm(pde FDE) = 0. Δ 0 Commet: Cosstecy deals wth how well the FDE approxmates the PDE. Stablty For a stable umercal scheme, the errors from ay source wll ot grow uboudedly wth tme. Commets: A cocept that s applcable oly to marchg (tmetegrato) problems. Geerally we are much more cocered wth stablty tha cosstecy. Some hard wor s ofte eeded to establsh aalytcally the stablty of a scheme. Covergece It meas that the soluto to a FDE approaches the true soluto to the PDE as both grd terval ad tme step sze are reduced.
2 Lax's Equvalece Theorem For a wellposed, lear tal value problem, the ecessary ad suffcet codto for covergece s that the FDE s stable ad cosstet. The theorem has bee proved for tal value problems govered by lear PDE's (Rchtmyer ad Morto 967). We wll dscuss the three cocepts oe by oe..3.. Cosstecy Cosstecy meas PDE FDE 0 whe Δx 0 ad Δt 0. Clearly cosstecy s the ecessary codto for covergece. Example: Cosder a D dffuso equato: u t u = K ( K > 0 ad costat) x We use the forwardtme ad ceteredspace (FTCS) scheme:
3 u u u u + u Δt Δx + + = K To show cosstecy, we eed to determe the trucato error τ. Usg Taylor seres expaso method, u ( Δt) u ( Δt) u u = u +Δ t t! t 3! t 3 3 u ( Δx) u ( Δx) u u± = u ±Δ x + ± +... x 3! x 3! x Substtutg to the FDE, we have u Δt u u + + O( Δ t ) = K + O( Δ x ) +... t t x therefore Δt u τ = + Δ +Δ t O( t x ) τ 0 whe Δx 0 ad Δt 0 => the scheme s cosstet.. A couter example: The DufortFrael method for the same dffuso equatos: 3
4 + + u u ( u+ + u ) ( u + u ) = K. Δt Δx It's a ceteredtme scheme that s dorder accurate both space ad tme. We ca fd aga the trucato error (do t yourself!) 4 3 K( Δx) u Δt u ( Δt) u τ = K HRT x Δx t 6 t We ca see that lm τ 0 except for the secod term. Δx, Δt 0 If Δ t lm = 0 Δx Δx, Δt 0 the the scheme s cosstet therefore Δt must approaches zero faster the Δx. If they approaches zero at the same rate, the lm Δx, Δt 0 Δ t = β, the Δx lm Δx, Δt 0 τ = K u β, t our equato becomes t t x u u u + K β = K 4
5 Thus, we are solvg the wrog equato. I fact ths equato s hyperbolc stead of parabolc. Δt Note that f ~ Δx you mght see spurous waves your soluto, due to the hyperbolc ature of the "ew" PDE Covergece Geeral Dscusso Defto s gve earler. Symbolcally, t s lm u = u( x, t). Δx, Δt 0 Covergece s geerally hard to prove, especally for olear problems. The Lax's Theorem we preseted earler s very helpful uderstadg the covergece for lear systems, ad s ofte exteded to olear systems. We wll also dscuss umercal covergece ad methods for measurg soluto accuracy later. We wll frst show a covergece proof for a dffuso problem. Certa cocept troduced wll be useful later. Covergece proof for a D dffuso problem Cosder 5
6 u t u = K ( K > 0 ad costat) x for 0 x L whch has a tal codto: π x uxt (, = 0) = a s( ) = f( x). L = The B.C. s u(0, t) = u( L, t) = 0. Ths s a wellposed, lear tal value problem (otce the I.C. satsfes the B.C. as well). Frst, let's fd the aalytcal soluto to the PDE. Because the problem lear, we eed oly to exame the soluto for a sgle waveumber, ad ca assume a soluto of the form: π x u( x, t) = A( t)s( ) L Here A s the ampltude ad s gves the spatal structure ad the fal soluto should be the sum of all wave compoets. Note that ths soluto satsfes the boudary codtos. Substtutg the soluto to the PDE, we obta a ODE for the ampltude A : 6
7 da () t dt π = L KA dl( A ) π = K dt L A() t = A(0)exp K π / L t. ( ) It says that the ampltude of the solutos for all wave umbers decreases wth tme. From the I.C., A (0) = a, so we have (, ) exp ( / ) π x u x t = a K π L t s( ) L ad uxt (, ) = u ( xt, ) whch s the aalytcal soluto to the orgal dffuso equato. A umercal approxmato to the dffuso equato should coverge to ths soluto as Δx, Δt 0. Cosder the forwardtme ceteredspace (FTCS) scheme we derved earler. 7
8 Goal: Show that u t u(x,t) as Δx, Δt 0. Frst, fd the umercal soluto. Ths tme, we use the FDE ad substtute a dscrete Fourer seres to the equato. Let the I.C. be gve by π x f x u a for = 0,,,.., J J 0 ( ) = = s( ) = 0 L where J+ = total umber of grd pots used to represet the tal codto. The coeffcet a s gve by a dscrete Fourer trasform: J π x a = f( x)s( ) for =0,,,., J. J L = 0 Note : L = JΔx. As Δx 0, J, the dscrete Fourer seres becomes cotuous ad a a. Note : The umber of harmocs or Fourer wave compoets that ca be represeted s a fucto of the umber of grd pots (J+), whch s the umber of degrees of freedom. Spectral methods represet felds terms of spectral compoets, whose ampltudes are solved for. The waveumber for the wave compoets s π the above equatos. L Recall that wavelegth 8
9 π π L λ = = = w.. ( π / L) where s the umber (dex) of wave compoets. Logest wavelegth =, correspodg to waveumber zero (=0). Next logest wave = L ( = )... Shortest wave = L/J = JΔx/J = Δx. Commets: A Δx wave s the shortest wave that ca be resolved o ay grd ad t taes at least 3 pots to represet a wave. 9
10 Δx waves ofte have some specal propertes. They are also represeted most poorly by umercal methods recall that smooth felds are more accurately represeted by a fte umber of grd pots. As the cotuous case, we exame oly oe waveumber (the soluto s the sum of all waves), so for our dscrete problem, assume a soluto of the form u π x = A( )s L (It satsfes B.C.) ad s the tme level. We also have from I.C. A (0) = a. Substtutg ths to the FDE ad lettg u u u u + u Δt Δx + + = K, 0
11 S π x = s L, we have A( + ) S A( ) S KA( ) = [ S + S + S ]. Δt ( Δx) Sce x = Δx, x + = (+) Δx therefore S + π ( x+δx) = s L. Usg stadard trgoometrc dettes, we ca wrte the above the form of a recurso relato: π Δx A( + ) = A( ) 4μ s L KΔt where μ =. ( Δ x) π Δx If we let M() 4μ s L, the we have A ( + ) = M( ) A ( ).
