Multilevel Iterative Methods

Transcription

1 Mltleel Iterate Methods Erc Yn Adsor: Professor Randy Bank, UCSD

2 Contents Introdcton.. 3 Basc Mltgrd Algorthm. Mltgrd fondatons 3. Mltgrd algorthm 4 3 Epermentaton 5 4 Algebrac Mltgrd (AMG) 4. Introdcton to AMG Implementng AMG Epermentaton Acknowledgements References 7

3 Introdcton Ths paper noles the analyss and stdy of mltleel terate methods, otherwse known as Mltgrd. Mltgrd s generally sed to appromate the solton of ellptc partal dfferental eqatons. The defnton of a partal dfferental eqaton (PDE) s the relatons nolng an nknown fncton of seeral ndependent arables and ts partal derates wth respect to those arables. [6] An ellptc PDE s a specfc branch of PDEs whch wll be descrbed n the followng secton. Whle there are many methods that can drectly sole PDEs (.e. ntegral transform, separaton of arables, and Green s fncton), real lfe problems are often too comple to be soled drectly. Ths appromaton methods sch as SSOR, ILU, Congate Gradents, and Mltgrd are sed. In ths paper, we dscss aros aspects of Mltgrd. We wll look at the adantages of Mltgrd (as opposed to the other appromaton methods), break down the aros components that make p Mltgrd, descrbe the Mltgrd algorthm as a whole, do some epermentaton, and then eamne Algebrac Mltgrd, a more comple araton of the basc Mltgrd. Backgrond. Mltgrd fondatons Mltgrd s an terate method that can be sed to sole A = f (.) A s a sqare matr, f s a ector termed the rght hand sde, and s a ector for whch we are tryng to sole. An terate method s defned as a method that attempts to sole a problem by fndng sccesse appromatons to the solton startng from an ntal gess. [6] Mltgrd s prmarly sed to sole ellptc partal dfferental eqatons. Ths branch of problems can be represented n general terms as: a(, + b(, + c(, + d(, + e(, + g(, f (, where b ac < 0 y yy y =, and Ω (the doman) and Ω (the bondares) are proded. There are other terate methods sed to sole ellptc PDEs bt the rate of conergence and cost of Mltgrd s what makes the algorthm sgnfcant. Below s a chart that depcts the aros appromaton methods and ther respecte spectral rads (whch measres the rate of conergence) and cost. 3

4 Method Cost Spectral rads Jacob / N N log N ch SOR/SSOR/ILU / N N log N ch Jacob-CG / N N log N SSOR-CG/ILU-CG / N N log N Mltgrd N c < Fg.. where d s the nmber of dmensons of the problem, N represents the sze of the problem, c s a constant whch ares by method, and h=/n (length of each step sze).. Mltgrd Algorthm There est seeral key components of Mltgrd. These are: smoothng, restrcton, and prolongaton. Pror to mplementng the Mltgrd algorthm, t s necessary to dscretze the problem frst. The dscretzaton of ellptc partal dfferental eqatons leads to solng a problem of the form (.). Ths matr problem s what Mltgrd wll be mplemented on. So the ery frst thng we wll dscss s how to dscretze a problem. Dscretzaton: Dscretzaton by defnton s the process of transferrng contnos models and eqatons nto dscrete conterparts. [6] Whle dscretzaton takes away from the contnty of the problem, the dea behnd t s that the ntal problem s well-posed: Condtons to beng well-posed:. A solton ests. The solton s nqe 3. If the data are changed only slghtly, then the resltng solton changes only slghtly (contnos dependence of solton on data). When sng a nmercal algorthm to appromate a solton for a problem t s mportant to determne f a problem s well-posed. Frst, f no solton ests then there s no pont n applyng the algorthm to the problem. Second, f mltple soltons ests, one mst determne whch compted solton s of nterest. Thrd, f the frst two condtons are satsfed bt the solton does not depend contnosly on the data, then the appromaton to the eact solton wll most lkely be of no ale. Ths s becase n appromaton technqes, the comptatons noled do not nole drectly solng the problem, bt nstead a problem that has been slghtly altered. The modfed problem may hae one solton bt that solton may be sgnfcantly dfferent than the tre solton of the orgnal problem f t does not depend contnosly on the data. There are aros methods of dscretzng a PDE. Some of the more common technqes nclde: 4

