Some Recent Advances on Spectral Methods for Unbounded Domains


 Ashley Glenn
 2 years ago
 Views:
Transcription
1 COMMUICATIOS I COMPUTATIOAL PHYSICS Vol. 5, o. 24, pp Commun. Comput. Phy. February 29 REVIEW ARTICLE Some Recent Advance on Spectral Method for Unbounded Domain Jie Shen 1, and LiLian Wang 2 1 Department of Mathematic, Purdue Univerity, Wet Lafayette, I, 4797, USA. 2 Diviion of Mathematical Science, School of Phyical and Mathematical Science, anyang Technological Univerity, , Singapore. Received 13 January 28; Accepted (in revied verion) 8 June 28 Available online 1 Augut 28 Abtract. We preent in thi paper a unified framework for analyzing the pectral method in unbounded domain uing mapped Jacobi, Laguerre and Hermite function. A detailed comparion of the convergence rate of thee pectral method for olution with typical decay behavior i carried out, both theoretically and computationally. A brief review on ome of the recent advance in the pectral method for unbounded domain i alo preented. AMS ubect claification: 6535, 6522,65F5, 35J5 Key word: Spectral method, unbounded domain, orthogonal polynomial, rational function, Hermite function, Laguerre function. Content 1 Introduction Mapped Jacobi method Laguerre pectral method 21 4 Hermite pectral method Implementation, numerical reult and dicuion 23 6 Micellaneou iue and extenion Concluding remark 237 Correponding author. addree: (J. Shen), (L. Wang) 195 c 29 GlobalScience Pre
2 196 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp Introduction Spectral method for olving PDE on unbounded domain can be eentially claified into four approache: (i) Domain truncation: truncate unbounded domain to bounded domain and olve the PDE on bounded domain upplemented with artificial or tranparent boundary condition (ee, e.g., [17, 21, 22, 25, 44, 51]); (ii) Approximation by claical orthogonal ytem on unbounded domain, e.g., Laguerre or Hermite polynomial/function (ee, e.g., [7, 14, 2, 3, 31, 36, 43, 47]); (iii) Approximation by other, nonclaical orthogonal ytem (ee, e.g., [14]), or by mapped orthogonal ytem, e.g., image of claical Jacobi polynomial though a uitable mapping (ee, e.g. [32, 34, 35, 54]); (iv) Mapping: map unbounded domain to bounded domain and ue tandard pectral method to olve the mapped PDE in the bounded domain (ee, e.g., [9 12, 15, 24, 26]). Boyd provided in [11] an excellent review on general propertie and practical implementation for many of thee approache. In general, the domain truncation approach i only a viable option for problem with rapidly (exponentially) decaying olution or when accurate nonreflecting or exact boundary condition are available at the truncated boundary. On the other hand, with proper choice of mapping and/or caling parameter, the other three approache can all be effectively applied to a variety of problem with rapid or low decaying (or even growing) olution. Since there i a vat literature on domain truncation, particularly for Helmholtz equation and Maxwell equation for cattering problem and the analyi involved i very different from the other three approache, the domain truncation approach will not be addreed in thi paper. We note that the lat two approache are mathematically equivalent (ee Section for more detail) but their computational implementation are different. More preciely, the lat approach involve olving the mapped PDE (which are often cumberome to deal with) uing claical Jacobi polynomial while the third approach olve the original PDE uing the mapped Jacobi polynomial. The main advantage of the lat approach i that it can be implemented and analyzed uing tandard procedure and approximation reult, but it main diadvantage i that the tranformed equation i uually very complicated which, in many cae, make it implementation and analyi unuually cumberome. On the other hand, we work on the original PDE in the third approach and approximate it olution by uing a new family of orthogonal function which are image of claical Jacobi polynomial under a uitable mapping. The analyi of thi approach will require approximation reult by the new family of orthogonal function. The main advantage i that once thee approximation reult are etablihed, they can be directly applied to a large cla of problem. Thu, we hall mainly concentrate on the econd and third approache, and provide a general framework for the analyi of thee pectral method. While pectral method have been ued for olving PDE on unbounded domain
3 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp for over thirty year, and there have been everal iolated effort in the early year on the error analyi of thee method (ee, e.g. [6, 7, 15, 2, 23, 42]), it i only in the lat ten year or o that the baic approximation propertie of thee orthogonal ytem, and their application to PDE, were ytematically tudied (cf. [13] for a brief account). However, many of thee analye ue different approache and involve complicated Sobolev pace, making it hard for nonexpert to extract ueful information from thee error etimate and to carry out error analyi for their application. The main purpoe of thi paper are three fold: (i) to preent a unified framework, for the analyi of mapped Jacobi, Laguerre and Hermite pectral method, which lead to more concie reult (than thoe appeared in the literature) and optimal approximation reult in mot ituation; (ii) to make a detailed comparion on the convergence rate of different method for everal typical olution; and (iii) to provide a brief (by no mean complete) review on ome of the recent work for the analyi and application of pectral method in unbounded domain. Thi paper i organized a follow. In the next ection, we conider the mapped pectral method and preent a unified framework to tudy their convergence propertie. In Section 3, we conider the approximation by the (generalized) Laguerre polynomial/function, and Section 4 i devoted to the approximation by the Hermite polynomial/function. Thee three ection are preented with a unified tyle and encompa mot of the important approximation reult on thee orthogonal ytem developed in the lat few year. In Section 5, we provide ome implementation detail and compare the performance of different method with two typical example. In Section 6, we dicu variou extenion and other iue related to the application of thee pectral method. We end thi paper with a few concluding remark. We now introduce ome notation. Let ω(x) be a certain weight function in Ω:=(a,b), where a or b could be infinite. We hall ue the weighted Sobolev pace H r ω(ω) (r =,1,2, ), whoe inner product, norm and eminorm are denoted by (, ) r,ω, r,ω and r,ω, repectively. For real r>, we define the pace H r ω(ω) by pace interpolation. In particular, the norm and inner product of L 2 ω(ω) = H ω(ω) are denoted by ω and (, ) ω, repectively. The ubcript ω will be omitted from the notation in cae of ω 1. For notational convenience, we denote k x=d k /dx k, k 1, and for any nonnegative integer, let P be the et of all algebraic polynomial of degree. We denote by c a generic poitive contant independent of any function and, and ue the expreion A B to mean that there exit a generic poitive contant c uch that A cb. 2 Mapped Jacobi method A common and effective trategy in dealing with an unbounded domain i to ue a uitable mapping that tranform an infinite domain to a finite domain. Then, image of claical orthogonal polynomial under the invere mapping will form a et of orthogonal bai function which can be ued to approximate olution of PDE in the infinite domain.