12 Wrte t out for =0,,,., : A () = M( ) A (0) = M( ) a () = ( ) () = [ ( )] A M A M a... A ( ) = M( ) A ( ) = [ M( )] a we therefore have the soluto of u for wave mode : π x ( u ) = A( ) S = a [ M( )] s. L Defto: M() s ow as the amplfcato factor, ad f M(), the soluto wll ot grow tme as. Ths had better to be the case because the ampltude of the aalytcal soluto s supposed to always decrease wth tme. For our problem we ca see that M() meas π Δx L 4μ s. If we tae the maxmum possble value of s ( ) =, 4μ
13 μ /. Ths codto eeds to be met for all to prevet soluto growth. Based o the defto of μ, the codto becomes ( Δx) Δt. K Ths mposes a upper boud o the Δt that ca be used for a gve value of Δx, ad such a codto s uow as the Stablty Costrat. Now we have our soluto, let's chec covergece for sgle mode. By defto of covergece, we tae 4 π Δ lm ( u ) lm a t s s Δx, Δt 0 Δx, Δt 0 K x πx = Δ ( Δx) L L 4K π Δx or f we let f(δx) s ( Δx) L, π x lm ( u ) = lm a [ Δt f( Δx) ] s. L Δx, Δt 0 Δx, Δt 0 3
14 It ca be show that f f(x) s a complexvalued fucto of a real argumet, say Δx, such that lm f ( Δ x) = a, the, Δ x 0 lm [ ±Δtf( Δ x)] = e ± at (we wll show ths later). Δx, Δt 0 s( y) We ow thatlm =, therefore y 0 y π Δx s K( π) L π lm f ( Δ x) = lm = K a L π x = Δ L L Δ x 0 Δ x 0 Also lm a Δ x 0 = a, therefore Δx, Δt 0 π π x j lm ( u ) = aexp K t s. L L Iterestgly, ths soluto s detcal to our aalytcal soluto derved earler! Therefore the umercal soluto coverges to the PDE soluto whe Δx, Δt 0. 4
15 Fally, we show here (otg that Δt = t) ( ) ( )( )! 3! 3 a a 3 = ± at + ( Δ t) ± ( Δ t) +...! 3! lm( ± fδ t) = ± fδ t+ ( fδ t) + 3 ( fδ t) +... Δ x 0 3 ( at) ( at) = ± at + ± +...! 3! ± at = e.3.4. Numercal Covergece Readg: Fletcher (hadout), Sectos 4.., 4.., Numercal Covergece Covergece s ofte hard to demostrate theoretcally. True aalytcal soluto s hard or eve mpossble to fd s oe of the reasos. We ca, however, fd out the covergece of a gve scheme umercally. We compute solutos at successvely hgher resolutos ad see how the error chages wth the resoluto. Does τ 0 whe Δx 0? 5
16 Ad how fast τ decreases? The procedure ca be very expesve (remember the cost factor crease as Δ doubles). A typcal measure of error s the L orm or RMS error: L [ u Δ u ] = x where u s a true soluto or a 'coverged' umercal soluto whe exact soluto s ot avalable. Example: D dffuso equato usg FTCS scheme: 4 Δt u ( Δx) u τ = K t x Mag use of 4 u u u u u u = K = K K K = =. 4 t x t t x x t x we have 4 K( Δx) u τ = s + O Δ x +Δt 4 6 x 4 ( ). K t where s = Δ ( Δ x) 6
17 Table 4. (from Fletcher) shows the error reducto wth Δx for two values of s. The above fgure shows plots of log 0 (err) as a fucto of log 0 (Δx). 7
18 Recall that τ = A (Δx) log τ = log A + log Δx. Ths s a straght le loglog dagram wth a slope of ad tercept A. Thus the slope of the le gves the rate of covergece. I the above fgure, we see that whe s = /6, the error le has a steeper slope ad the error s smaller for all Δx. Ths s because for ths value of s, the frst term τ drops out ad the scheme because 4thorder accurate. The scheme s secodorder accurate for all other values of s. Note that you ca choose Δx ad Δt such that s=/6 oly whe K s costat the etre doma. I cases where o exact solutos s avalable, a socalled 'grdcovergece' or referece soluto s ofte sought ad ths soluto ca be used the place of true soluto the estmatg the soluto error. 8
19 A example from Straa et al (993). It shows a referece soluto obtaed at Δx=5 m, for a desty curret resultg from a droppg cold bubble. 9
20 Fgure (from Straa et al 993). Graph of θ' L orms ( C) from selfcovergece tests wth the compressble referece model (REFC). The bold sold le labeled wth 'selfcovergece solutos' represets the L orms for spatal trucato errors of solutos made wth Δt = costat ad varyg grd spacgs. The L orms were computed agast a 5.0 m referece soluto. The bold dashed les labeled wth, for example, '00.0 m solutos' represet L orms for temporal trucato errors of solutos made wth Δx =costat (e.g m) ad varyg tme steps. The referece solutos for these computatos were made usg a tme step cosstet wth Δt =.5 s tmes a costat each of the cases. The sold fes labeled O(l) ad O() represet frst ad secodorder covergece, respectvely. 0
21 Rchardso Extrapolato As the grd becomes very fe, the error behaves much le that predcted from the leadg terms τ. Further refemets are expesve, so we use aother techque to mprove the soluto Rchardso Extrapolato. Cosder two umercal solutos obtaed at Δx a ad Δx b. Wth FTCS scheme (assumg s /6), τ a 4 K( Δxa ) u 4 = s + O Δ x +Δt 6 x 4 ( a ) 4 K( Δxb ) u 4 τ b = s + O Δ x +Δt 6 x 4 ( b ) Fd a lear combato of the two solutos, u a ad u b u c = a u a + b u b where a + b = ad a ad b are chose so that the leadg terms τ a ad τ b cacel ad the scheme becomes 4thorder accurate (Of course ths assumes that u x s the same both case, whch s reasoably assumpto oly 4 4 at relatvely hgh resolutos whe the soluto s well resolved). If Δx b = Δx a /, the 4a + b = 0 wth a + b = a = /3, b = 4/3 ad u c = /3 u a + 4/3 u b.
