Different General Algorithms for Solving Poisson Equation

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1 Dffeent Geneal Algotms fo Solvng Posson Equaton Me Yn SUMMARY Te objectve of ts tess s to dscuss te applcaton of dffeent geneal algotms to te soluton of Posson Equaton subject to Dclet bounday condton on a squae doman: u π u Γ 0 sn( π* xsn( π* y wee Ω x, y. I adopt two nds of gds, one wt a coase mes and one wt a fne mes. Accodng to dffeent needs, I dscuss bot te advantages and dsadvantages of tese algotms. Some of te deas mgt be novel and a few mpovements ae offeed. *Undelyng ( J denotes te numbe of unnowns, wee J equals 50 (J and 00 (J, espectvely. INTRODUCTION Fnte Dffeence Metod (FDM s a pmay numecal metod fo solvng Posson Equatons. As electonc dgtal computes ae only capable of andlng fnte data and opeatons, any numecal metod equng te use of computes must fst be dscetzed. Te steps ae as follows: Mes patton s employed and fnte gd nodes ae constucted on a contnuous egon. Dffeentaton opeatos ae dscetzed to tansfe te ognal poblem to fnte sets of lnea equatons. THE ALGORITHM u f (x y Consde Posson Equaton u Γ α(x y wee Ω x, y. Adopt a squae gd wt x and y mes lengts and daw a bunc of paallel lnes to te x * axes. Te ntesectons (, j ae denoted node (, j. y j* Employ second-ode cental dffeence along x and y dectons to substtute fo u xx and u yy, we ave at fve-pont fnte dffeence fomula: u + j u j + u j u j+ u j + u + u Fo smplfcaton, mae j f ( j u and f stand fo mes functons, wee u (x y j u j, f (x y f f (x y. Te above Posson Equaton s ten ewtten as u f. j j j Page

2 Fom Taylo Expanson, we now tat te cutoff eo of s O(. Hee s a setc map fo llustaton: At last, we can set up te appoxmate sets of lnea equatons: Wee L B I I B I O I B I s a (M- OOOOOOO KKKK I B 4 4 B 0 4 s also a (M- matx ( M /. OOOOOOO KKKK 4 L u f matx, I s a (M- dentty matx and As L s a uge spase matx and tee ae at most fve nonzeo elements on evey lne, t can be seen as a band matx wt alf band wdt of N and band wdt of N+. Restcted by compute memoy, t can not be nput by odnay means and MATLAB command "spase" s adopted. Undelyng s te constucton pocess of 50*50 mes: J50; /J; aspase(:(j-^,:(j-^,4*ones(,(j-^,(j-^,(j-^; aspase(j:j^-*j+,:j^-3*j+,-ones(,j^-3*j+,(j-^,(j-^; fo :J^-*J s(-; end fo :J- s(*(j-0; end %sub-dagonal zeo elements ae consdeed wen establsng s a3spase(:j^-*j+,:j^-*j,s,(j-^,(j-^; Aa+a+a'+a3+a3'; Now we tae a loo at ow to solve te sets of lnea equatons wt coeffcent matx A. Undelyng s te man pocess of Steepest Descent (SD Metod: wle (nom>tol Page

3 p A * ; alpa (nom^/(p'*; %fx alpa x x + alpa * ; %fom new teate b - A * x; tete+; ys(tenom; nomnom(; end And te man pocess of Conjugate Gadent (CG Metod: wle (nom>tol q A * p; alpa (nom^/(p'*q; %fx alpa x x + alpa * p; %fom new teate - alpa * q; beta (nom(^/(nom^; %fx beta ; p + beta * p; tete+; yc(tenom; nomnom(; end Wen A s mobd ( K >>, convegence s evolvng vey slowly. To tacle ts poblem, pecondton metod s ntoduced to educe A s condton numbe. Te smplest way s to coose a dagonal matx (Jacoban Pecondton, oweve t mgt not always be effectve. Mspase(:n,:n,0.5*ones(,n,n,n; zm*; pz; wle (nom>tol q A * p; alpa (z'*/(p'*q; %fx alpa x x + alpa * p; %fom new teate - alpa * q; zm*; beta (z'*/(z'*; %fx beta zz; ; p z + beta * p; tete+; yp(tenom; nomnom(; end Tee-dagonal matx M seems to be a nce coce, wc can be solved by Tomas Algotm. adag(dag(a; Page 3