4 198 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp Early practitioner of thi approach include Groch & Orzag [24] and Boyd [8]. The book by Boyd [11] contain an extenive review on many practical apect of the mapped pectral method. In the lat couple of year, a erie of paper have been devoted to the convergence analyi of the mapped pectral method (ee, e.g., [34, 35, 39, 54]). We preent below a general framework for the analyi and implementation of the mapped pectral method. To tudy the propertie of the mapped Jacobi approximation, we recall ome baic propertie and reult for the claical Jacobi polynomial J α,β n (y), y I :=( 1,1), n. 2.1 Some reult on Jacobi approximation Let ω α,β (y)=(1 y) α (1+y) β be the Jacobi weight function. For α,β> 1, the Jacobi polynomial are mutually orthogonal in L 2 ω α,β (I), i.e., I J α,β n where δ n,m i the Kronecker function, and (y)jm α,β (y)ω α,β (y)dy=γn α,β δ n,m, (2.1) γn α,β 2 = α+β+1 Γ(n+α+1)Γ(n+β+1) (2n+α+β+1)Γ(n+1)Γ(n+α+β+1). (2.2) They are eigenfunction of the SturmLiouville problem: with the eigenvalue: y ((1 y) α+1 (1+y) β+1 y J α,β n (y))+λn α,β (1 y) α (1+y) β Jn α,β (y)=, (2.3) λ α,β n = n(n+α+β+1), n, α,β> 1. (2.4) ow, we define the L 2 (I) orthogonal proection: ˆπ α,β ω α,β : L2 (I) P ω α,β, uch that Define the weighted pace ( ˆπ α,β v v,v ) ω α,β =, v P. (2.5) ˆB m α,β (I) := {v L2 ω α,β (I) : k y v L2 ω α+k,β+k (I), k m}. (2.6) The following reult wa proved in [19] (ee alo [3, 38]): Lemma 2.1. l y( ˆπ α,β v v) ω α+l,β+l l m m y v ω α+m,β+m, l m, v ˆB α,β m (I). (2.7) Let I α,β be the JacobiGau or JacobiGauRadau interpolation operator. The following interpolation approximation reult can be found in [38]. Lemma 2.2. For any v ˆB α,β m (I) with m 1, y (I α,β v v) α,β ωα+1,β+1+ I v v ω α,β 1 m m y v ω α+m,β+m. (2.8)
5 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp Mapping Let u conider a family of mapping of the form: x= g(y;), >, y I :=( 1,1), x Λ :=(,+ )or(,+ ), (2.9) uch that dx dy = g (y;)>, >, y I, g( 1;)=, g(1;)=+, g(±1;)=±, if Λ=(,+ ). if Λ=(,+ ), (2.1) In thi onetoone tranform, the parameter i a poitive caling factor. Without lo of generality, we further aume that the mapping i explicitly invertible, and denote it invere mapping by y= g 1 (x;) := h(x;), x Λ, y I, >. (2.11) Several typical mapping that have been propoed and ued in practice are of the above type (ee, e.g., [11] and the reference therein): Mapping between x Λ=(,+ ) and y I =( 1,1) with >: Algebraic mapping: x= y 1 y 2, y= x x (2.12) Logarithmic mapping: Exponential mapping: x=arctanh(y)= 1+y ln 2 1 y, y=tanh( 1 x). (2.13) x=inh(y), y= 1 ln( x+ x 2 +1 ), y ( 1,1), x ( L,L ), (2.14) where L =inh(). Mapping between x Λ=(,+ ) and y I =( 1,1) with >: Algebraic mapping: x= (1+y) x, y= 1 y x+. (2.15)
6 2 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp Logarithmic mapping: ( y+1 ) x=arctanh = 3+y ln y, y=1 2tanh( 1 x). (2.16) Exponential mapping: ( ) x=inh 2 (1+y), y= 2 ln( x+ x 2 +1 ) 1, (2.17) where y ( 1,1) and x (,L ) with L =inh(). The pecial feature which ditinguihe thee mapping i that, a y ±1, x varie algebraically, logarithmically or exponentially for algebraic, logarithmic or exponential mapping, repectively. The parameter i a caling/tretching factor which can be ued to fine tune the pacing of collocation point. We alo notice that the image of the exponential mapping (2.14) and (2.17) i a finite interval, o they combine both mapping and domain truncation. 2.3 Mapped Jacobi approximation Given a mapping x=g(y;) atifying (2.9) (2.11) and a family of orthogonal polynomial {p k (y)} with y I = ( 1,1), { p k (h(x;)) } form a new family of orthogonal function in Λ = (, ) or (, ). For example, the algebraic mapping (2.12) or (2.15) with the Chebyhev or Legendre polynomial lead to orthogonal rational bai function which have been tudied in [8, 9, 14, 34, 35, 4]. For the ake of generality, we conider the mapped Jacobi approximation. Let J α,β k (y) (α, β > 1) be the kth degree claical Jacobi polynomial whoe propertie are ummarized in the Appendix. We define the mapped Jacobi polynomial a,n α,β (x) := Jn α,β (y)= Jn α,β (h(x;)), x Λ, y I. (2.18) We infer from (2.1) that (2.18) define a new family of orthogonal function {,n α,β } in L 2 (Λ), i.e., ω α,β,n α,β (x),m(x)ω α,β α,β (x)dx=γn α,β δ m,n, (2.19) where the contant γ α,β n Λ i given in (2.2), and the weight function ω α,β (x)=ω α,β (y) dy dx = ωα,β (y)(g (y;)) 1 >, (2.2) with y=h(x;) and ω α,β (y)=(1 y) α (1+y) β. We now preent ome approximation propertie of thee mapped Jacobi polynomial. Let u define the finite dimenional approximation pace V α,β, =pan{α,β,n (x) : n=,1,,}, (2.21)
7 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp and conider the orthogonal proection π α,β ( α,β π, u u,v ) Thank to the orthogonality, we can write where, : L2 ω α,β ω α,β ( α,β π, u) (x)= (Λ) V α,β, uch that =, v V α,β,. (2.22) n= û α,β,n = 1 γn α,β u(x) α,β Λ û α,β,n α,β,n (x), (2.23),n (x)ω α,β (x)dx. We now introduce a weighted pace which i particularly uitable to decribe the L 2 proection error. Given a mapping atifying (2.9) (2.11), we et a (x) := dx dy (>), U (y) := u(x)=u(g(y;)). (2.24) The key to expre the error etimate in a concie form i to introduce an operator One verifie readily that du dy = a du dx = D xu, and an induction argument lead to d k U dy k D x u := a du dx. d 2 U dy 2 = a d ( dx a du dx = a d ( d ( ( a } dx dx {{ } k 1 parenthee ) = D 2 xu, du ) ) a := D k dx xu. (2.25) Let u define B α,β m (Λ)={ u : u i meaurable in Λ and u B < α,β m } equipped with the norm and eminorm ( m u B = α,β m k= D k xu 2 ω α+k,β+k ) 1 2, u B m α,β = Dx m u α+m,β+m ω, where the weight function ω α+k,β+k i defined in (2.2). We have the following fundamental reult for the mapped Jacobi approximation.
8 22 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp Theorem 2.1. If u B α,β m (Λ), we have that for m, π α,β, u u ω α,β m Dx m u α+m,β+m ω, (2.26) and for m 1, where x (π α,β, u u) ω α,β 1 m Dx m u α+m,β+m ω, (2.27) ω α,β (x)=ω α+1,β+1 (y)g (y;), y=h(x;). Proof. Let U (y) = u(x) = u(h(y;)) whoe Jacobi expanion i U (y) = n=ûα,β n (y). Then, by the definition (2.18), we have the relation between the coefficient of the Jacobi and mapped Jacobi expanion: û α,β,n = 1 γn α,β (u, α,β,n ) α,β ω = 1 γ α,β n,n J α,β (U,Jn α,β ) ω α,β =Û,n α,β. (2.28) Let ˆπ α,β be the L2 orthogonal proection operator aociated with the Jacobi polynomial (cf. (2.5)). By (2.1), (2.19) and Lemma ω α,β 2.1, π α,β, u u 2 ω α,β = (û,n) α,β 2 γ α,β n = (Û α,β n=+1 n=+1,n ) 2 γn α,β = ˆπ α,β U U 2 2m m ω α,β y U 2 ω α+m,β+m 2m Dx m u 2. (2.29) ω α+m,β+m ext, we deduce from (2.18) and the orthogonality of { y Jn α,β } that { x,n α,β } i L orthogonal, and x α,β,n α,β 2 ω = y J α,β n 2 = λ α,β ω α+1,β+1 n γn α,β, where λ α,β n i the eigenvalue of the Jacobi SturmLiouville problem (cf. (2.4)). Therefore, by (2.28) and Lemma 2.1, x (π α,β, Thi end the proof. u u) 2 ω α,β = n=+1 λ α,β n γn α,β (û α,β,n) 2 = n=+1 λ α,β n γn α,β (Û,n α,β ) 2 = y (π α,β U U ) 2 ω α+1,β+1 2(1 m) m y U 2 ω α+m,β+m α,β 2 ω 2(1 m) Dx m u 2. (2.3) ω α+m,β+m
9 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp Remark 2.1. It hould be pointed out that under the above general etting, the approximation reult on the higherorder proection, uch a the H 1 (Λ) orthogonal proec ω α,β tion π 1,α,β, : H1 ω α,β (Λ) V α,β,, can be etablihed by uing the exiting Jacobi approximation reult (ee, e.g., [38]) and a imilar argument a above. In particular, applying the above reult with α=β=, 1/2 to the algebraic mapping (2.12) and (2.15) lead to more concie and in ome cae improved, Chebyhev and Legendre rational approximation reult which were developed eparately in [34, 35, 39, 54]. The error etimate in the above theorem look very imilar to the uual pectral error etimate in a finite interval (cf. Lemma 2.1). Firt of all, it i clear from the above theorem that the proection error converge fater than any algebraic rate if a function decay exponentially fat at infinity. For a function with ingularitie inide the domain, the above theorem and Lemma 2.1 lead to the ame order of convergence, auming that the function decay ufficiently fat at infinity. However, for a given mooth function, they may lead to very different convergence rate due to the difference in the norm ued to meaure the regularity. We now determine the convergence rate for three et of function with typical decay propertie: Set 1. Exponential decay with ocillation at infinity u(x)=inkxe x for x (, ) or u(x)=inkxe x2 for x (, ). (2.31) Set 2. Algebraic decay without ocillation at infinity u(x)=(1+x) h for x (, ) or u(x)=(1+x 2 ) h for x (, ). (2.32) Set 3. Algebraic decay with ocillation at infinity u(x)= inkx inkx for x (, ) or u(x)= (1+x) h (1+x 2 for x (, ). (2.33) ) h Conider firt the mapping (2.15). Then, D x = ( dy ) 1 d dx dx = (x+)2 d 2 dx, ( 2 ) k ( 2x ) l 2 ωk,l (x)= x+ x+ (x+) 2. Hence, for u(x)=(1+x) h, it can be eaily checked that Dx mu ω α+m,β+m < if m<2h+α+1, which implie that u π α,β u ω α,β (2h+α+1) (u(x)=(1+x) h ). (2.34)
