( ) α is determined to be a solution of the one-dimensional minimization problem: = 2. min = 2

Transcription

1 Homewo (Patal Solton) Posted on Mach, 999 MEAM 5 Deental Eqaton Methods n Mechancs. Sole the ollowng mat eqaton A b by () Steepest Descent Method and/o Pecondtoned SD Method Snce the coecent mat A s symmetc, solng the system o lnea eqaton s eqalent to the mnmzaton poblem: mn (), () A b Sppose that S s a poste dente mat. hen the steepest descent method s based on the teaton scheme: ( A b) ( ) ( ) ( ) ( ) α S whee ( ) α s detemned to be a solton o the one-dmensonal mnmzaton poblem: mn ( α ) ( ) ( ). Notng that ( ) ( ) ( ) ( ) ( ) α S ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ α S ] A[ α S ] [ α S ] b

2 whee the esdal ( ) ( ) A b s dened, we hae the necessay condton o the one-dmensonal mnmzaton poblem: α α ( ) ( ) ( ) S ( ) ( ) ( ) ( ) [ S ] α S ( ) ( ) ( ) [ S ] ( ) ( ) [ S ] A[ S ] ( ) ( ) ( ) ( ) [ ] A[ α S ] [ S ] ( ) [ ] A[ S ] b Sppose that A s not symmetc. hat s, we cannot se the nctonal we hae dened n aboe. We mst se the least sqae type nctonal: ( ) ( ) ( ) ( ) ( ) α S ( ) ( ) ( ) ( ) ( ) ( ) { A[ α S ] b} { A[ α S ] b} and then α α ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) { } { α } ( ) { } ( ) ( ) ( ) { } { } () Conjgate Gadent Method and/o Pe-condtoned CG Method

3 Conjgate gadent method s based on the teaton scheme ( ) ( ) ( ) ( ) α ( ) ( ) β S ( A b) whee the paametes ( ) ( ) α, β ae the soltons o the two-dmensonal mnmzaton poblem mn ( ), ( α β ) ( ) ( ) Notng that ( ) ( ) ( ) A ( ) ( b) ( A b) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( A( α d β S ) b) A α d β S ( b) whee ( ) ( ) ( ) ( ) A b and d s the esdal, we hae

4 β α β α β α β β α α () Gass Elmnaton Method sch as LU decomposton o sng MALAB Whee......, b A Snce the aboe poblem possesses a symmetc coecent mat, we shall consde a nsymmetc mat:

5 ......, b A Usng the ollowng MALAB pogam Homewo #()_99W MEAM 5 Wnte 999 set p the coecent mat A and the ght hand sde b Azeos(); bzeos(,); o : A(,); >, A(-,)-;, end <, A(,)-;, end end A(7,7);A(7,8)-;A(8,7)-;A(8,8);A(8,9)-;A(9,8)-;A(9,9);A(9,)-;A(,9)- ;A(,); A b[,,,,-.,-.,-.,-.,-.,-.]' set p a pecondtoned mat S

6 o : S(,)A(,); end S(A'A)/; Snn(S); steepest descent method omega; zeos(,); eo; teaton; toleance^-5; mate; eoh[]; ala[]; whle eo>toleance A*-b; A*Sn*; ala('*)/('*); -omega*ala*sn*; teatonteaton eosqt((-)'*(-))/sqt('*) eoh(teaton)eo; ala(teaton)ala; ; teaton>mate, bea, end end plot(eoh) label('teaton') ylabel('elate eo') ttle('conegence o Steepest Descent Method')

7 pase Conjgate Gadent Method omega; zeos(,); m; eo; teaton; toleance^-5; mate; eoh[]; ala[]; beta[]; whle eo>toleance A*-b; d-m; m; A*Sn*; A*d; CM['*,'*;'*,'*]; alabeta-pn(cm)*['*;'*]; alaalabeta(); betaalabeta();; ala*dbeta*sn*; teatonteaton eosqt((-)'*(-))/sqt('*) eoh(teaton)eo; ala(teaton)ala; beta(teaton)beta; ; teaton>mate, bea, end end