4 adag(dag(a,-,-; Ma+a+a'; %tee-dagonal matx M Ax d b c a b c A O OOOO c n KKKKK a n b n d d d d d M M n n Pefom LU Decomposton of A, u b a l u u b lc.3...n Ax d s tus tansfomed nto Ly d and Ux y. ( u 0 y y d d l y.3...n x x n y u y n n cx u + n.n... It wll be even bette to coose M as te pecondton matx fo SSOR, wc wll educe A s condton numbe to ts squae oot. Especally n te case of w, symmetc GS teaton wll aceve good esults. atl(a; aspase(:n,:n,0.5*ones(,n,n,n; Sa*a; MS*S'; Two-gd metod s a vey popula paallel algotm tese days, wc seves to educe te amount of computaton needed to pocess te data. M u M v M L M u M f M d M Mv M d M Man Pncple: u + L wee v M s te modfe. Messmoot teaton s embedded n-between. (Te oveall sequence of teaton on fne mes, estcton, teaton on te coase mes, ntepolaton and ten fute teaton on te fne mes epesents one cycle of teaton. Geneal teaton fomula s adapted to ts poblem and computaton s fute educed. (J- lnes n te font and bac ae dscussed sepaately and ote lnes ae vewed as a wole. Besdes, sub-dagonal nonzeo elements ae also taen nto consdeaton. educe late computaton Page 4

5 compute (J- lnes n te font ote lnes ae vewed as a wole compute (J- lnes n te bac Pogammng used fo estcton and ntepolaton: Vectos ae tansfomed nto nodes on te gd, wc ae fst aayed ozontally and ten vetcally. Hee s a setc map fo llustaton: Defne estcton opeato I H and ntepolaton opeato I H : I H 6 4 fom fne mes to coase mes Page 5

6 I H 4 fom coase mes to fne mes (dstbuton accodng to wegt coeffcent 4 It s qute stagtfowad to desgn I H : fo ::J- fo j::j- spacng s enlaged u0.065*(a(+,j-+*a(+,j+a(+,j++*a(,j-+4*a(,j+*a(,j++a(-,j-+*a (-,j+a(-,j+; evaluate accodng to wegt coeffcent +; end end Wen dealng wt te desgn of I H, te stuaton becomes complcated and bounday and ntenal ponts need to be consdeed sepaately. Moeove, dffeent elatve postons between fne mes nodes and coase mes nodes wll lead to vaed wegt coeffcents. coase mes nodes fne mes nodes on dffeent ows and dffeent columns fne mes nodes on dffeent ows but same columns fne mes nodes on dffeent columns but same ows fou vetces fne mes nodes on same ows (fst and last column fne mes nodes on dffeent ows (fst and last column fne mes nodes on same columns (fst and last ow fne mes nodes on dffeent columns (fst and last ow mes nodes ae tansfomed nto vectos STRUCTURE PROGRAMMING Data Flow Gap Input Evaluate x Output nom>tol Steepest Descent (SD Metod: Intal value of x0 b Ax α ( (A Eo x x + α Page 6

7 Conjugate Gadent (CG Metod: Eo Intal value of x0 p b Ax α ( (p Ap p α Ap x x + α p + β p β ( ( Pecondton Conjugate Gadent (PCG Metod: Eo Intal value of x0 p b Ax z M α ( (p Ap p + β p z M α Ap x x + α p β (z (z z Page 7

8 Two-gd Metod: Relaxaton on fne mes Modfcaton on coase mes Restcton Eo Modfcaton on fne mes Intepolaton Relaxaton on coase mes Module Stuctue Man M-Fle PCG Algotm Tomas Algotm Two-gd Metod Intepolaton Restcton Sedel PCG Tomas RESULTS AND FURTHER DISCUSSION Caactestcs of te dffeence fomat: u j f j ( x y j D u j α j ( x y j D Suppose u s a functon defned on te doman D j D, we ave: u j 0, ( x y j D ten u j 0, ( x y j D ten max u max u j D D max u max u j D D j j α as a unque soluton, max u j max u j + max u j wee α s te x-sde lengt of D. D D D Page 8