22 .3.5. Stablty Aalyss Readg: Taehll et al. Secto 3.6. Stablty For a stable umercal scheme, the errors the tal codto wll ot grow uboudedly wth tme. I ths secto, we dscuss the methods for determg the stablty of F.D. schemes. Ths s very mportat whe desgg a F.D. scheme ad for uderstadg ts behavor. There are several methods: Eergy method vo Neuma method Matrx method (for systems of equatos) Dscrete perturbato method (wll ot dscuss) Note: Stablty refers to the F.D. Does ot volve B.C. or I.C. Refers to tmematchg problems oly The eergy method Read Durra Secto. (hadout, see web l)
23 Ths method s used much less ofte tha the vo Neuma method. It's attractve because t wors for olear problem ad problems wthout perodc B.C. The ey s to show that a postve defte quatty le ( u ) s bouded for all. u u We llustrate ths method usg the upstreamforward scheme for wave (advecto) equato + c = 0 : t x u u u u + = Δt Δx + c 0 Let μ = cδt/δx u = ( μ) u + μu + Squarg both sdes ad summg over all grd pots: ( u ) = [( μ) ( u ) + ( μ) μu u + μ ( u ) ] () + Assumg perodc B.C. ( u ) = ( u ) ad usg the Schwarz equalty (whch says that for two vectors U ad V, U V U V ), 3
24 uu ( u) ( u ) = ( u). If μ( μ) 0, all coeffcets of RHS terms () are postve ad we have ( u ) [( μ) + ( μ) μ + μ ] ( u ) = ( u ), +.e., the L orm at + s o greater tha that at, therefore the scheme s stable! The codto μ( μ) 0 gves μ = c Δt/Δx whch s the stablty codto for ths scheme. As we dscussed earler, t says that waves caot propagate more tha oe grd terval durg oe Δt order to mata stablty. vo Neuma method Read Taehll et al, Secto I a sese, we have already used ths method to fd stablty of the D dffuso equato whe we were provg the covergece of the soluto usg FTCS scheme. We foud the u + = [ M(t) ] + u 0 4
25 where M(t) = amplfcato factor. If M(t) by some measure (M ca be a matrx or a complex umber), the u + u ad the soluto caot grow tme the scheme s stable. Essetally, vo Neuma method expads the F.D.E. a Fourer seres, fds the amplfcato factor ad determes uder what codto the factor s less tha or equal to for stablty. Assumptos:. The equato has to be Lear wth costat coeffcets.. It s assumed that the soluto s perodc. Wth ths method, the depedet varable s decomposed to a complex (or a real) Fourer seres: uxyzt (,,, ) = Ulm (,, )exp[ x ( + ly+ mz ωt)] () lm,, where U s the complex ampltude ad ω = ω R + ω Ι s the complex frequecy. I fact ω R gves the wave propagato speed ad ω Ι gves the growth ad decayg rate. e = e = e e ωt [ ω + ω ] t ω t ω t R I R I Rt e ω I e ω t  phase fucto of Fourer compoets  growth or decay rate 5
26 If ω I >0, the soluto wll grow expoetally tme. Example : D dffuso equato wth FTCS scheme. u u = μ( u u + u ) (3) + + KΔt where μ =. ( Δ x) Let's exame a sgle wave : u = U e ( x ωt) + ( x ωt) ωδt = = ( x ωt) ± Δx ± u U e e u U e e Substtute the above to (3) Ue ( e ) = μue [ e + e ] ( x ωt) ωδt ( x ωt) Δx Δx ( x ωt ) ω t Ue [ e Δ μ(cosδx )] = 0 ( x ωt) ω t Ue e μ x [ Δ + 4 s ( Δ /)] = 0 For otrval soluto, we requre 6
27 e ω Δ t + 4μ s ( Δ x/) = 0 e ω Δ t = 4μ s ( Δ x/) Here, t e ω Δ s actually the amplfcato factor, the same as the M dscussed earler. λ t e ω Δ = u + /u  the amplfcato factor. For stablty we requre λ 4μ s ( Δx/) μ / as before! I practce, whe μ=/, the soluto (amplfcato factor) swtches betwee ad + for Δx waves ( s ( Δ x/ ) = ) very other step, whch s urealstc. The stadard requremet s therefore μ /4. Therefore, do ot avely th dffuso terms a umercal model does ot cause umercal stablty! Whe tegrated stably, the dffuso term CFD models teds to stablze the soluto by llg off/dampg small scale waves, but whe stablty codto s ot met, t tem tself wll cause problem! Read Pele (984) secto 0.. (hadout, see web l). 7
28 Shorthad otatos for dscrete/fte dfferece operators ad dscretzato dettes Notatos: Idettes: A + A x A = Aj A δxx = Δx Aj+ Aj δ + xx = Δx Aj Aj δ xx = Δx j+ / j / + / j / δ A δ A δ A x x x x x x Aδ B δ ( A B) Bδ A x x x x x A δxa δx( A /) 8
29 .3.6. Implct Methods Read Taehll et al, secod part of secto So far, we have dealt wth oly explct schemes whch have the form of + u = f( u, u,...). Wth these schemes, the future state at each grd pot s oly depedet o the curret ad past tme levels, therefore the soluto ca be obtaed drectly or explctly. c.f., explct fuctos such as y = x. Implct scheme volves varables of the future tme level at more tha oe grd pot (ofte resultg from fte dfferece of varable(s) at the future tme level). Mathematcally t ca be expressed as + + u = f( u, u, u,...). Ths s aalogous to mplct fuctos such as x = s(x). As oe ca mage, mplct schemes are more dffcult to solve. Usually matrx verso s volved. We wll frst loo at the stablty property of a mplct scheme. Example. Cosder the D dffuso equato u t = K u xx aga. It s approxmated by the followg F.D. scheme: δ+ = αδ + α δ (4) + tu K[ xxu ( ) xxu ] 9
30 (Note: shorthad otatos for F.D. are used. See Appedx. e.g., δ u t ( u + u )/ t + = Δ ). Whe α = 0, the scheme s explct, ad s the FTCS scheme dscussed earler. Whe α = /, t s mplct ad s called CraNcolso scheme. For other values of α, t s a geeral mplct scheme. We ca show that 4 u ( Δx) τ = K KΔt α O x t 4 + Δ +Δ x (show f for yourself!). 4 ( ) We ca see that whe α = /, t s dorder accurate tme ad space. Otherwse, t's frstorder tme whch s expected for ucetered tmedfferecg scheme (whe α=/, the rght had sde s a averaged betwee the curret ad future tme levels vald at +/. Relatve to ths RHS, the LHS tme dfferece becomes cetered tme. We ow that the smplest cetered dfferece scheme s secodorder accurate). Whe [ ] = 0, the scheme becomes fourthorder space. Let's perform stablty aalyss o (4) usg vo Neuma method. u = U e = U e e U λ e (5) ( x ωt) ωδt jδx jδx j Hereλ t e ω Δ. Note that we are ow usg j as the grd pot dex. 30
31 Substtute (5) to (4) jδ x + jδ x + Δx Δx Ue ( λ λ ) = μue αλ + ( α) λ ( e + e ) Dvdg Ue jδx λ o both sdes, ad rearragg 4 s ( x/ )[ ( )] λ = μ Δ αλ + α λ = 4( αμ ) s ( Δx/) + 4αμ s ( Δx/) Loo at several cases: Case I: α = 0, λ = 4μs ( Δx/) μ / as before. The scheme s codtoally stable. Case II: α = / (CraNcolso) μ s ( Δx/) + μ s ( Δx/) λ = for all values of μ, therefore the scheme s absolutely or ucodtoally stable. Case III: α =, the tme dfferece s bacward, relatve to the RHS terms. 3
32 λ =, aga for all values of μ, + 4μ s ( Δx/) therefore the scheme s also absolutely stable. However, ths scheme s oly frstorder accurate tme, as dscussed earler (cosstet wth the tme dfferece scheme beg ucetered). I geeral, whe 0 α < /, t s requred that μ /(  4α ), therefore the scheme s codtoally stable. Whe / α, the scheme s ucodtoally stable (t s sometmes referred to as the forwardbased scheme). I the ARPS, the mplct dffuso scheme s a opto for treatg the vertcal turbulet mxg terms. Ths treatmet s ecessary order to remove the severe stablty costrat from these terms whe vertcal mxg s strog sde the plaetary boudary layer (PBL). The latter occurs whe the PBL s covectvely ustably ad the olocal PBL mxg s voed wth the Su ad Chag (986) parameterzato. Parameter alfcoef arps.put correspods to α here (see hadout). Fally, we ote that for multtme level schemes, there s usually multple solutos for the amplfcato factor λ. some of them mght represet spurous computatoal modes due to the use of extra (artfcal) tal codtos. The expresso of λ ca be too complcated so that a graphc plottg s eeded to uderstad ts depedecy o wave umber. λ has to be o greater tha for all possble waves. The shortest wave resolvable o a grd has a wavelegth of Δ, ad the logest s L, where L s the doma wdth. 3