10 24 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp On the other hand, for u(x) = inkx (1+x) h, it can alo be eaily checked that Dx mu < ω α+m,β+m if m< 2h+α+1 3, which implie that ( u π α,β u ω α,β (2h+α+1)/3 u(x)= inkx ) (1+x) h. (2.35) ext, we conider the mapping (2.12) which lead to ( dy ) 1 d D x = dx dx = (x2 + 2 ) 3/2 2 ω k,l (x)= d dx, ( x x x2 + 2 ) k ( x x x2 + 2 ) l 2 (x ) 3/2. Hence, for u(x)=(1+x 2 ) h, we have Dx m u α+m,β+m ω < if m<2h+α+1, which implie that u π α,β u ω α,β (2h+α+1) (u(x)=(1+x 2 ) h ). (2.36) On the other hand, for u(x)=inkx (1+x 2 ) h, we have Dx m u α+m,β+m ω < if m< 2h+α+1 2, which implie that u π α,β u ω α,β (2h+α+1)/2 ( u(x)= inkx (1+x 2 ) h ). (2.37) A few remark are in order: (i) If h i a poitive integer, then u(x) = (1+x) h and u(x)=(1+x 2 ) h are rational function and they can be expreed exactly by a finite um of mapped rational function; (ii) For other cae, only algebraic convergence rate are achievable even though the function are mooth; (iii) the convergence rate for olution with ocillation at infinitie i much lower than that for olution without ocillation at infinitie; and (iv) For olution with exponential decay at infinity, the convergence rate will be fater than any algebraic rate; numerical reult in [34, 35, 54] (ee alo [11]) indicate that the convergence rate i ubgeometrical a e c ; and (v) numerical reult performed in [34, 35, 39, 54] are conitent with the etimate in (2.34)(2.37). 2.4 Mapped Jacobi interpolation approximation We now conider the Gau and GauRadau quadrature formula on unbounded domain baed on the mapped Jacobi polynomial. To fix the idea, we only conider the Gau quadrature, ince the GauRadau quadrature (which i ueful in the emiinfinite interval) can be treated in exactly the ame fahion. Let { ξ α,β,,},ωα,β be the Jacobi= Gau node and weight, and there hold 1 1 φ(y)ω α,β (y)dy= = φ(ξ α,β, )ωα,β,, φ P 2+1. (2.38)
11 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp Applying a mapping (2.9) to the above lead to the mapped JacobiGau quadrature: Λ u(x)ω α,β (x)dx = = u(ζ α,β,, )ρα,β,,, u Vα,β,2+1, (2.39) where ζ α,β,, := g(ξα,β, ;), ρα,β,, := ωα,β,, (2.4) are the mapped JacobiGau node and weight. Accordingly, we can define the dicrete inner product and dicrete norm: (u,v) α,β = ω, u(ζ α,β,, )v(ζα,β,, )ρα,β,,, = u ω α,β =(u,u) 1 2, ω α,β,, u,v C(Λ). The mapped JacobiGau interpolation operator I α,β, : C(Λ) Vα,β,, i defined by I α,β, u Vα,β, α,β uch that (I, u)(ζα,β,, )=u(ζα,β,, ), =,1,,. (2.41) be the JacobiGau (or JacobiGauRadau) interpolation operator. By defini Let I α,β tion, we have I α,β α,β,u(x)=(i U )(y)=(i α,β U )(h(x;)). (2.42) Then, we can eaily derive the following reult by combining Lemma 2.2 and Theorem 2.1. Theorem 2.2. If u B α,β m (Λ) with m 1, then x (I α,β, u u) ω α,β + I α,β, u u ω α,β 1 m Dx m u α+m,β+m ω. (2.43) We now examine how the mapping parameter affect the ditribution of the node. Aume that the node { ζ,,} α,β are arranged in acending order. We firt oberve that = by the mean value theorem, ζ α,β,,+1 ζα,β,, = g (ξ;)(ξ α,β,,+1 ξα,β,, ), (2.44) for certain ξ (ξ α,β,,,ξα,β,,+1 ). Hence, the intenity of tretching eentially depend on the derivative value of the mapping. For the mapping (2.13), (2.12), (2.16) and (2.15), we have dx dy = g (y;)= 1 y 2, (1 y 2 ) 3/2, 2 (3+y)(1 y), repectively. Therefore, the grid i tretched more and more a increae. 2 (1 y) 2, (2.45)
12 26 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp (a) Ditribution of x (, ) n=2 n=16 n=8 n= x (c) Ditribution of x (, ) n=8 n=12 n=16 n=2 (b) Effect of caling factor =2. =.5 =.1 =1. = x (d) Effect of caling factor =2., m=13 =1.5, m=15 =1., m=17 =.5, m=2 =.1, m= x x Figure 1: (a) HermiteGau point ( ) v. mapped LegendreGau point uing the algebraic map (2.12) with =1 ( ) for variou n; (b) Mapped LegendreGau point with n=16 and variou caling factor ; (c) LaguerreGauRadau point ( ) v. mapped LegendreGauRadau point uing the algebraic map (2.15) with =1 ( ) for variou n; (d) Mapped LegendreGauRadau point with n =32 and variou caling factor (m i the number of point in the ubinterval [,1)). In Fig. 1, we plot ample grid ditribution for different caling factor with variou number of node for the mapped Legendre Gau (or GauRadau) point (ee the caption for detail). A comparion with HermiteGau point i alo preented in Fig. 1(a). We notice that the mapped LegendreGau point are more clutered near the origin and pread further, while the HermiteGau point are more evenly ditributed. It hould be oberved that the ditribution of mapped LegendreGau point i more favorable ince a much larger effective interval i covered. However, it can be hown that in both cae,
13 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp the mallet ditance between neighboring point i O( 1 ), a oppoed to O( 2 ) for JacobiGau type node in a finite interval. A comparion of mapped Legendre and Laguerre GauRadau node i hown in Fig. 1(c). The mapped LegendreGauRadau point are much more clutered near the origin, and one can check that the mallet ditance between neighboring point i O( 2 ), a oppoed to O( 1 ) for the Laguerre GauRadau node. Hence, the ditribution of mapped LegendreGauRadau point i more favorable a far a reolution/accuracy i concerned but it will lead to a more retrictive CFL condition if explicit cheme are ued for timedependent problem. 2.5 umerical method uing mapped Jacobi polynomial A generic example Conider the model equation γu x (a(x) x u)= f, x Λ=(,+ ), γ, (2.46) with uitable decay condition at ± which will depend on the weight function in the weighted variational formulation. For a given mapping x=g(y;) with x Λ and y ( 1,1), we recall that the mapped Jacobi polynomial are mutually orthogonal in L 2 ω α,β for (2.46) i to find u V α,β, uch that γ(u,v ) ω α,β + ( a(x) x u, x (v ω α,β ) ) =(I α,β (Λ). Hence, the mapped Jacobi method, f,v ) ω α,β, v V α,β,. (2.47) Let u now conider the econd approach decribed in the introduction. Here, Eq. (2.46) i firt tranformed into γu 1 ( a(g(y;)) ) g (y;) y g (y;) yu = F, (2.48) where U (y)=u(g(y;)) and F (y)= f(g(y;)). Then, let ˆω α,β (y)=ω α,β (y)g (y;), the Jacobi pectral method for (2.48) i to find ũ P uch that ( a(g(y;)) ) γ(ũ,ṽ ) ω α,β+ g (y;) yũ, y (ṽ ˆω α,β ) =(I α,β F,ṽ ) ω α,β, ṽ P. (2.49) One can verify eaily that ũ (y)=u (g(y;)). Hence, the two approache are mathematically equivalent. We remark that the formulation (2.49) i in general more difficult to analyze due to the ingular nature of g (y;), while the analyi for the formulation (2.47) become tandard once we etablih the baic approximation propertie of the mapped Jacobi polynomial.
14 28 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp On the other hand, Eq. (2.48) can be eaily implemented uing the tandard Jacobicollocation (or more pecifically Chebyhevcollocation) method. Indeed, let {h, (y)} 1 be the Lagrange polynomial aociated with the JacobiGau point {y } 1, the Jacobicollocation approximation to (2.48) i to find U, (y) = =1 u h, (y) uch that Let u denote ( 1 ( a(g(y;)) )) γu, (y ) g (y;) y g (y;) yu, (y )= F (y ), 1. (2.5) u=(u 1,,u ) t, f=(f (y 1 ),,F (y )) t, D i = h (y i), D=(D i ), Λ i = a(g(y i;)) g (y i ;), Λ=diag(Λ i), Σ i = 1 g (y i ;), Then, (2.5) lead to the matrix ytem (γi ΣDΛD)u=f, Σ=diag(Σ i). which can be eaily olved by uing a tandard linear algebra routine. ote that in the above procedure, we only need to compute the JacobiGau point {y } 1 and the aociated differentiation matrix D whoe entrie can be found for intance in [19] Error etimate for a model problem We conider the Jacobi rational approximation to the model problem γu(x) 2 xu(x)= f(x), x Λ=(, ), u()=, (2.51) with a uitable decay condition at infinity which i to be determined by the weak formulation of (2.51). For a given mapping, let ω=ω α,β be the weight function aociated with the mapped Jacobi polynomial, and denote We define a bilinear form H 1,ω(Λ)={u H 1 ω(λ) : u()=}. a ω (v,φ)=γ(v,φ) ω +( x v, x (φω)), u,v H 1,ω(Λ). Then, a weak formulation for (2.51) i to find u H,ω 1 (Λ) uch that a ω (u,v)=( f,v) ω, v H 1,ω(Λ), (2.52) for f ( H 1,ω (Λ)). ote that u H 1,ω (Λ) implie a decay condition for u at infinity.