8 plot(eoh) label('teaton') ylabel('elate eo') ttle('conegence o Conjgate Gadent Method') Dect Method n(a)*b we shall sole the poblem as ollows: Conegence o Steepest Descent Method Conegence o Conjgate Gadent Method elate eo.6.5. elate eo teaton teaton At 5 th teaton, the steepest descent method podes the solton

9 At the 6 th teaton, the conjgate gadent method podes a solton whle the dect Gassan elmnaton schemes the solton

10 hs, yo shold wo ot o the case o symmetc coecents by denng the nctonal () A b.. Sole the ollowng nonlnea deental eqaton d d ( d d) d d sn ( π) n (,) wth the bonday condton () (), by sng the Newton method o moded Newton method, ate t wold be appomated by FDM, FEM, o weghted esdal methods. We shall dee the Newton scheme o the aboe nonlnea deental eqaton by gong bac to the pncple. o ths end, we shall appomate the appomaton o the solton: ( ) ( ) Sbstttng ths nto the nonlnea deental eqaton d d ( ) ( ) ( d d d d) d d ( ) ( ) d d ( ) ( ) ( ) sn( π) Neglectng the hghe ode tems than the lnea n ( ), we hae

11 d d d d sn ( π) ( ) d d ( ) ( d d) ( ) d d ( ) ( d d) ( ) ( ) ( ) ( d d) d d ( ) ( ) ( ) ( ) that s, the ncement ( ) o the appomaton s a solton o the deental eqaton d d d d ( ) ( ) ( d d) ( ) d d d d ( ) ( d d) ( ) ( d d) ( ) ( ) sn( π) d d ( ) ( ) ( ) ( ) Applyng the nte deence appomaton, we hae

12 π sn Reaangng ths om, we hae the nte deence eqaton o the ncement :

13 a a a a a a π sn whee Usng the MALAB pogam Homewo # : Poblem / hwp MEAM 5 Wnte 999 Newton's Method o a Nonlnea Deental Eqaton n; d/(n-); :d:;

14 zeos(n,); eo; eoh[]; toleance^-5; teaton; mate; KMzeos(n); bzeos(n,); KM(,); KM(n,n); whle eo>toleance o :n- dp(()-())/d; dm(()-(-))/d; ap-dp/(sqt(dp^))^/sqt(dp^); am-dm/(sqt(dm^))^/sqt(dm^); *()^; b()(dp/sqt(dp^))/d-(dm/sqt(dm^))/d-()^sn(p*()); KM(-,)-am/d^; KM(,)am/d^ap/d^; KM(,)-ap/d^; end KM(,);KM(n,n-); dkm b; d; teatonteaton; eosqt(d'*d)/('*); eoh(teaton)eo; teaton>mate, bea, end end

15 teaton eo plot(,) label('') ylabel('') ttle('solton o a Nonlnea Deental Eqaton') we hae conegence o the Newton's method: teaton 9 eo.669e-6 and the solton. Solton o a Nonlnea Deental Eqaton Fnd the best appomaton h H o a gen ncton () ( ) ep() n the Sobole

16 space { },,, L H V wth the nne podct ()() () () sn, d π whee () { },,,,,, R c c c c c c V H. Let the best appomaton H h o be epesented by { } N h 5 whle an abtay element H h be smlaly epesented by N h. hen the best appomaton solton s detemned by H h h h,, hs, we hae sn sn, sn, d d d h h N N NN K N N b K b N N N N π π π

17 hs means that the coecent can be obtaned by solng the mat eqaton K b whee K s a symmetc 55 mat, and b s a e component ecto. Usng the ollowng MAHEMAICA pogam N{,,^,^,^}; dnd[n,]; (-^)*Ep[]; dd[,]; a.5*sn[p*]; Kmable[,{,,5},{j,,5}]; bable[,{,,5}]; Do[b[[]]NIntegate[N[[]]*dN[[]]*a*d,{,-,}]; Do[Km[[,j]]NIntegate[N[[]]*N[[j]]dN[[]]*a*dN[[j]],{,-,}],{j,, 5}],{,,5}] LneaSole[Km,b]; hn. Plot[{h,},{,-,}] We hae and

18 I we consde the best appomaton by sng the L nne podct: ( ) { ()() }, d then the best appomaton becomes

19 It s clea that the L nne podct podes the appomaton wth lage appomaton eo n the semnom o the Sobole space V. Hee the semnom o the eo s dened by () d