33 Trdagoal Solver D mplct method ofte leads to trdagoal systems of lear algebrac equatos. (I the ARPS, ths appears twce oce whe soud waves are treated mplctly the vertcal drecto ad oce whe the vertcal turbulece mxg s treated mplctly). For example, Eq. (4) ca be rewrtte as u u u u + u u u + u Δt Δx Δx = αk + ( α) K (6) It ca be rearraged to ΔtK ΔtK u α ( u u + u ) = ( α) ( u u + u ) + u Δx Δx ΔtK ΔtK ΔtK α u + ( + α u ) α u = d Δx Δx Δx u u + u+ where d =Δt( α) K + u Δx αδtk αδtk Let A = C =, B ( ) Δ = +, we the have x Δ x Au + Bu + Cu = d, (7) 33
34 for =,,, N, assumg the boudares are at =0 ad N. If we have Drchlet boudary codtos,.e., u at =0 ad N are ow, the for =, the equato becomes B u + Cu = d Au (8) ad for = N, the equato s A u + B u = d C u. (9) N N N N N N N For =, 3,., N, the equato remas of the form Eq.(7). If we wrte the equatos (79) a matrx form, we have B C u D A B C u D..... A B C u = D..... AN BN CN un DN A B u D N N N N (0) where D = d for =,, N, 34
35 D = d Au +, 0 N N N N D = d C u +. If we have Neuma boudary codtos,.e., we ow the gradet of u at the boudares whch dscretzed form are u u0 = L ad un un = R. Plug these relatos to Eq.(7) for = ad =N, we obta equatos smlar to (8) ad (9): ( A + B) u + Cu = d + AL () N N ( N N ) N N N A u + B + C u = d C R. () I ths case, the fal coeffcets (0) are dfferet for the frst ad last equato. Sce each except for the frst ad last row of the coeffcet matrx, oly three elemets are ozero ad the ozero elemets of the matrx are alged alog the dagoal axs, ths system s called trdagoal system of equatos. It ca be solved effcetly usg Thomas Algorthm. The procedure cossts of two parts. Frst, Eq. (7) s mapulated to the followg form: C ' u D' C ' u D'..... C ' u = D'..... C ' N un D' N u D' N N (3) 35
36 whch the subdagoal coeffcets A are elmated ad the dagoal coeffcets are ormalzed. For the frst equato C D C' =, D' =. (4a) B B For the geeral equatos: C D AD' C' =, D' = B AC B AC ' ' for =,, N. (4b) Equatos (4) represet a forward sweep step (see fgure below). It s followed by a bacward substtuto step that fds soluto u from (3). The soluto s: u = D' N N u = D' u C' for from N to. + (5) 36
37 Note both (4) ad (5) volve reducto, the algorthm s heretly oparallelzable. Fortuately, for multdmesoal problems, multple systems of equatos ofte eed to be solved, ad oe ca explot parallelsm alog other dmesos (e.g., j stead of drecto). Read Secto of Taehll et al ad Appedx A. 37
38 .3.7. Stablty Aalyss for Systems of Equatos Whe we are dealg wth a system of equatos, we ca also apply the vo Neuma method to fd the stablty property of a gve F.D. scheme. As wth sgle equatos, vo Neuma ca oly be used for lear systems of equatos. For olear systems, learzato has to be performed frst. Wthout gog to detals, we pot out that a system of lear equatos ca be expressed a matrx form le u u + A = 0 (6) t x The equato s frst dscretzed usg certa F.D. scheme, u ca be wrtte terms of a dscrete Fourer seres ad the wave compoet s the substtuted to the dscrete equato to obta somethg le: U + = M( Δ t, Δ x) U (7) where U s the ampltude vector for wave at tme level, ad M s called the amplfcato matrx. The scheme s stable whe the maxmum absolute egevalue of M s o greater tha. Why the maxmum absolute egevalue? Because as you saw earler ( Chapter ) that a system of equato le (7) ca be trasformed to a system of decoupled equatos, ad the egevalues of M become the amplfcato factors for each of the ew depedet varables, v (the elemet of vector V),.e., we ca obta from (7) 38
39 V N V, + = where N s a dagoal matrx wth the egevalues of M as ts dagoal elemets. T  M T = N T U = T MTT U + Let V = T U, we have V NV. + = Therefore + = where λ are the egevalues. ( v) λ( v) Sce the system s stable oly whe all depedet varables rema bouded, the absolute value of the maxmum egevalue has to be o greater tha. Read secto 3.6. of Taehll et al. 39
End of Finite Volume Methods Cartesian grids. Solution of the NavierStokes Equations. REVIEW Lecture 17: Higher order (interpolation) schemes
REVIEW Lecture 17: Numercal Flud Mechacs Sprg 2015 Lecture 18 Ed of Fte Volume Methods Cartesa grds Hgher order (terpolato) schemes Soluto of the NaverStokes Equatos Dscretzato of the covectve ad vscous
More informationFunctions of Random Variables
Fuctos of Radom Varables Chapter Fve Fuctos of Radom Varables 5. Itroducto A geeral egeerg aalyss model s show Fg. 5.. The model output (respose) cotas the performaces of a system or product, such as weght,
More informationCHAPTER 4 RADICAL EXPRESSIONS
6 CHAPTER RADICAL EXPRESSIONS. The th Root of a Real Number A real umber a s called the th root of a real umber b f Thus, for example: s a square root of sce. s also a square root of sce ( ). s a cube
More informationLecture 12 APPROXIMATION OF FIRST ORDER DERIVATIVES
FDM: Appromato of Frst Order Dervatves Lecture APPROXIMATION OF FIRST ORDER DERIVATIVES. INTRODUCTION Covectve term coservato equatos volve frst order dervatves. The smplest possble approach for dscretzato
More information2006 Jamie Trahan, Autar Kaw, Kevin Martin University of South Florida United States of America
SOLUTION OF SYSTEMS OF SIMULTANEOUS LINEAR EQUATIONS GaussSedel Method 006 Jame Traha, Autar Kaw, Kev Mart Uversty of South Florda Uted States of Amerca kaw@eg.usf.edu Itroducto Ths worksheet demostrates
More information1 Lyapunov Stability Theory
Lyapuov Stablty heory I ths secto we cosder proofs of stablty of equlbra of autoomous systems. hs s stadard theory for olear systems, ad oe of the most mportat tools the aalyss of olear systems. It may
More informationOrdinary Least Squares Regression. Simple Regression. Algebra and Assumptions.
Ordary Least Squares egresso. Smple egresso. Algebra ad Assumptos. I ths part of the course we are gog to study a techque for aalysg the lear relatoshp betwee two varables Y ad X. We have pars of observatos
More informationPGE 310: Formulation and Solution in Geosystems Engineering. Dr. Balhoff. Interpolation
PGE 30: Formulato ad Soluto Geosystems Egeerg Dr. Balhoff Iterpolato Numercal Methods wth MATLAB, Recktewald, Chapter 0 ad Numercal Methods for Egeers, Chapra ad Caale, 5 th Ed., Part Fve, Chapter 8 ad
More information1 Mixed Quantum State. 2 Density Matrix. CS Density Matrices, von Neumann Entropy 3/7/07 Spring 2007 Lecture 13. ψ = α x x. ρ = p i ψ i ψ i.