15 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp Let u denote X ={u V α,β, :u()=}. We can then define the Galerkin approximation of (2.52) by the mapped Jacobi polynomial a follow: For f L 2 ω(λ) C( Λ), find u X uch that a ω (u,v )=(I α,β, f,v ) ω, v X. (2.53) Unlike the tandard pectral method in a finite domain, the wellpoedne of (2.52) and of (2.53) i not guaranteed for all cae with γ. A general reult for the wellpoedne of an abtract equation of the form (2.52) i etablihed in [49]. For the reader convenience, we recall thi reult below (cf. Lemma 2.3 in [49]): Lemma 2.3. We aume that d 1 =max x Λ ω 1 (x) x ω(x), d 2 =max ω 1 (x) 2 x ω(x) x Λ are finite and that u 2 (x)ω (x) x Λ = for u H 1,ω (Λ). Then, for any u,v H1 ω(λ), we have that a ω (u,v) (d 1 +1) u 1,ω v 1,ω +γ u ω v ω, (2.54) and for any v H 1,ω (Λ), a (ν) ω (v,v) v 2 1,ω +(γ d 2/2) v 2 ω. (2.55) Remark 2.2. The inequality (2.55) i derived under a general framework. For a pecific problem, the contant γ d 2 /2 can often be replaced by a larger contant. Thank to the above lemma, it i then traightforward to prove the following general reult: Theorem 2.3. Aume that the condition of Lemma (2.3) are atified and γ d 2 /2>. Then the problem (2.52) (rep. (2.53)) admit a unique olution. Furthermore, we have the error etimate: u u 1,ω inf v X u v 1,ω + f I α,β, f ω. (2.56) Remark 2.3. With a change of variable x to x/c (c>) for Eq. (2.46), the retriction on γ can be relaxed to γ>. Hence, given a mapping and a pair of Jacobi parameter (α,β), we ut need to compute upper bound for d 1 and d 2, verify that the condition of Theorem 2.3 are atified, and apply the approximation reult in Theorem 2.1 and 2.2 to (2.56) to get the deired error etimate. Conider for example the mapped Legendre method for (2.52) with the mapping (2.15). It can be hown that for thi mapping, we have d 1 2 and d 2 6. Applying Theorem 2.1 and 2.2 to (2.56) with (α,β)=(,) lead to the following reult:
16 21 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp Corollary 2.1. Let u and u be the olution of (2.52) and (2.53) with (α,β) = (,) and the mapping (2.15) with =1. Aume that u B, m (Λ), f B, k (Λ) and γ>3. We have u u 1,ω, 1 1 m D m x u ω m,m 1 + k Dx k f ω k,k. (2.57) 1 We note that a lightly improved condition on γ wa derived in [35] uing a refined etimate for (2.55). A imilar procedure can be applied to the mapped Chebyhev method for (2.52) with the mapping (2.15). ote however that in thi cae we have d 1,d 2 =. everthele, one can till how that a ω (, ) i continuou and coercive (cf. [34]). Applying Theorem 2.1 and 2.2 to (2.56) with (α,β)=( 1 2, 1 2 ) lead to the following reult (cf. [34]): Corollary 2.2. Let u and u be the olution of (2.52) and (2.53) with (α,β)=( 1 2, 1 2 ) and the mapping (2.15) with =1. Auming that u B 1/2, 1/2 m (Λ) and f B 1/2, 1/2 k (Λ) and that γ> We have u u 1,ω 1/2, 1/2 1 1 m D m x u ω m 1/2,m 1/2 1 + k Dx k f ω k 1/2,k 1/2. (2.58) 1 Remark 2.4. Error etimate which are eentially equivalent to (2.57) and (2.58) but in different form were derived in [34, 35]. The ame procedure can be ued to derive error etimate on mapped Jacobi method for problem in the whole line (cf. [39, 54]). 3 Laguerre pectral method For problem in a emiinfinite interval, it i natural to ue (generalized) Laguerre polynomial/function which form orthonormal bai in (weighted) Sobolev pace. 3.1 Generalized Laguerre approximation We firt recall ome baic propertie of generalized Laguerre polynomial/function Generalized Laguerre polynomial Let Λ := (, ). The generalized Laguerre polynomial (GLP), denoted by L (α) n (x)(α > 1), are the eigenfunction of the SturmLiouville problem x α e x ( x x α+1 e x x L (α) n (x) ) +λ n L (α) n (x)=, x Λ, (3.1) with the eigenvalue λ n = n. Compared with the Jacobi polynomial in a finite interval, the linear growth of λ n for the Laguerre polynomial indicate, on the one hand, a lower convergence rate of the Laguerre expanion, but on the other hand, lead to better invere inequalitie and conequently milder CFL condition for time dependent problem.
17 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp The GLP are mutually orthogonal in L 2 ω α (Λ) with the weight function ω α (x)=x α e x, i.e., + L (α) n (x)l (α) m (x)ω α (x)dx=γ (α) n δ mn withγ (α) n = Γ(n+α+1). (3.2) Γ(n+1) The threeterm recurrence formula of the GLP read (n+1)l (α) n+1 (x)=(2n+α+1 x)l(α) n (x) (n+α)l (α) n 1 (x), L (α) (x)=1, L (α) 1 (x) = α+1 x. (3.3) We infer from (3.1) and (3.2) that + x L (α) n (x) x L (α) m (x)xω α (x)dx=λ n γ (α) n δ mn. (3.4) An important property of the GLP i the following derivative relation: x L (α) n (x)= L (α+1) n 1 n 1 (x)= L (α) k (x). (3.5) k= The cae α= lead to the claical Laguerre polynomial, which are ued mot frequently in practice and will imply be denoted by L n (x). A in the finite interval cae, it i actually eaier to tudy the whole family of generalized Laguerre polynomial, rather than the Laguerre polynomial alone Approximation reult by generalized Laguerre polynomial We begin by analyzing the approximation propertie of the L 2 ω α orthogonal proection π,α : L 2 ω α (Λ) P, defined by (u π,α u,v ) ωα =, v P. (3.6) It i clear that the polynomial π,α u i the bet approximation u in L 2 ω α (Λ), and with π,α u(x)= n= û (α) n L (α) n (x), û (α) n = 1 + γ (α) u(x)l (α) n (x)ω α (x)dx, n. n Similar to the Jacobi approximation, we define B m α (Λ) := { u : k xu L 2 ω α+k (Λ), k m }, (3.7)
18 212 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp equipped with the norm and eminorm ( m ) 1/2, u B m α = k x u 2 ω u B m α+k α = m x u 2 ω α+m. k= In particular, we omit the ubcript α, when α =. In contrat to the uual weighted Sobolev pace H m ω α (Λ), the weight function correponding to derivative of different order i different in B m α (Λ). We oberve from (3.5) that k xl (α) n (x)=( 1) k L (α+k) n k (x), n k, (3.8) and o { k xl (α) } n are orthogonal with repect to the weight ωα+k, i.e., + k xl (α) l By (3.2) and the Stirling formula, we have Summing (3.9) over k m lead to (x) k xl (α) n (x)ω α+k (x)dx=γ (α+k) n k δ ln. (3.9) Γ(x+1) 2πx x+1/2 e x, x 1, (3.1) γ (α+k) n k = Γ(n+α+1) Γ(n k+1) nα+k, for n 1. m k= ( k x L (α) l, k xl (α) ) n =, ω α+k if l = n and k>min{l,n} which implie that { L (α) } n are orthogonal in the pace B m α (Λ). The fundamental generalized Laguerre approximation reult i tated below (ee, e.g., [19]). Theorem 3.1. For any u B m α (Λ) and m, Proof. Obviouly, we have that l x(π,α u u) ωα+l (l m)/2 m x u ωα+m, l m. (3.11) l x(π,α u u)= û (α) n n=+1 l xl (α) (x). n
19 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp Hence, by the orthogonality and the Stirling formula (3.1), l x(π,α u u) 2 ω α+l = Thi complete the proof. n=+1 max n> { γ (α+l) n l û (α) n 2 γ (α+l) n l /γ (α+m) n m l m m x u 2 ω α+m. } γ (α+m) n m û(α) n 2 n=+1 ext, we conider the approximation reult for the H 1 type orthogonal proection. For implicity, we conider only the uual Laguerre cae, i.e., α=. Hereafter, let ω(x)= e x be the uual Laguerre weight function, and denote H 1,ω(Λ)= { u H 1 ω(λ) : u()= }, P ={ φ P : φ()= }. (3.12) Conider the orthogonal proection π 1, : H1,ω (Λ) P, defined by ( (u π 1, u),v ) ω =, v P. (3.13) Theorem 3.2. If u H 1,ω (Λ) and xu B m 1 (Λ), then for m 1, Proof. Let π 1, u u 1,ω 1 2 m 2 m x u ωm 1. (3.14) x φ(x)= π 1, u (y)dy. Then u φ H,ω 1 (Λ). Thank to the imbedding inequality (ee, e.g., [31]) and Theorem 3.1 with α=, we find that Thi end the proof. u ω x u ω, π 1, u u 1,ω φ u 1,ω x (φ u) ω 1 2 m 2 m x u ωm 1. We note that in general Laguerre polynomial are not good candidate for approximation in infinite domain due to their wild behavior at infinity. Thi fact i alo reflected in the error etimate in Theorem 3.1 and 3.2. Although thee error etimate are alo of pectral type, but due to the exponential decay weight in the norm, they only imply meaningful pointwie approximation for a hort interval. Hence, the GLP are only uitable for the approximation of function with fat algebraic (or exponential) growth at infinity. For problem with ome decay propertie at infinity, it i more appropriate to ue the o called generalized Laguerre function (GLF) which we hall conider below.