5 . Fnte dfferences. Fnte olmes 3. Fnte elements It wold be too mch to go nto eery sngle method of dscretzaton, so nstead, below s an eample of how to dscretze a problem sng fnte dfferences. Also nclded are salzatons of a problem sng fnte dscretzaton. Eample: Fnte dfferences dscretzaton We frst ntrodce the followng notaton: Ω s the doman n whch we are workng on. P =, y ) s a collecton of grd ponts for nteger ales of and wth ( = h, y = y k o + o + where ( o, yo ) s a chosen specfc pont and the stepszes h = and k = y are fed mesh wdths. (note: does not denote the Laplacan bt nstead a chosen stepsze for both the and y drecton respectel The dscrete pont set Ω h, k = { P = (, y ) = o + h, y = yo + k for all ntegers, wth P Ω} prodes a dscrete representaton of Ω. Let = (, y ) for (, y ) Ω h, k For a smooth fncton = (, and for any small nmber, take the Taylor epanson wth respect to to get: 3 ( +, = (, + (, + ( ) (, + ( ) (, + ϑ (( ) 3! 4 where ϑ (( ) ) refers to a bonded qantty: ϑ ( h) c h where c s a constant and n ths case, h = ( ). ϑ(h) s termed the trncaton error. By mong term s of the Taylor epanson arond to get the frst derate of wth respect to solated, we get the followng: (, = [ ( +, (, ] (, + ϑ ( ) 4 ) 5

6 or n grd notaton wth h = and replacng h (, y ) = [ +,, ] + ϑ( h) (.) h by we can get h (, y ) = [,, ] + + ϑ( h) (.3) h Sbtractng formlas (.) from (.3) and solatng : (, y ) [,,, ] ( h ) ϑ (.4) h = and sng a common stepsze h=k, a correspondng dscretzaton wth respect to y : yy (, y ) = [,,, ] ( h ) ϑ (.5) h Vsal eample: Fnte dfferences dscretzaton of, y ) + (, y ) ( yy Usng the resltng eqatons (.4) and (.5) and addng them together we can get the followng: (, y ) + yy (, y ) = [ +, +, +, + +, h 4 ] + ϑ(, h ) In the followng fgre, = a +, b =, c =, + d =, = e, 6

7 Fg.. As shown n Fgre., tlzng fnte dscretzaton, n order to get a better appromaton of at pont e for the problem (, y ) + yy (, y ) sng any ntal gess, t s necessary to mltply that pont by -4 and add t to the for srrondng ales arond t. Smoothng/Relaaton Process: Upon dscretzng the ntal problem, the problem then becomes one of solng a qeston of the form (.). Ths can be shown by the followng: Let s eamne From.4, (, y ) = f (, y ) = [ +,, +, ] + ϑ ( h ) = h f ( ) 7

8 For conenence, we wll drop off the h and ϑ ( h ) terms, so the problem now becomes:, y ) [ + ] = f ( ) ( = +,,, Frthermore, n ths problem, we are only workng n one-dmenson () so for frther smplcty, we can drop ot the y terms and ts correspondng ales: ) = [ + + ] = f ( ) (.6) ( (.6) only refers to one solated ale,, on the grd. So to represent ) = [ + + ] = f ( ) where n ( meanng there are n grd ponts, we get the followng system of lnear eqatons (n matr form): n n f f. =. f n f n Notce that for the frst and last eqatons, the frst term and last term respectely are dropped off. Ths s becase data ponts on those correspondng ales are ot of the bonds of the problems. In ths case, n (.) the matr A s the trdagonal matr shown aboe, and the ectors and f correspond to the and f ectors. We begn by talkng abot the frst part of the Mltgrd algorthm termed relaaton. Let s term are crrent appromaton as 0. Let s also pretend that we hae the real solton. These two ales lead to the followng relatonshp: error = 0 (Note: we wll only be sng 0 when dscssng the Mltgrd algorthm, (the eact solton) and the error are only assmed to be known n order to be tlzed as a fondaton for comparson) As a sal eample to help eplan the smoothng process, let s say or error takes on the followng sal form: 8