CS 94 Desty Matrces, vo Neuma Etropy 3/7/07 Sprg 007 Lecture 3 I ths lecture, we wll dscuss the bascs of quatum formato theory I partcular, we wll dscuss mxed quatum states, desty matrces, vo Neuma etropy
More informationEconometric Methods. Review of Estimation
Ecoometrc Methods Revew of Estmato Estmatg the populato mea Radom samplg Pot ad terval estmators Lear estmators Ubased estmators Lear Ubased Estmators (LUEs) Effcecy (mmum varace) ad Best Lear Ubased Estmators
More informationA Primer on Summation Notation George H Olson, Ph. D. Doctoral Program in Educational Leadership Appalachian State University Spring 2010
Summato Operator A Prmer o Summato otato George H Olso Ph D Doctoral Program Educatoal Leadershp Appalacha State Uversty Sprg 00 The summato operator ( ) {Greek letter captal sgma} s a structo to sum over
More informationWe have already referred to a certain reaction, which takes place at high temperature after rich combustion.
ME 41 Day 13 Topcs Chemcal Equlbrum  Theory Chemcal Equlbrum Example #1 Equlbrum Costats Chemcal Equlbrum Example #2 Chemcal Equlbrum of Hot Bured Gas 1. Chemcal Equlbrum We have already referred to a
More informationRecall MLR 5 Homskedasticity error u has the same variance given any values of the explanatory variables Var(u x1,...,xk) = 2 or E(UU ) = 2 I
Chapter 8 Heterosedastcty Recall MLR 5 Homsedastcty error u has the same varace gve ay values of the eplaatory varables Varu,..., = or EUU = I Suppose other GM assumptos hold but have heterosedastcty.
More informationHomework 1: Solutions Sid Banerjee Problem 1: (Practice with Asymptotic Notation) ORIE 4520: Stochastics at Scale Fall 2015
Fall 05 Homework : Solutos Problem : (Practce wth Asymptotc Notato) A essetal requremet for uderstadg scalg behavor s comfort wth asymptotc (or bgo ) otato. I ths problem, you wll prove some basc facts
More information1 Onto functions and bijections Applications to Counting
1 Oto fuctos ad bectos Applcatos to Coutg Now we move o to a ew topc. Defto 1.1 (Surecto. A fucto f : A B s sad to be surectve or oto f for each b B there s some a A so that f(a B. What are examples of
More information( q Modal Analysis. Eigenvectors = Mode Shapes? Eigenproblem (cont) = x x 2 u 2. u 1. x 1 (4.55) vector and M and K are matrices.
4.3  Modal Aalyss Physcal coordates are ot always the easest to work Egevectors provde a coveet trasformato to modal coordates Modal coordates are lear combato of physcal coordates Say we have physcal
More informationAnalysis of System Performance IN2072 Chapter 5 Analysis of Non Markov Systems
Char for Network Archtectures ad Servces Prof. Carle Departmet of Computer Scece U Müche Aalyss of System Performace IN2072 Chapter 5 Aalyss of No Markov Systems Dr. Alexader Kle Prof. Dr.Ig. Georg Carle
More informationEngineering Vibration 1. Introduction
Egeerg Vbrato. Itroducto he study of the moto of physcal systems resultg from the appled forces s referred to as dyamcs. Oe type of dyamcs of physcal systems s vbrato, whch the system oscllates about certa
More informationChapter 11 Systematic Sampling
Chapter stematc amplg The sstematc samplg techue s operatoall more coveet tha the smple radom samplg. It also esures at the same tme that each ut has eual probablt of cluso the sample. I ths method of
More informationAlgorithms Theory, Solution for Assignment 2
JuorProf. Dr. Robert Elsässer, Marco Muñz, Phllp Hedegger WS 2009/200 Algorthms Theory, Soluto for Assgmet 2 http://lak.formatk.ufreburg.de/lak_teachg/ws09_0/algo090.php Exercse 2.  Fast Fourer Trasform
More information8.1 Hashing Algorithms
CS787: Advaced Algorthms Scrbe: Mayak Maheshwar, Chrs Hrchs Lecturer: Shuch Chawla Topc: Hashg ad NPCompleteess Date: September 21 2007 Prevously we looked at applcatos of radomzed algorthms, ad bega
More informationFor combinatorial problems we might need to generate all permutations, combinations, or subsets of a set.
Addtoal Decrease ad Coquer Algorthms For combatoral problems we mght eed to geerate all permutatos, combatos, or subsets of a set. Geeratg Permutatos If we have a set f elemets: { a 1, a 2, a 3, a } the
More information12.2 Estimating Model parameters Assumptions: ox and y are related according to the simple linear regression model
1. Estmatg Model parameters Assumptos: ox ad y are related accordg to the smple lear regresso model (The lear regresso model s the model that says that x ad y are related a lear fasho, but the observed
More informationNumerical Solution of Linear Second Order Ordinary Differential Equations with Mixed Boundary Conditions by Galerkin Method
Mathematcs ad Computer Scece 7; (5: 6678 http://www.scecepublshggroup.com//mcs do:.648/.mcs.75. Numercal Soluto of Lear Secod Order Ordary Dfferetal Equatos wth Mxed Boudary Codtos by Galer Method Aalu
More informationDiscrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand DIS 10b
CS 70 Dscrete Mathematcs ad Probablty Theory Fall 206 Sesha ad Walrad DIS 0b. Wll I Get My Package? Seaky delvery guy of some compay s out delverg packages to customers. Not oly does he had a radom package
More informationUnsupervised Learning and Other Neural Networks
CSE 53 Soft Computg NOT PART OF THE FINAL Usupervsed Learg ad Other Neural Networs Itroducto Mture Destes ad Idetfablty ML Estmates Applcato to Normal Mtures Other Neural Networs Itroducto Prevously, all
More informationArithmetic Mean and Geometric Mean
Acta Mathematca Ntresa Vol, No, p 43 48 ISSN 4536083 Arthmetc Mea ad Geometrc Mea Mare Varga a * Peter Mchalča b a Departmet of Mathematcs, Faculty of Natural Sceces, Costate the Phlosopher Uversty Ntra,
More information3D Geometry for Computer Graphics. Lesson 2: PCA & SVD
3D Geometry for Computer Graphcs Lesso 2: PCA & SVD Last week  egedecomposto We wat to lear how the matrx A works: A 2 Last week  egedecomposto If we look at arbtrary vectors, t does t tell us much.
More informationComputational Geometry
Problem efto omputatoal eometry hapter 6 Pot Locato Preprocess a plaar map S. ve a query pot p, report the face of S cotag p. oal: O()sze data structure that eables O(log ) query tme. pplcato: Whch state
More informationOn the characteristics of partial differential equations
Sur les caractérstques des équatos au dérvées artelles Bull Soc Math Frace 5 (897) 8 O the characterstcs of artal dfferetal equatos By JULES BEUDON Traslated by D H Delhech I a ote that was reseted to
More informationThe numerical simulation of compressible flow in a Shubin nozzle using schemes of BeanWarming and flux vector splitting
The umercal smulato of compressble flow a Shub ozzle usg schemes of BeaWarmg ad flux vector splttg Gh. Paygaeh a, A. Hadd b,*, M. Hallaj b ad N. Garjas b a Departmet of Mechacal Egeerg, Shahd Rajaee Teacher
More information( ) 2 2. MultiLayer Refraction Problem Rafael Espericueta, Bakersfield College, November, 2006
MultLayer Refracto Problem Rafael Espercueta, Bakersfeld College, November, 006 Lght travels at dfferet speeds through dfferet meda, but refracts at layer boudares order to traverse the leasttme path.