20 214 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp Generalized Laguerre function The generalized Laguerre function (GLF) are defined by L (α) n (x) := e x/2 L (α) n (x), α> 1, x Λ. (3.15) It i clear that by (3.2), the GLF are orthogonal with repect to the weight function ˆω α = x α, i.e., + L (α) n (x) L (α) m (x) ˆω α (x)dx=γ (α) n δ mn, (3.16) where the contant γ (α) n i given in (3.2). In particular, the uual Laguerre function L n (x)=e x/2 L n (x), n, x Λ, (3.17) are orthonormal with repect to the uniform weight ˆω 1. A in the lat ection, we introduce an operator ˆ x = x which implie that xl (α) n (x)=e x/2 ˆ x L (α) n (x). (3.18) It i traightforward to check that the GLF atify the following propertie: Threeterm recurrence relation (n+1) L (α) n+1 =(2n+α+1 x) L (α) n (n+α) L (α) n 1, L (α) = e x/2, L (α) 1 =(α+1 x)e x/2. The SturmLiouville equation: ( x α e x/2 x x α+1 e x/2 ˆ x L (α) n (x) ) (3.19) +n L (α) n (x)=. (3.2) Orthogonality of the derivative: ˆ x L (α) n (x)ˆ x L (α) m (x) ˆω α+1 (x)dx=λ n γ (α) n δ mn. (3.21) Some recurrence formula: Λ ˆ x L (α) n L (α) n xˆ x L (α) n (x)= L (α+1) n 1 n 1 (x)= L (α) k (x), (3.22a) k= (x)= ˆ x L (α) n (x) ˆ x L (α) n+1 (x), (3.22b) (x)=n L (α) n (x) (n+α) L (α) n 1 (x). (3.22c) We plot in Fig. 2 ome ample graph of GLP and GLF. In contrat to the GLP, the GLF are wellbehaved with the decay property (ee Fig. 2 (d)): L (α) n (x), a x +. (3.23) Therefore, the GLP are uitable for approximation of function which decay at infinity.
Assessing the Discriminatory Power of Credit Scores
Aeing the Dicriminatory Power of Credit Score Holger Kraft 1, Gerald Kroiandt 1, Marlene Müller 1,2 1 Fraunhofer Intitut für Techno und Wirtchaftmathematik (ITWM) GottliebDaimlerStr. 49, 67663 Kaierlautern,
More informationOptical Illusion. Sara Bolouki, Roger Grosse, Honglak Lee, Andrew Ng
Optical Illuion Sara Bolouki, Roger Groe, Honglak Lee, Andrew Ng. Introduction The goal of thi proect i to explain ome of the illuory phenomena uing pare coding and whitening model. Intead of the pare
More informationOriginal Article: TOWARDS FLUID DYNAMICS EQUATIONS
Peer Reviewed, Open Acce, Free Online Journal Publihed monthly : ISSN: 88X Iue 4(5); April 15 Original Article: TOWARDS FLUID DYNAMICS EQUATIONS Citation Zaytev M.L., Akkerman V.B., Toward Fluid Dynamic
More informationRedesigning Ratings: Assessing the Discriminatory Power of Credit Scores under Censoring
Redeigning Rating: Aeing the Dicriminatory Power of Credit Score under Cenoring Holger Kraft, Gerald Kroiandt, Marlene Müller Fraunhofer Intitut für Techno und Wirtchaftmathematik (ITWM) Thi verion: June
More informationQueueing systems with scheduled arrivals, i.e., appointment systems, are typical for frontal service systems,
MANAGEMENT SCIENCE Vol. 54, No. 3, March 28, pp. 565 572 in 25199 ein 1526551 8 543 565 inform doi 1.1287/mnc.17.82 28 INFORMS Scheduling Arrival to Queue: A SingleServer Model with NoShow INFORMS
More informationUnit 11 Using Linear Regression to Describe Relationships
Unit 11 Uing Linear Regreion to Decribe Relationhip Objective: To obtain and interpret the lope and intercept of the leat quare line for predicting a quantitative repone variable from a quantitative explanatory
More informationTwo Dimensional FEM Simulation of Ultrasonic Wave Propagation in Isotropic Solid Media using COMSOL
Excerpt from the Proceeding of the COMSO Conference 0 India Two Dimenional FEM Simulation of Ultraonic Wave Propagation in Iotropic Solid Media uing COMSO Bikah Ghoe *, Krihnan Balaubramaniam *, C V Krihnamurthy
More informationControl of Wireless Networks with Flow Level Dynamics under Constant Time Scheduling
Control of Wirele Network with Flow Level Dynamic under Contant Time Scheduling Long Le and Ravi R. Mazumdar Department of Electrical and Computer Engineering Univerity of Waterloo,Waterloo, ON, Canada
More informationPartial optimal labeling search for a NPhard subclass of (max,+) problems
Partial optimal labeling earch for a NPhard ubcla of (max,+) problem Ivan Kovtun International Reearch and Training Center of Information Technologie and Sytem, Kiev, Uraine, ovtun@image.iev.ua Dreden
More informationMECH 2110  Statics & Dynamics
Chapter D Problem 3 Solution 1/7/8 1:8 PM MECH 11  Static & Dynamic Chapter D Problem 3 Solution Page 7, Engineering Mechanic  Dynamic, 4th Edition, Meriam and Kraige Given: Particle moving along a traight
More informationA technical guide to 2014 key stage 2 to key stage 4 value added measures
A technical guide to 2014 key tage 2 to key tage 4 value added meaure CONTENTS Introduction: PAGE NO. What i value added? 2 Change to value added methodology in 2014 4 Interpretation: Interpreting chool
More informationSolution of the Heat Equation for transient conduction by LaPlace Transform
Solution of the Heat Equation for tranient conduction by LaPlace Tranform Thi notebook ha been written in Mathematica by Mark J. McCready Profeor and Chair of Chemical Engineering Univerity of Notre Dame
More informationMixed Method of Model Reduction for Uncertain Systems
SERBIAN JOURNAL OF ELECTRICAL ENGINEERING Vol 4 No June Mixed Method of Model Reduction for Uncertain Sytem N Selvaganean Abtract: A mixed method for reducing a higher order uncertain ytem to a table reduced
More information1 Introduction. Reza Shokri* Privacy Games: Optimal UserCentric Data Obfuscation
Proceeding on Privacy Enhancing Technologie 2015; 2015 (2):1 17 Reza Shokri* Privacy Game: Optimal UerCentric Data Obfucation Abtract: Conider uer who hare their data (e.g., location) with an untruted
More informationHOMOTOPY PERTURBATION METHOD FOR SOLVING A MODEL FOR HIV INFECTION OF CD4 + T CELLS
İtanbul icaret Üniveritei Fen Bilimleri Dergii Yıl: 6 Sayı: Güz 7/. 95 HOMOOPY PERURBAION MEHOD FOR SOLVING A MODEL FOR HIV INFECION OF CD4 + CELLS Mehmet MERDAN ABSRAC In thi article, homotopy perturbation
More informationDMA Departamento de Matemática e Aplicações Universidade do Minho
Univeridade do Minho DMA Departamento de Matemática e Aplicaçõe Univeridade do Minho Campu de Gualtar 4757 Braga Portugal www.math.uminho.pt Univeridade do Minho Ecola de Ciência Departamento de Matemática
More informationA note on profit maximization and monotonicity for inbound call centers
A note on profit maximization and monotonicity for inbound call center Ger Koole & Aue Pot Department of Mathematic, Vrije Univeriteit Amterdam, The Netherland 23rd December 2005 Abtract We conider an
More informationLinear energypreserving integrators for Poisson systems
BIT manucript No. (will be inerted by the editor Linear energypreerving integrator for Poion ytem David Cohen Ernt Hairer Received: date / Accepted: date Abtract For Hamiltonian ytem with noncanonical
More informationv = x t = x 2 x 1 t 2 t 1 The average speed of the particle is absolute value of the average velocity and is given Distance travelled t
Chapter 2 Motion in One Dimenion 2.1 The Important Stuff 2.1.1 Poition, Time and Diplacement We begin our tudy of motion by conidering object which are very mall in comparion to the ize of their movement
More informationA Resolution Approach to a Hierarchical Multiobjective Routing Model for MPLS Networks
