Waveform Transmission Method, A New Waveform-relaxation Based Algorithm. to Solve Ordinary Differential Equations in Parallel

Transcription

1 Waveform Transmission Mehod, A New Waveform-relaxaion Based Algorihm o Solve Ordinary Differenial Equaions in Parallel Fei Wei Huazhong Yang Deparmen of Elecronic Engineering, Tsinghua Universiy, Beijing, China Technical Repor Preprined a arxiv.org Absrac Waveform Relaxaion mehod (WR) is a disribued algorihm o solve Ordinary Differenial Equaions (ODEs). In his paper, we propose a new disribued algorihm, named Waveform Transmission Mehod (WTM), by virually insering waveform ransmission lines ino he dynamical sysem o achieve disribued compuing of ODEs. WTM is convergen o solve linear SPD ODEs.. Inroducion Waveform-relaxaion mehod was a daring aemp o solve ODEs exraced from circui by he ransien analysis []. WR maes use of a piece of waveform o ierae among he circuis, insead of a single value a one ime poin. Virual Transmission Mehod (VTM) is a new disribued algorihm o solve sparse linear sysems [3]. Is physical bacground is microwave newor and lossless ransmission line. If we replace each unnown of VTM by a piece of waveform, VTM is expanded ino a WR based algorihm, called Waveform Transmission Mehod (WTM). By marrying WR wih VTM, WTM is born. WTM is a disribued algorihm o solve sparse ODEs. The waveforms of WTM are ransferred in a bidireced way, similar o Gauss-Jacobi.. This paper is organized as follows. In Secion we define he Waveform Transmission Line (WTL). In Secion 3 we describe he basic seps of WTM. Secion 4 gives a simple example. Finally we conclude his paper in Secion 5.

2 . Waveform Transmission Line (WTL) Waveform Transmission Line (WTL) is exended from he virual ransmission line [4]. The mahemaical descripion of he virual ransmission line is shown in (.). The diagram of he virual ransmission line is shown in Fig.. (.) is called Transmission Delay Equaions. U( p) + ZI( p) = U( p ρ) ZI( p ρ) (.) U( p) + ZI( p) = U( p ρ) ZI( p ρ) where U ( p ) and U ( ) p represen he poenial of he wo pors of he virual ransmission line, respecively, while I ( p ) and I ( p ) represen he ouflow curren. p is he virual ime variable and ρ is he virual ransmission delay. Z is he characerisic impedance, which is posiive. Figure. The diagram of he virual ransmission line. If we replace each unnown of he ransmission line by a piece of waveform, we ge he Waveform Transmission Line (WTL). The mahemaical descripion of he WTL is shown in (.). The diagram of he WTL is shown in Fig.. (.) is called Waveform Transmission Delay Equaions. U(, p) + ZI () (, p) = U(, p ρ) ZI () (, p ρ) U(, p) + ZI () (, p) = U(, p ρ) ZI () (, p ρ) here is he physical ime variable. p is he virual ime variable. ρ is he virual ransmission delay. U (, p ) and (, ) U p,, represen he poenial waveforms of he wo pors of he waveform ransmission line a he virual ime p, respecively. I (, p ) and (, ) ouflow curren waveforms. () Z, I p,, represen he, is he characerisic impedance (.)

3 waveform, which should be posiive, i.e. Z ) > 0, [ T,. Z() could be considered as a precondiioner for WTM. ( Figure. The diagram of he Waveform Transmission Line If he delay ρ of all he WTLs are same, hen (.) could be simplified ino a discree ieraive form, as shown in (.3). Consequenly, (.) is he coninuous ieraive form of (.3). 3. Waveform Transmission Mehod U + ZI () = U ZI () U+ ZI () = U ZI () (.3) The mahemaic descripion of he linear ODEs is (3.). If boh C and A are Symmeric Posiive Definie (SPD), his ind of ODEs is called SPD ODEs. In his paper, we mainly focus on how o solve SPD ODEs. dx() C + A x = b, x(0) = x 0 (3.) d The basic sep of WTM is similar o VTM [3, 4].. Map he ODEs ino an elecric graph, which conains a weighed graph of C, a weighed graph of A, and b as he verex source vecor.. Se he verex spliing boundary, and perform he Elecric Verex Spliing (EVS) [3, 4].. Each verex on boundary is spli ino a pair of win verices.. The weighed graphs of C and A are elecrically spli along he boundary, as well as he verex source b..3 Add inflow currens ino he win verices. As he resul, he original elecric graph for ODEs is spli ino n subgraghs. 3. Add one WTL beween each pair of win verices. 4. Se he ime window [T, T ] for he waveform. Se he iniial waveforms for 3