9 Fve-pont fnte dffeence metod conveges wt max u u(x y O( + Numecal esults: (Intal value of x0ones(n, Iteaton Steps Steepest Descent J50 D j CG PCG Revsed PCG j Revsed PCG Two-gd Metod Iteaton Steps Steepest Descent J00 CG PCG Revsed PCG Revsed PCG Two-gd Metod Fo Steepest Descent (SD Metod, Solvng of ϕ ( x (Ax x (b x. Stat fom an abtay A * x b s equvalent to fndng te mnmum pont x and fnd te decton along wc ϕ(x ϕ(x deceases geatest,.e. ϕ( x b Ax. If 0, fnd α tat maes ϕ (x + α least n value. Ts epesents one cycle of teaton. It can be fute poved tat lm x > * * λ λ n * x A b x x A x x A λ + λ n 0.5 A (Au u u Wee λ and λ n ae A s maxmum and mnmum egenvalues, espectvely. Convegence s extemely slow wen λ >> λ n, so ts metod s not of pactcal use. CG s a vaatonal metod. It fst appeaed n te 950 s and ganed populaty wt te development of PCG n te 980 s and s now a pmay metod fo solvng uge spase matces.. In essence, t calls fo ϕ(x mn ϕ(y. Teoetcally, ts metod can fnd te ( y span{p...p } soluton afte at most n teaton steps. Howeve, because of te exstence of oundng eo, ts goal mgt not be aceved so easly. We ave * K * x x A x x A wee K K cond (A. + Wen A s mobd ( K >>, convegence s evolvng vey slowly and pecondton metod s ntoduced to educe A s condton numbe. T Consde Colesy Decomposton M LL and A M N, wee M s symmetc postve defnte and N s as nea zeo as possble. Te smplest way s to coose a dagonal matx (Jacoban Pecondton and my attaced PCG pogamme adopts ts dea. My Revsed PCG cooses tee-dagonal matx M as te Page 9

10 pecondton matx and my Revsed PCG cooses M as te pecondton matx fo SSOR T ( M SS 0.5 wees (D LD. Fo Two-gd Metod: Egenfuncton ϕ x sn(m π* x sn(m π* x m ( And egenvalue λ 0.5*(cos(m π * + cos(m π*. m Undelyng x denotes teaton steps and y denotes eo. J50 Pecson: 0.0 J50 Pecson: 0.00 J00 Pecson: 0.0 J00 Pecson: 0.00 J50 Pecson: 0.0 J50 Pecson: 0.00 Page 0

11 J00 Pecson: 0.0 J00 Pecson: 0.00 Pecson: 0.0 Pecson: 0.00 Multple Gd (MG Metod s nealy always te best and as been wdely used snce 977 fo solvng ellptcal bounday condton poblems. Hee s a setc map fo one cycle of MG teaton. Page

12 Undelyng s a compason table. Dmenson Metod Jacoban Gauss-Sedel SOR MG Two O N 4 O N 4 O N 3 O N Tee O N 5 O N 5 O N 4 O N 3 A LIST OF RELATED PROGRAMS spasepaametes 50*50 mes spasepaametes 00*00 mes steepestdescent Steepest Descent Metod cg CG Metod pcg PCG Metod (Jacoban Pecondton pcgg PCG Metod (tee-dagonal pecondton matx, solved by embedded system pogamme evsedpcg PCG Metod (tee-dagonal pecondton matx, solved by Tomas Algotm Tomas Tomas Algotm evsedpcg PCG Metod (pecondton matx fo SSOR gd Two-gd Metod Page

13 estct Restcton Algotm ntepolate Intepolaton Algotm sedel Messmoot Algotm REFERENCES Yang Huazong, Wang Hu Numecal Metods & C Language Scence Pess 996 Da Jazun, Qu Janxan Numecal Metods fo Dffeental Equatons Souteast Unvesty Pess 00 L Rongua, Feng Guocen Numecal Metods fo Dffeental Equatons People s Educaton Pess 980 L Qngyang, Guan Z, Ba Fengsan Numecal Computaton Tsngua Unvesty Pess 000 Tang Huamn, Hu Janwe Numecal Metods fo Dffeental Equatons Nana Unvesty Pess 990 Wang Moan MATLAB & Scentfc Computaton Publsng House of Electoncs Industy 004 Lu Jnfu, Guan Z Numecal Metods fo Patal Dffeental Equatons Tsngua Unvesty Pess 987 Guan Z, Lu Jnfu Intoducton to Numecal Analyss Hge Educaton Pess 998 Page 3