A Reolution Approach to a Hierarchical Multiobjective Routing Model for MPLS Networ Joé Craveirinha a,c, Rita GirãoSilva a,c, João Clímaco b,c, Lúcia Martin a,c a b c DEECFCTUC FEUC INESCCoimbra International
More informationDISTRIBUTED DATA PARALLEL TECHNIQUES FOR CONTENTMATCHING INTRUSION DETECTION SYSTEMS. G. Chapman J. Cleese E. Idle
DISTRIBUTED DATA PARALLEL TECHNIQUES FOR CONTENTMATCHING INTRUSION DETECTION SYSTEMS G. Chapman J. Cleee E. Idle ABSTRACT Content matching i a neceary component of any ignaturebaed network Intruion Detection
More information2. METHOD DATA COLLECTION
Key to learning in pecific ubject area of engineering education an example from electrical engineering AnnaKarin Cartenen,, and Jonte Bernhard, School of Engineering, Jönköping Univerity, S Jönköping,
More informationDISTRIBUTED DATA PARALLEL TECHNIQUES FOR CONTENTMATCHING INTRUSION DETECTION SYSTEMS
DISTRIBUTED DATA PARALLEL TECHNIQUES FOR CONTENTMATCHING INTRUSION DETECTION SYSTEMS Chritopher V. Kopek Department of Computer Science Wake Foret Univerity WintonSalem, NC, 2709 Email: kopekcv@gmail.com
More informationSolutions to Sample Problems for Test 3
22 Differential Equation Intructor: Petronela Radu November 8 25 Solution to Sample Problem for Tet 3 For each of the linear ytem below find an interval in which the general olution i defined (a) x = x
More informationA Note on Profit Maximization and Monotonicity for Inbound Call Centers
OPERATIONS RESEARCH Vol. 59, No. 5, September October 2011, pp. 1304 1308 in 0030364X ein 15265463 11 5905 1304 http://dx.doi.org/10.1287/opre.1110.0990 2011 INFORMS TECHNICAL NOTE INFORMS hold copyright
More informationRisk Management for a Global Supply Chain Planning under Uncertainty: Models and Algorithms
Rik Management for a Global Supply Chain Planning under Uncertainty: Model and Algorithm Fengqi You 1, John M. Waick 2, Ignacio E. Gromann 1* 1 Dept. of Chemical Engineering, Carnegie Mellon Univerity,
More informationThus far. Inferences When Comparing Two Means. Testing differences between two means or proportions
Inference When Comparing Two Mean Dr. Tom Ilvento FREC 48 Thu far We have made an inference from a ingle ample mean and proportion to a population, uing The ample mean (or proportion) The ample tandard
More informationTurbulent Mixing and Chemical Reaction in Stirred Tanks
Turbulent Mixing and Chemical Reaction in Stirred Tank André Bakker Julian B. Faano Blend time and chemical product ditribution in turbulent agitated veel can be predicted with the aid of Computational
More informationLinear Momentum and Collisions
Chapter 7 Linear Momentum and Colliion 7.1 The Important Stuff 7.1.1 Linear Momentum The linear momentum of a particle with ma m moving with velocity v i defined a p = mv (7.1) Linear momentum i a vector.
More informationDesign of Compound Hyperchaotic System with Application in Secure Data Transmission Systems
Deign of Compound Hyperchaotic Sytem with Application in Secure Data Tranmiion Sytem D. Chantov Key Word. Lyapunov exponent; hyperchaotic ytem; chaotic ynchronization; chaotic witching. Abtract. In thi
More informationHeat transfer to or from a fluid flowing through a tube
Heat tranfer to or from a fluid flowing through a tube R. Shankar Subramanian A common ituation encountered by the chemical engineer i heat tranfer to fluid flowing through a tube. Thi can occur in heat
More informationDUE to the small size and low cost of a sensor node, a
1992 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 14, NO. 10, OCTOBER 2015 A Networ Coding Baed Energy Efficient Data Bacup in SurvivabilityHeterogeneou Senor Networ Jie Tian, Tan Yan, and Guiling Wang
More informationProject Management Basics
Project Management Baic A Guide to undertanding the baic component of effective project management and the key to ucce 1 Content 1.0 Who hould read thi Guide... 3 1.1 Overview... 3 1.2 Project Management
More informationIntroduction to the article Degrees of Freedom.
Introduction to the article Degree of Freedom. The article by Walker, H. W. Degree of Freedom. Journal of Educational Pychology. 3(4) (940) 5369, wa trancribed from the original by Chri Olen, George Wahington
More informationSimulation of Power Systems Dynamics using Dynamic Phasor Models. Power Systems Laboratory. ETH Zürich Switzerland
X SEPOPE 2 a 25 de maio de 26 May 2 rt to 25 th 26 FLORIANÓPOLIS (SC) BRASIL X SIMPÓSIO DE ESPECIALISTAS EM PLANEJAMENTO DA OPERAÇÃO E EXPANSÃO ELÉTRICA X SYMPOSIUM OF SPECIALISTS IN ELECTRIC OPERATIONAL
More informationClusterAware Cache for Network Attached Storage *
CluterAware Cache for Network Attached Storage * Bin Cai, Changheng Xie, and Qiang Cao National Storage Sytem Laboratory, Department of Computer Science, Huazhong Univerity of Science and Technology,
More informationYou may use a scientific calculator (nongraphing, nonprogrammable) during testing.
TECEP Tet Decription College Algebra MATTE Thi TECEP tet algebraic concept, procee, and practical application. Topic include: linear equation and inequalitie; quadratic equation; ytem of equation and
More informationExposure Metering Relating Subject Lighting to Film Exposure
Expoure Metering Relating Subject Lighting to Film Expoure By Jeff Conrad A photographic expoure meter meaure ubject lighting and indicate camera etting that nominally reult in the bet expoure of the film.
More informationCASE STUDY ALLOCATE SOFTWARE
CASE STUDY ALLOCATE SOFTWARE allocate caetud y TABLE OF CONTENTS #1 ABOUT THE CLIENT #2 OUR ROLE #3 EFFECTS OF OUR COOPERATION #4 BUSINESS PROBLEM THAT WE SOLVED #5 CHALLENGES #6 WORKING IN SCRUM #7 WHAT
More informationQuadrilaterals. Learning Objectives. PreActivity
Section 3.4 PreActivity Preparation Quadrilateral Intereting geometric hape and pattern are all around u when we tart looking for them. Examine a row of fencing or the tiling deign at the wimming pool.
More informationBUILTIN DUAL FREQUENCY ANTENNA WITH AN EMBEDDED CAMERA AND A VERTICAL GROUND PLANE
Progre In Electromagnetic Reearch Letter, Vol. 3, 51, 08 BUILTIN DUAL FREQUENCY ANTENNA WITH AN EMBEDDED CAMERA AND A VERTICAL GROUND PLANE S. H. ZainudDeen Faculty of Electronic Engineering Menoufia
More informationReview of Multiple Regression Richard Williams, University of Notre Dame, http://www3.nd.edu/~rwilliam/ Last revised January 13, 2015
Review of Multiple Regreion Richard William, Univerity of Notre Dame, http://www3.nd.edu/~rwilliam/ Lat revied January 13, 015 Aumption about prior nowledge. Thi handout attempt to ummarize and yntheize
More informationChapter 10 Stocks and Their Valuation ANSWERS TO ENDOFCHAPTER QUESTIONS
Chapter Stoc and Their Valuation ANSWERS TO ENOFCHAPTER QUESTIONS  a. A proxy i a document giving one peron the authority to act for another, typically the power to vote hare of common toc. If earning
More informationNETWORK TRAFFIC ENGINEERING WITH VARIED LEVELS OF PROTECTION IN THE NEXT GENERATION INTERNET
Chapter 1 NETWORK TRAFFIC ENGINEERING WITH VARIED LEVELS OF PROTECTION IN THE NEXT GENERATION INTERNET S. Srivatava Univerity of Miouri Kana City, USA hekhar@conrel.ice.umkc.edu S. R. Thirumalaetty now
More informationREDUCTION OF TOTAL SUPPLY CHAIN CYCLE TIME IN INTERNAL BUSINESS PROCESS OF REAMER USING DOE AND TAGUCHI METHODOLOGY. Abstract. 1.