More informationThe E vs k diagrams are in general a function of the k space direction in a crystal
vs dagram p m m he parameter s called the crystal mometum ad s a parameter that results from applyg Schrödger wave equato to a sglecrystal lattce. lectros travelg dfferet drectos ecouter dfferet potetal
More informationAnswer key to problem set # 2 ECON 342 J. Marcelo Ochoa Spring, 2009
Aswer key to problem set # ECON 34 J. Marcelo Ochoa Sprg, 009 Problem. For T cosder the stadard pael data model: y t x t β + α + ǫ t a Numercally compare the fxed effect ad frst dfferece estmates. b Compare
More informationA Helmholtz energy equation of state for calculating the thermodynamic properties of fluid mixtures
A Helmholtz eergy equato of state for calculatg the thermodyamc propertes of flud mxtures Erc W. Lemmo, Reer TllerRoth Abstract New Approach based o hghly accurate EOS for the pure compoets combed at
More informationLogistic regression (continued)
STAT562 page 138 Logstc regresso (cotued) Suppose we ow cosder more complex models to descrbe the relatoshp betwee a categorcal respose varable (Y) that takes o two (2) possble outcomes ad a set of p explaatory
More information13. Artificial Neural Networks for Function Approximation
Lecture 7 3. Artfcal eural etworks for Fucto Approxmato Motvato. A typcal cotrol desg process starts wth modelg, whch s bascally the process of costructg a mathematcal descrpto (such as a set of ODEs)
More informationLecture IV : The HartreeFock method
Lecture IV : The HartreeFock method I. THE HARTREE METHOD We have see the prevous lecture that the maybody Hamltoa for a electroc system may be wrtte atomc uts as Ĥ = N e N e N I Z I r R I + N e N e
More informationChapter 3. Linear Equations and Matrices
Vector Spaces Physcs 8/6/05 hapter Lear Equatos ad Matrces wde varety of physcal problems volve solvg systems of smultaeous lear equatos These systems of lear equatos ca be ecoomcally descrbed ad effcetly
More informationDIFFERENTIAL GEOMETRIC APPROACH TO HAMILTONIAN MECHANICS
DIFFERENTIAL GEOMETRIC APPROACH TO HAMILTONIAN MECHANICS Course Project: Classcal Mechacs (PHY 40) Suja Dabholkar (Y430) Sul Yeshwath (Y444). Itroducto Hamltoa mechacs s geometry phase space. It deals
More informationIntroduction to Matrices and Matrix Approach to Simple Linear Regression
Itroducto to Matrces ad Matrx Approach to Smple Lear Regresso Matrces Defto: A matrx s a rectagular array of umbers or symbolc elemets I may applcatos, the rows of a matrx wll represet dvduals cases (people,
More information9.1 Introduction to the probit and logit models
EC3000 Ecoometrcs Lecture 9 Probt & Logt Aalss 9. Itroducto to the probt ad logt models 9. The logt model 9.3 The probt model Appedx 9. Itroducto to the probt ad logt models These models are used regressos
More informationConvergence of the Desroziers scheme and its relation to the lag innovation diagnostic
Covergece of the Desrozers scheme ad ts relato to the lag ovato dagostc chard Méard Evromet Caada, Ar Qualty esearch Dvso World Weather Ope Scece Coferece Motreal, August 9, 04 o t t O x x x y x y Oservato
More informationMultiple Regression. More than 2 variables! Grade on Final. Multiple Regression 11/21/2012. Exam 2 Grades. Exam 2 Regrades
STAT 101 Dr. Kar Lock Morga 11/20/12 Exam 2 Grades Multple Regresso SECTIONS 9.2, 10.1, 10.2 Multple explaatory varables (10.1) Parttog varablty R 2, ANOVA (9.2) Codtos resdual plot (10.2) Trasformatos
More information4 Inner Product Spaces
11.MH1 LINEAR ALGEBRA Summary Notes 4 Ier Product Spaces Ier product s the abstracto to geeral vector spaces of the famlar dea of the scalar product of two vectors or 3. I what follows, keep these key
More informationRisk management of hazardous material transportation
Maagemet of atural Resources, Sustaable Developmet ad Ecologcal azards 393 Rs maagemet of hazardous materal trasportato J. Auguts, E. Uspuras & V. Matuzas Lthuaa Eergy Isttute, Lthuaa Abstract I recet
More informationStatistics MINITAB  Lab 5
Statstcs 10010 MINITAB  Lab 5 PART I: The Correlato Coeffcet Qute ofte statstcs we are preseted wth data that suggests that a lear relatoshp exsts betwee two varables. For example the plot below s of
More information(b) By independence, the probability that the string 1011 is received correctly is
Soluto to Problem 1.31. (a) Let A be the evet that a 0 s trasmtted. Usg the total probablty theorem, the desred probablty s P(A)(1 ɛ ( 0)+ 1 P(A) ) (1 ɛ 1)=p(1 ɛ 0)+(1 p)(1 ɛ 1). (b) By depedece, the probablty
More informationLecture Notes Types of economic variables
Lecture Notes 3 1. Types of ecoomc varables () Cotuous varable takes o a cotuum the sample space, such as all pots o a le or all real umbers Example: GDP, Polluto cocetrato, etc. () Dscrete varables fte
More informationSignal,autocorrelation 0.6
Sgal,autocorrelato Phase ose p/.9.3.7. .5 5 5 5 Tme Sgal,autocorrelato Phase ose p/.5..7.3 . .5 5 5 5 Tme Sgal,autocorrelato. Phase ose p/.9.3.7. .5 5 5 5 Tme Sgal,autocorrelato. Phase ose p/.8..6.
More informationContinuous Distributions
7//3 Cotuous Dstrbutos Radom Varables of the Cotuous Type Desty Curve Percet Desty fucto, f (x) A smooth curve that ft the dstrbuto 3 4 5 6 7 8 9 Test scores Desty Curve Percet Probablty Desty Fucto, f
More informationQualifying Exam Statistical Theory Problem Solutions August 2005
Qualfyg Exam Statstcal Theory Problem Solutos August 5. Let X, X,..., X be d uform U(,),
More informationOn Fuzzy Arithmetic, Possibility Theory and Theory of Evidence
O Fuzzy rthmetc, Possblty Theory ad Theory of Evdece suco P. Cucala, Jose Vllar Isttute of Research Techology Uversdad Potfca Comllas C/ Sata Cruz de Marceado 6 8 Madrd. Spa bstract Ths paper explores
More informationStatistics: Unlocking the Power of Data Lock 5
STAT 0 Dr. Kar Lock Morga Exam 2 Grades: I Class Multple Regresso SECTIONS 9.2, 0., 0.2 Multple explaatory varables (0.) Parttog varablty R 2, ANOVA (9.2) Codtos resdual plot (0.2) Exam 2 Re grades Re
More informationThe Effect of Distance between OpenLoop Poles and ClosedLoop Poles on the Numerical Accuracy of Pole Assignment
Proceedgs of the 5th Medterraea Coferece o Cotrol & Automato, July 79, 007, Athes  Greece T900 The Effect of Dstace betwee OpeLoop Poles ad ClosedLoop Poles o the Numercal Accuracy of Pole Assgmet
More informationDebabrata Dey and Atanu Lahiri
RESEARCH ARTICLE QUALITY COMPETITION AND MARKET SEGMENTATION IN THE SECURITY SOFTWARE MARKET Debabrata Dey ad Atau Lahr Mchael G. Foster School of Busess, Uersty of Washgto, Seattle, Seattle, WA 9895 U.S.A.