4 he poenials and inflow currens of he win verices. 5. Locae each subgragh on a processor, and perform he disribued ieraion on n processors. Generally, we poin ha, if he elecric graph of an SPD ODEs (3.) is pariioned ino n subgraghs, and all hese subgraghs are SNND, hen for posiive characerisic impedance waveforms of he waveform ransmission lines, WTM converges o he soluion of he original sysem. This conclusion is similar o he convergence heory of VTM [3]. 4. Example Fig 3A is a simple capaciy-resisor newor, whose mahemaic descripion is shown in (4.). du() C + G u() = b d (4.) u(0) = u0 here C = 3, G =.5, u 0 = 0. Then, we spli his newor ino wo subgraghs by EVS, and inser a zero resisor, as shown in Fig 3B. (4.) and (4.3) are he mahemaic descripions for hese wo subgraghs, respecively. du() C + G u = b+ i() d (4.) u(0) = u0 here C =, G = 0.5. du() C + G u = b + i() d u(0) = u0 (4.3) here C =, G =. Afer ha, we replace he zero resisor by one WTL. For simpliciy, here we se he characerisic impedance waveform o be a consan waveform, Z ( ) =.5, T, T, T = 0, T =.0. u + Z i = u Z i u + Z i = u Z i (4.4) 4

5 Figure 3. Illusraion of he Elecric Verex Spliing. (A) The original capaciy-resisor newor. (B) Elecrically spli he newor, and a zero resisor is insered beween hem. C = C +C. G = G +G. b = b +b. (C) Replace he zero resisor by a waveform ransmission line. Laer, combine (4.) and (4.4), we ge he descripion for subgragh, as below: du C + G u = b+ i d u + Z i = u Z i (4.5) T, T (4.5) is hen simplified ino (4.6): 5

6 du C ( G Z ) u Z u i b d i = Z u + Z u i + + () = () () () + (4.6) Similarly, we combine (4.3) and (4.4), and ge he descripion for subgragh, as below: du C + G u = b + i() d u + Z i = u Z i (4.7) T, T Reforma (4.7) ino (4.8): du C + ( G + Z ) u = Z u i + b d i = Z u + Z u i T, T (4.8) Finally, we locae (4.6) on processor, and (4.8) on processor, hen do he disribued ieraion. Numerical experimens show ha WTM is convergen. To illusrae he error of WTM afer ieraions, we define he max error of waveform, as in (4.9) and (4.0). Fig 4 gives he convergen curve for his example. ( ) max _ err _ u = Max u u, T, T (4.9) ( ) max _ err _ u = Max u u, T, T (4.0) 6

7 5. Conclusion and Fuure Wor. Figure 4. Max error curve of waveforms. WTM is a new disribued algorihm o solve large ODEs. I is based on WR and VTM. If he propagaion delays of WTLs are differen, WTM would urn o an asynchronous algorihm, similar o DTM [5]. Experimens show ha WTM is convergen o solve SPD ODEs. However, is convergence speed and precision are no so impressive, compared o VTM. To speedup WTM, he characerisic impedance waveform Z() of WTL should be carefully seleced. Furher, WTM migh be used o solve he nonlinear ODEs exraced from large scale nonlinear dynamical sysems. Franly speaing, i is sill a doub wheher WTM would be a fas enough algorihm o be acceped by he applied mah indusry, or jus lie WR, beauiful bu impracical. Reference [] E. Lelarasmee, A. Ruehli, A. Sangiovanni-Vincenelli. The waveform relaxaion mehod for ime domain analysis of large scale inegraed circuis. TCAD, July 98. [] E. Lelarasmee. The waveform relaxaion mehod for he ime domain analysis of large scale nonlinear dynamical sysems. Ph.D. disseraion of UC Bereley. 98. [3] Fei Wei, Huazhong Yang. Virual Transmission Mehod, a new disribued algorihm o solve sparse linear sysems. NCM 008, and arxiv.org. [4] Fei Wei, Huazhong Yang. From devil o angel, ransmission lines boos parallel compuing of linear resisor newor, arxiv.org, 009. [5] Fei Wei, Huazhong Yang. Direced Transmission Mehod, a fully asynchronous approach o solve sparse linear sysems in parallel. SPAA 008, and arxiv.org. 7