International Journal of Advanced Technology & Engineering Reearch (IJATER) REDUCTION OF TOTAL SUPPLY CHAIN CYCLE TIME IN INTERNAL BUSINESS PROCESS OF REAMER USING DOE AND Abtract TAGUCHI METHODOLOGY Mr.
More informationChapter 32. OPTICAL IMAGES 32.1 Mirrors
Chapter 32 OPTICAL IMAGES 32.1 Mirror The point P i called the image or the virtual image of P (light doe not emanate from it) The leftright reveral in the mirror i alo called the depth inverion (the
More informationTRADING rules are widely used in financial market as
Complex Stock Trading Strategy Baed on Particle Swarm Optimization Fei Wang, Philip L.H. Yu and David W. Cheung Abtract Trading rule have been utilized in the tock market to make profit for more than a
More informationMSc Financial Economics: International Finance. Bubbles in the Foreign Exchange Market. Anne Sibert. Revised Spring 2013. Contents
MSc Financial Economic: International Finance Bubble in the Foreign Exchange Market Anne Sibert Revied Spring 203 Content Introduction................................................. 2 The Mone Market.............................................
More informationA model for the relationship between tropical precipitation and column water vapor
Click Here for Full Article GEOPHYSICAL RESEARCH LETTERS, VOL. 36, L16804, doi:10.1029/2009gl039667, 2009 A model for the relationhip between tropical precipitation and column water vapor Caroline J. Muller,
More informationProfitability of Loyalty Programs in the Presence of Uncertainty in Customers Valuations
Proceeding of the 0 Indutrial Engineering Reearch Conference T. Doolen and E. Van Aken, ed. Profitability of Loyalty Program in the Preence of Uncertainty in Cutomer Valuation Amir Gandomi and Saeed Zolfaghari
More informationScheduling of Jobs and Maintenance Activities on Parallel Machines
Scheduling of Job and Maintenance Activitie on Parallel Machine ChungYee Lee* Department of Indutrial Engineering Texa A&M Univerity College Station, TX 778433131 cylee@ac.tamu.edu ZhiLong Chen** Department
More informationQueueing Models for Multiclass Call Centers with RealTime Anticipated Delays
Queueing Model for Multicla Call Center with RealTime Anticipated Delay Oualid Jouini Yve Dallery Zeynep Akşin Ecole Centrale Pari Koç Univerity Laboratoire Génie Indutriel College of Adminitrative Science
More informationBlock Diagrams, StateVariable Models, and Simulation Methods
5 C H A P T E R Block Diagram, StateVariable Model, and Simulation Method CHAPTER OUTLINE CHAPTER OBJECTIVES Part I. Model Form 25 5. Tranfer Function and Block Diagram Model 25 5.2 StateVariable Model
More informationCASE STUDY BRIDGE. www.futureprocessing.com
CASE STUDY BRIDGE TABLE OF CONTENTS #1 ABOUT THE CLIENT 3 #2 ABOUT THE PROJECT 4 #3 OUR ROLE 5 #4 RESULT OF OUR COLLABORATION 67 #5 THE BUSINESS PROBLEM THAT WE SOLVED 8 #6 CHALLENGES 9 #7 VISUAL IDENTIFICATION
More informationAvailability of WDM Multi Ring Networks
Paper Availability of WDM Multi Ring Network Ivan Rado and Katarina Rado H d.o.o. Motar, Motar, Bonia and Herzegovina Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, Univerity
More informationTIME SERIES ANALYSIS AND TRENDS BY USING SPSS PROGRAMME
TIME SERIES ANALYSIS AND TRENDS BY USING SPSS PROGRAMME RADMILA KOCURKOVÁ Sileian Univerity in Opava School of Buine Adminitration in Karviná Department of Mathematical Method in Economic Czech Republic
More informationBidding for Representative Allocations for Display Advertising
Bidding for Repreentative Allocation for Diplay Advertiing Arpita Ghoh, Preton McAfee, Kihore Papineni, and Sergei Vailvitkii Yahoo! Reearch. {arpita, mcafee, kpapi, ergei}@yahooinc.com Abtract. Diplay
More informationSENSING IMAGES. School of Remote Sensing and Information Engineering, Wuhan University, 129# Luoyu Road, Wuhan, China,ych@whu.edu.
International Archive of the Photogrammetry, Remote Sening and Spatial Information Science, Volume X/W, 3 8th International Sympoium on Spatial Data Quality, 3 May  June 3, Hong Kong COUD DETECTION METHOD
More informationTransient turbulent flow in a pipe
Tranient turbulent flow in a pipe M. S. Ghidaoui A. A. Kolyhkin Rémi Vaillancourt CRM3176 January 25 Thi work wa upported in part by the Latvian Council of Science, project 4.1239, the Natural Science
More informationSocially Optimal Pricing of Cloud Computing Resources
Socially Optimal Pricing of Cloud Computing Reource Ihai Menache Microoft Reearch New England Cambridge, MA 02142 timena@microoft.com Auman Ozdaglar Laboratory for Information and Deciion Sytem Maachuett
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationEXISTENCE AND NONEXISTENCE OF SOLUTIONS TO ELLIPTIC EQUATIONS WITH A GENERAL CONVECTION TERM
EXISTENCE AND NONEXISTENCE OF SOLUTIONS TO ELLIPTIC EQUATIONS WITH A GENERAL CONVECTION TERM SALOMÓN ALARCÓN, JORGE GARCÍAMELIÁN AND ALEXANDER QUAAS Abtract. In thi paper we conider the nonlinear elliptic
More informationStochasticity in Transcriptional Regulation: Origins, Consequences, and Mathematical Representations
36 Biophyical Journal Volume 8 December 200 36 336 Stochaticity in Trancriptional Regulation: Origin, Conequence, and Mathematical Repreentation Thoma B. Kepler* and Timothy C. Elton *Santa Fe Intitute,
More informationA Spam Message Filtering Method: focus on run time
, pp.2933 http://dx.doi.org/10.14257/atl.2014.76.08 A Spam Meage Filtering Method: focu on run time SinEon Kim 1, JungTae Jo 2, SangHyun Choi 3 1 Department of Information Security Management 2 Department
More informationA family of chaotic pure analog coding schemes based on baker s map function
Liu et al. EURASIP Journal on Advance in Signal Proceing 5 5:58 DOI.86/36345439 RESEARCH Open Acce A family of chaotic pure analog coding cheme baed on baker map function Yang Liu * JingLi Xuanxuan
More informationResearch Article An (s, S) Production Inventory Controlled SelfService Queuing System
Probability and Statitic Volume 5, Article ID 558, 8 page http://dxdoiorg/55/5/558 Reearch Article An (, S) Production Inventory Controlled SelfService Queuing Sytem Anoop N Nair and M J Jacob Department
More informationSimulation of Sensorless Speed Control of Induction Motor Using APFO Technique
International Journal of Computer and Electrical Engineering, Vol. 4, No. 4, Augut 2012 Simulation of Senorle Speed Control of Induction Motor Uing APFO Technique T. Raghu, J. Sriniva Rao, and S. Chandra
More informationCHAPTER 5 BROADBAND CLASSE AMPLIFIER
CHAPTER 5 BROADBAND CLASSE AMPLIFIER 5.0 Introduction ClaE amplifier wa firt preented by Sokal in 1975. The application of cla E amplifier were limited to the VHF band. At thi range of frequency, clae
More informationGrowing SelfOrganizing Maps for Surface Reconstruction from Unstructured Point Clouds
Growing SelfOrganizing Map for Surface Recontruction from Untructured Point Cloud Renata L. M. E. do Rêgo, Aluizio F. R. Araújo, and Fernando B.de Lima Neto Abtract Thi work introduce a new method for
More informationSupport Vector Machine Based Electricity Price Forecasting For Electricity Markets utilising Projected Assessment of System Adequacy Data.