More informationAnalyzing Control Structures
Aalyzg Cotrol Strutures sequeg P, P : two fragmets of a algo. t, t : the tme they tae the tme requred to ompute P ;P s t t Θmaxt,t For loops for to m do P t: the tme requred to ompute P total tme requred
More informationExample. Row Hydrogen Carbon
SMAM 39 Least Squares Example. Heatg ad combusto aalyses were performed order to study the composto of moo rocks collected by Apollo 4 ad 5 crews. Recorded c ad c of the Mtab output are the determatos
More informationOn the construction of symmetric nonnegative matrix with prescribed Ritz values
Joural of Lear ad Topologcal Algebra Vol. 3, No., 14, 6166 O the costructo of symmetrc oegatve matrx wth prescrbed Rtz values A. M. Nazar a, E. Afshar b a Departmet of Mathematcs, Arak Uversty, P.O. Box
More informationEntropy ISSN by MDPI
Etropy 2003, 5, 233238 Etropy ISSN 10994300 2003 by MDPI www.mdp.org/etropy O the Measure Etropy of Addtve Cellular Automata Hasa Aı Arts ad Sceces Faculty, Departmet of Mathematcs, Harra Uversty; 63100,
More informationChapter 7. Solution of Ordinary Differential Equations
Ordar Dfferetal Equatos  4 Chapter 7. Soluto of Ordar Dfferetal Equatos 7.. Itroducto The damc behavor of ma relevat sstems ad materals ca be descrbed wth ordar dfferetal equatos ODEs. I ths chapter,
More informationChapter Business Statistics: A First Course Fifth Edition. Learning Objectives. Correlation vs. Regression. In this chapter, you learn:
Chapter 3 3 Busess Statstcs: A Frst Course Ffth Edto Chapter 2 Correlato ad Smple Lear Regresso Busess Statstcs: A Frst Course, 5e 29 PretceHall, Ic. Chap 2 Learg Objectves I ths chapter, you lear:
More informationChapter 2 Simple Random Sampling
Chapter  Smple Radom Samplg Smple radom samplg (SRS) s a method of selecto of a sample comprsg of umber of samplg uts out of the populato havg umber of samplg uts such that every samplg ut has a equal
More informationSolution of General Dual Fuzzy Linear Systems. Using ABS Algorithm
Appled Mathematcal Sceces, Vol 6, 0, o 4, 637 Soluto of Geeral Dual Fuzzy Lear Systems Usg ABS Algorthm M A Farborz Aragh * ad M M ossezadeh Departmet of Mathematcs, Islamc Azad Uversty Cetral ehra Brach,
More informationStatistical modelling and latent variables (2)
Statstcal modellg ad latet varables (2 Mxg latet varables ad parameters statstcal erece Trod Reta (Dvso o statstcs ad surace mathematcs, Departmet o Mathematcs, Uversty o Oslo State spaces We typcally
More informationLinear Regression Linear Regression with Shrinkage. Some slides are due to Tommi Jaakkola, MIT AI Lab
Lear Regresso Lear Regresso th Shrkage Some sldes are due to Tomm Jaakkola, MIT AI Lab Itroducto The goal of regresso s to make quattatve real valued predctos o the bass of a vector of features or attrbutes.
More informationPseudorandom Functions
Pseudoradom Fuctos Debdeep Mukhopadhyay IIT Kharagpur We have see the costructo of PRG (pseudoradom geerators) beg costructed from ay oeway fuctos. Now we shall cosder a related cocept: Pseudoradom
More information2. Independence and Bernoulli Trials
. Ideedece ad Beroull Trals Ideedece: Evets ad B are deedet f B B.  It s easy to show that, B deedet mles, B;, B are all deedet ars. For examle, ad so that B or B B B B B φ,.e., ad B are deedet evets.,
More informationv 1 periodic 2exponents of SU(2 e ) and SU(2 e + 1)
Joural of Pure ad Appled Algebra 216 (2012) 1268 1272 Cotets lsts avalable at ScVerse SceceDrect Joural of Pure ad Appled Algebra joural homepage: www.elsever.com/locate/jpaa v 1 perodc 2expoets of SU(2
More informationChapter 10 Two Stage Sampling (Subsampling)
Chapter 0 To tage amplg (usamplg) I cluster samplg, all the elemets the selected clusters are surveyed oreover, the effcecy cluster samplg depeds o sze of the cluster As the sze creases, the effcecy decreases
More informationCOMPROMISE HYPERSPHERE FOR STOCHASTIC DOMINANCE MODEL
Sebasta Starz COMPROMISE HYPERSPHERE FOR STOCHASTIC DOMINANCE MODEL Abstract The am of the work s to preset a method of rakg a fte set of dscrete radom varables. The proposed method s based o two approaches:
More informationPoint Estimation: definition of estimators
Pot Estmato: defto of estmators Pot estmator: ay fucto W (X,..., X ) of a data sample. The exercse of pot estmato s to use partcular fuctos of the data order to estmate certa ukow populato parameters.
More informationQuantitative analysis requires : sound knowledge of chemistry : possibility of interferences WHY do we need to use STATISTICS in Anal. Chem.?
Ch 4. Statstcs 4.1 Quattatve aalyss requres : soud kowledge of chemstry : possblty of terfereces WHY do we eed to use STATISTICS Aal. Chem.? ucertaty ests. wll we accept ucertaty always? f ot, from how
More informationNotes on Censored EL, and Harzard
Notes o Cesored EL, ad Harzard Ma Zhou I survval aalyss, the statstcs volvg the hazard fuctos are usually easer to hadle mathematcally the those volvg the dstrbutos. For example, t s easer to show the
More informationPseudorandom Functions. PRG vs PRF
Pseudoradom Fuctos Debdeep Muhopadhyay IIT Kharagpur PRG vs PRF We have see the costructo of PRG (pseudoradom geerators) beg costructed from ay oeway fuctos. Now we shall cosder a related cocept: Pseudoradom
More informationMatrix Algebra Tutorial With Examples in Matlab
Matr Algebra Tutoral Wth Eamples Matlab by Klaus Moelter Departmet of Agrcultural ad Appled Ecoomcs Vrga Tech emal: moelter@vt.edu web: http://faculty.ageco.vt.edu/moelter/ Specfcally desged as a / day
More informationECE 559: Wireless Communication Project Report Diversity Multiplexing Tradeoff in MIMO Channels with partial CSIT. Hoa Pham
ECE 559: Wreless Commucato Project Report Dversty Multplexg Tradeoff MIMO Chaels wth partal CSIT Hoa Pham. Summary I ths project, I have studed the performace ga of MIMO systems. There are two types of
More informationBarycentric Interpolators for Continuous. Space & Time Reinforcement Learning. Robotics Institute, Carnegie Mellon University
Barycetrc Iterpolators for Cotuous Space & Tme Reforcemet Learg Rem Muos & Adrew Moore Robotcs Isttute, Carege Mello Uversty Pttsburgh, PA 15213, USA. Emal:fmuos, awmg@cs.cmu.edu Category : Reforcemet
More informationON THE LOGARITHMIC INTEGRAL
Hacettepe Joural of Mathematcs ad Statstcs Volume 39(3) (21), 393 41 ON THE LOGARITHMIC INTEGRAL Bra Fsher ad Bljaa JolevskaTueska Receved 29:9 :29 : Accepted 2 :3 :21 Abstract The logarthmc tegral l(x)
More informationBootstrap Method for Testing of Equality of Several Coefficients of Variation
Cloud Publcatos Iteratoal Joural of Advaced Mathematcs ad Statstcs Volume, pp. 6, Artcle ID Sc Research Artcle Ope Access Bootstrap Method for Testg of Equalty of Several Coeffcets of Varato Dr. Navee
More informationKernelbased Methods and Support Vector Machines
Kerelbased Methods ad Support Vector Maches Larr Holder CptS 570 Mache Learg School of Electrcal Egeerg ad Computer Scece Washgto State Uverst Refereces Muller et al. A Itroducto to KerelBased Learg
More informationChapter Statistics Background of Regression Analysis
Chapter 06.0 Statstcs Backgroud of Regresso Aalyss After readg ths chapter, you should be able to:. revew the statstcs backgroud eeded for learg regresso, ad. kow a bref hstory of regresso. Revew of Statstcal
More informationSTA 105M BASIC STATISTICS (This is a multiple choice paper.)