The Sixth International Power Engineering Conference (IPEC23, 2729 November 23, Singapore Support Vector Machine Baed Electricity Price Forecating For Electricity Maret utiliing Projected Aement of Sytem
More informationUnusual Option Market Activity and the Terrorist Attacks of September 11, 2001*
Allen M. Potehman Univerity of Illinoi at UrbanaChampaign Unuual Option Market Activity and the Terrorit Attack of September 11, 2001* I. Introduction In the aftermath of the terrorit attack on the World
More informationAnalysis of Mesostructure Unit Cells Comprised of Octettruss Structures
Analyi of Meotructure Unit Cell Compried of Octettru Structure Scott R. Johnton *, Marque Reed *, Hongqing V. Wang, and David W. Roen * * The George W. Woodruff School of Mechanical Engineering, Georgia
More informationG*Power 3: A flexible statistical power analysis program for the social, behavioral, and biomedical sciences
Behavior Reearch Method 007, 39 (), 759 G*Power 3: A flexible tatitical power analyi program for the ocial, behavioral, and biomedical cience FRAZ FAUL ChritianAlbrechtUniverität Kiel, Kiel, Germany
More informationBiObjective Optimization for the Clinical Trial Supply Chain Management
Ian David Lockhart Bogle and Michael Fairweather (Editor), Proceeding of the 22nd European Sympoium on Computer Aided Proce Engineering, 1720 June 2012, London. 2012 Elevier B.V. All right reerved. BiObjective
More informationAdaptive Window Size Image Denoising Based on Intersection of Confidence Intervals (ICI) Rule
Journal of Mathematical Imaging and Viion 16: 223±235, 2002 # 2002 Kluwer Academic Publiher. Manufactured in The Netherland. Adaptive Window Size Image Denoiing Baed on Interection of Confidence Interval
More informationRISK MANAGEMENT POLICY
RISK MANAGEMENT POLICY The practice of foreign exchange (FX) rik management i an area thrut into the potlight due to the market volatility that ha prevailed for ome time. A a conequence, many corporation
More informationAuction Theory. Jonathan Levin. October 2004
Auction Theory Jonathan Levin October 2004 Our next topic i auction. Our objective will be to cover a few of the main idea and highlight. Auction theory can be approached from different angle from the
More informationName: SID: Instructions
CS168 Fall 2014 Homework 1 Aigned: Wedneday, 10 September 2014 Due: Monday, 22 September 2014 Name: SID: Dicuion Section (Day/Time): Intruction  Submit thi homework uing Pandagrader/GradeScope(http://www.gradecope.com/
More informationJanuary 21, 2015. Abstract
T S U I I E P : T R M C S J. R January 21, 2015 Abtract Thi paper evaluate the trategic behavior of a monopolit to influence environmental policy, either with taxe or with tandard, comparing two alternative
More informationSenior Thesis. Horse Play. Optimal Wagers and the Kelly Criterion. Author: Courtney Kempton. Supervisor: Professor Jim Morrow
Senior Thei Hore Play Optimal Wager and the Kelly Criterion Author: Courtney Kempton Supervior: Profeor Jim Morrow June 7, 20 Introduction The fundamental problem in gambling i to find betting opportunitie
More informationDistributed Monitoring and Aggregation in Wireless Sensor Networks
Ditributed Monitoring and Aggregation in Wirele Senor Network Changlei Liu and Guohong Cao Department of Computer Science & Engineering The Pennylvania State Univerity Email: {chaliu, gcao}@ce.pu.edu
More informationA COMPARATIVE STUDY OF THREEPHASE AND SINGLEPHASE PLL ALGORITHMS FOR GRIDCONNECTED SYSTEMS
A COMPARATIE STUDY OF THREEPHASE AND SINGLEPHASE PLL ALGORITHMS FOR GRIDCONNECTED SYSTEMS Ruben Marco do Santo Filho Centro Federal de Educação Tecnológica CEFETMG Coord. Eletrônica Av. Amazona 553 Belo
More informationSCM integration: organiational, managerial and technological iue M. Caridi 1 and A. Sianei 2 Dipartimento di Economia e Produzione, Politecnico di Milano, Italy Email: maria.caridi@polimi.it Itituto
More informationAccelerationDisplacement Crash Pulse Optimisation A New Methodology to Optimise Vehicle Response for Multiple Impact Speeds
AccelerationDiplacement Crah Pule Optimiation A New Methodology to Optimie Vehicle Repone for Multiple Impact Speed D. Gildfind 1 and D. Ree 2 1 RMIT Univerity, Department of Aeropace Engineering 2 Holden
More informationSELFMANAGING PERFORMANCE IN APPLICATION SERVERS MODELLING AND DATA ARCHITECTURE
SELFMANAGING PERFORMANCE IN APPLICATION SERVERS MODELLING AND DATA ARCHITECTURE RAVI KUMAR G 1, C.MUTHUSAMY 2 & A.VINAYA BABU 3 1 HP Bangalore, Reearch Scholar JNTUH, Hyderabad, India, 2 Yahoo, Bangalore,
More informationReport 46681b 30.10.2010. Measurement report. Sylomer  field test
Report 46681b Meaurement report Sylomer  field tet Report 46681b 2(16) Contet 1 Introduction... 3 1.1 Cutomer... 3 1.2 The ite and purpoe of the meaurement... 3 2 Meaurement... 6 2.1 Attenuation of
More informationHUMAN CAPITAL AND THE FUTURE OF TRANSITION ECONOMIES * Michael Spagat Royal Holloway, University of London, CEPR and Davidson Institute.
HUMAN CAPITAL AND THE FUTURE OF TRANSITION ECONOMIES * By Michael Spagat Royal Holloway, Univerity of London, CEPR and Davidon Intitute Abtract Tranition economie have an initial condition of high human
More informationOhm s Law. Ohmic relationship V=IR. Electric Power. Non Ohmic devises. Schematic representation. Electric Power
Ohm Law Ohmic relationhip V=IR Ohm law tate that current through the conductor i directly proportional to the voltage acro it if temperature and other phyical condition do not change. In many material,
More informationUsing Graph Analysis to Study Networks of Adaptive Agent
Uing Graph Analyi to Study Network of Adaptive Agent Sherief Abdallah Britih Univerity in Dubai, United Arab Emirate Univerity of Edinburgh, United Kingdom hario@ieee.org ABSTRACT Experimental analyi of
More informationAuctionBased Resource Allocation for Sharing Cloudlets in Mobile Cloud Computing
1 AuctionBaed Reource Allocation for Sharing Cloudlet in Mobile Cloud Computing ALong Jin, Wei Song, Senior Member, IEEE, and Weihua Zhuang, Fellow, IEEE Abtract Driven by pervaive mobile device and
More informationA New Optimum Jitter Protection for Conversational VoIP
Proc. Int. Conf. Wirele Commun., Signal Proceing (Nanjing, China), 5 pp., Nov. 2009 A New Optimum Jitter Protection for Converational VoIP Qipeng Gong, Peter Kabal Electrical & Computer Engineering, McGill
More informationMathematical Modeling of Molten Slag Granulation Using a Spinning Disk Atomizer (SDA)
Mathematical Modeling of Molten Slag Granulation Uing a Spinning Dik Atomizer (SDA) Hadi Purwanto and Tomohiro Akiyama Center for Advanced Reearch of Energy Converion Material, Hokkaido Univerity Kita
More informationINFORMATION Technology (IT) infrastructure management
IEEE TRANSACTIONS ON CLOUD COMPUTING, VOL. 2, NO. 1, MAY 214 1 BuineDriven Longterm Capacity Planning for SaaS Application David Candeia, Ricardo Araújo Santo and Raquel Lope Abtract Capacity Planning
More informationCHARACTERISTICS OF WAITING LINE MODELS THE INDICATORS OF THE CUSTOMER FLOW MANAGEMENT SYSTEMS EFFICIENCY
Annale Univeritati Apuleni Serie Oeconomica, 2(2), 200 CHARACTERISTICS OF WAITING LINE MODELS THE INDICATORS OF THE CUSTOMER FLOW MANAGEMENT SYSTEMS EFFICIENCY Sidonia Otilia Cernea Mihaela Jaradat 2 Mohammad
More informationPOSSIBILITIES OF INDIVIDUAL CLAIM RESERVE RISK MODELING
POSSIBILITIES OF INDIVIDUAL CLAIM RESERVE RISK MODELING Pavel Zimmermann * 1. Introduction A ignificant increae in demand for inurance and financial rik quantification ha occurred recently due to the fact
More information3.5 Practical measurement of ph in nonaqueous and mixed solvents
3.5 Practical meaurement of ph in nonaqueou and mixed olvent 3.5.1 Introduction Procedure analogou to thoe on which a practical ph cale for aqueou olution have been baed can be ued to etablih operational
More informationManaging Customer Arrivals in Service Systems with Multiple Servers
Managing Cutomer Arrival in Service Sytem with Multiple Server Chrito Zacharia Department of Management Science, School of Buine Adminitration, Univerity of Miami, Coral Gable, FL 3346. czacharia@bu.miami.edu
More informationChapter 10 Velocity, Acceleration, and Calculus
Chapter 10 Velocity, Acceleration, and Calculu The firt derivative of poition i velocity, and the econd derivative i acceleration. Thee derivative can be viewed in four way: phyically, numerically, ymbolically,
More informationFEDERATION OF ARAB SCIENTIFIC RESEARCH COUNCILS
Aignment Report RP/98983/5/0./03 Etablihment of cientific and technological information ervice for economic and ocial development FOR INTERNAL UE NOT FOR GENERAL DITRIBUTION FEDERATION OF ARAB CIENTIFIC
More informationUnobserved Heterogeneity and Risk in Wage Variance: Does Schooling Provide Earnings Insurance?
TI 011045/3 Tinbergen Intitute Dicuion Paper Unoberved Heterogeneity and Rik in Wage Variance: Doe Schooling Provide Earning Inurance? Jacopo Mazza Han van Ophem Joop Hartog * Univerity of Amterdam; *
More information