DCDM BUSINESS SCHOOL September Mock Eamatos STA 0M BASIC STATISTICS (Ths s a multple choce paper.) Tme: hours 0 mutes INSTRUCTIONS TO CANDIDATES Do ot ope ths questo paper utl you have bee told to do
More informationD KL (P Q) := p i ln p i q i
CheroffBouds 1 The Geeral Boud Let P 1,, m ) ad Q q 1,, q m ) be two dstrbutos o m elemets, e,, q 0, for 1,, m, ad m 1 m 1 q 1 The KullbackLebler dvergece or relatve etroy of P ad Q s defed as m D KL
More informationBoundary Elements and Other Mesh Reduction Methods XXIX 13
Boudary Elemets ad Other Mesh Reducto Methods XXIX 3 Nooverlappg doma decomposto scheme for the symmetrc radal bass fucto meshless approach wth double collocato at the subdoma terfaces H. Power, A. Heradez
More information15. Nanoparticle Optics in the Electrostatic Limit
5 aopartcle Optcs the lectrostatc Lmt I the prevous chapter, we were restrcted to aospheres because a full soluto of Maxwell s equato was eeded Such a aalytc calculato s oly possble for very smple geometres
More informationAbout a Fuzzy Distance between Two Fuzzy Partitions and Application in Attribute Reduction Problem
BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND IORMATION TECHNOLOGIES Volume 6, No 4 Sofa 206 Prt ISSN: 39702; Ole ISSN: 34408 DOI: 0.55/cat2060064 About a Fuzzy Dstace betwee Two Fuzzy Parttos ad Applcato
More informationStatistics Descriptive and Inferential Statistics. Instructor: Daisuke Nagakura
Statstcs Descrptve ad Iferetal Statstcs Istructor: Dasuke Nagakura (agakura@z7.keo.jp) 1 Today s topc Today, I talk about two categores of statstcal aalyses, descrptve statstcs ad feretal statstcs, ad
More informationarxiv: v1 [math.st] 24 Oct 2016
arxv:60.07554v [math.st] 24 Oct 206 Some Relatoshps ad Propertes of the Hypergeometrc Dstrbuto Peter H. Pesku, Departmet of Mathematcs ad Statstcs York Uversty, Toroto, Otaro M3J P3, Caada Emal: pesku@pascal.math.yorku.ca
More informationChannel Models with Memory. Channel Models with Memory. Channel Models with Memory. Channel Models with Memory
Chael Models wth Memory Chael Models wth Memory Hayder radha Electrcal ad Comuter Egeerg Mchga State Uversty I may ractcal etworkg scearos (cludg the Iteret ad wreless etworks), the uderlyg chaels are
More informationSTK3100 and STK4100 Autumn 2017
SK3 ad SK4 Autum 7 Geeralzed lear models Part III Covers the followg materal from chaters 4 ad 5: Sectos 4..5, 4.3.5, 4.3.6, 4.4., 4.4., ad 4.4.3 Sectos 5.., 5.., ad 5.5. Ørulf Borga Deartmet of Mathematcs
More informationJournal of Computational Physics
Joural of omputatoal Physcs 5 (013) 495 517 otets lsts avalable at ScVerse SceceDrect Joural of omputatoal Physcs www.elsever.com/locate/cp Fractoal Sturm Louvlle egeproblems: Theory ad umercal approxmato
More informationEP2200 Queueing theory and teletraffic systems. Queueing networks. Viktoria Fodor KTH EES/LCN KTH EES/LCN
EP2200 Queueg theory ad teletraffc systems Queueg etworks Vktora Fodor Ope ad closed queug etworks Queug etwork: etwork of queug systems E.g. data packets traversg the etwork from router to router Ope
More informationA Collocation Method for Solving Abel s Integral Equations of First and Second Kinds
A Collocato Method for Solvg Abel s Itegral Equatos of Frst ad Secod Kds Abbas Saadatmad a ad Mehd Dehgha b a Departmet of Mathematcs, Uversty of Kasha, Kasha, Ira b Departmet of Appled Mathematcs, Faculty
More information(Monte Carlo) Resampling Technique in Validity Testing and Reliability Testing
Iteratoal Joural of Computer Applcatos (0975 8887) (Mote Carlo) Resamplg Techque Valdty Testg ad Relablty Testg Ad Setawa Departmet of Mathematcs, Faculty of Scece ad Mathematcs, Satya Wacaa Chrsta Uversty
More informationTHE EFFICIENCY OF EMPIRICAL LIKELIHOOD WITH NUISANCE PARAMETERS
Joural of Mathematcs ad Statstcs (: 59, 4 ISSN: 5493644 4 Scece Publcatos do:.3844/jmssp.4.5.9 Publshed Ole ( 4 (http://www.thescpub.com/jmss.toc THE EFFICIENCY OF EMPIRICAL LIKELIHOOD WITH NUISANCE
More informationi 2 σ ) i = 1,2,...,n , and = 3.01 = 4.01
ECO 745, Homework 6 Le Cabrera. Assume that the followg data come from the lear model: ε ε ~ N, σ,,..., 6. .5 7. 6.9 . . .9. ..6.4.. .6 .7.7 Fd the mamum lkelhood estmates of,, ad σ ε s.6. 4. ε
More informationSupplementary Material for Limits on Sparse Support Recovery via Linear Sketching with Random Expander Matrices
Joata Scarlett ad Volka Cever Supplemetary Materal for Lmts o Sparse Support Recovery va Lear Sketcg wt Radom Expader Matrces (AISTATS 26, Joata Scarlett ad Volka Cever) Note tat all ctatos ere are to
More informationGenerating Multivariate Nonnormal Distribution Random Numbers Based on Copula Function
7659, Eglad, UK Joural of Iformato ad Computg Scece Vol. 2, No. 3, 2007, pp. 996 Geeratg Multvarate Noormal Dstrbuto Radom Numbers Based o Copula Fucto Xaopg Hu +, Jam He ad Hogsheg Ly School of Ecoomcs
More informationReliability Based Design Optimization with Correlated Input Variables
755 Relablty Based Desg Optmzato wth Correlated Iput Varables Copyrght 7 SAE Iteratoal Kyug K. Cho, Yoojeog Noh, ad Lu Du Departmet of Mechacal & Idustral Egeerg & Ceter for Computer Aded Desg, College
More information