9 The bg thng that shold be notced s that there are two types of oscllatons n ths error, hgh freqency error and low freqency error. The smoothng process helps get rd of the hgh freqency error sng the followng technqe. Wth 0 (whch we are hopng wll eentally appromate n (.)), smoothng noles the followng terate process: r k Bδ k = b A k = = r k k + δ k k where A and f are from (.) and B s a smplfed factorzaton of A termed the smoother (note: the terms for the smoothng and relaaton are synonymos). The nmber of teratons sed n the smoothng process s p to the ser, bt generally - 4 smoothng processes are adeqate (ths wll be dscssed later). Eamples of aros smoothers: t! B = ( L + D) D ( D + U ) P AP! B = D L,D,U represent the lower tranglar matr, dagonal, and pper tranglar matr of A respectely. P represents a permtaton matr. Vsal eample of the smoothng/relaaton process The followng represents the error before and after smoothng. Fgre.3 s the ntal error, whle Fgre.4 represents the error after the ntal gess has been pdated wthn the smoothng process. 9

10 Fg..3 Notce that most of the hgh freqency oscllatons n the top fgre are gone n bottom fgre (after smoothng). The bottom fgre s essentally smoother. Bt now, the qeston arses as to how to get rd of the low freqency oscllatons. Restrcton: Mltgrd, mch lke ts name, deals wth mltple grds. How are these grds related? The dea behnd Mltgrd s that throgh each sccesse grd, the error s magnfed so yo are able to dentfy and remoe t at a faster rate. Note: Ths wll only work when 0

11 dealng wth low freqency, bt keep n mnd, the smoothng process s etremely effecte n dealng wth the hgh freqency error. Three man qestons arse n dealng wth the mltple grd aspect of Mltgrd:. What are the dfferent types of grds?. How do yo get from one grd to another (.e. what are ther relatonshps)? 3. How does ths get rd of the low freqency error? We wll frst deal wth qestons & and then se sal eamples to eplan qeston 3. Let s begn by defnng certan termnology sed n dealng wth the dfferent classes of grds. In Mltgrd, the dscretzaton grds ether hae more grd ponts or less grd ponts. No two sccesse grds wll hae the same nmber of grd ponts (note: the key word s sccesse). A grd s termed to be more fne f t has more grd ponts. Lkewse, the coarser grds hae less grd ponts. The nterpolaton matr called the restrcton matr helps trn a fne grd nto a coarser grd. The nterpolatng matr that brngs the coarse grd back nto the fne grd agan s called the prolongaton matr. The restrcton process (from fne to coarse) takes on the followng generalzed method (restrcton/prolongaton can be ery dffclt to eplan so the se of eamples, both sally and comptatonally, wll be demonstrated followng ts ntrodcton). Let s say the crrent appromaton to yo hae s termed r, where n ths case, r s a ector. Frst t mst be determned what ponts wll be coarse ponts and what ponts wll be fne ponts. Coarse ponts are data ponts whose crrent ales are ales that yo want to keep n the net grd. The fne ponts are ponts n the crrent grd that yo want to somehow nterpolate nto the coarse ponts so that ther ales are agely represented n the coarse ponts. In other words, to get a coarser grd whle mantanng smlar propertes, fne pont ales wll be combned nto coarse ales. Let r f refer to the determned fne-grd ponts and r c refer to the coarse grd ponts. So, rf r = (.7) rc (Keep n mnd,.7 s not to be taken lterally n that the coarse ponts are the bottom porton of the ector and the fne ponts are st the top part of the ector..7 s sed to represent that ales n can be dded nto coarse ponts and fne ponts.) The restrcton matr takes on the block form: t ( I ) ω (.8)

12 Where t ω s an nterpolant that wll somehow transfer ales of fne-grd ponts nto the coarse grd. Applyng the restrcton matr to the orgnal fne grd r, the problem becomes: ( ) c c f t r r r I ˆ = ω where c rˆ represents the newly created coarse grd that mmedate proceeds the fne grd r. Ths precedng eplanaton may be a bt age, so hopeflly, the followng eamples wll help clarfy the concepts behnd restrcton. Eample: -D restrcton to a generc ector of length 7 Let s say we want to restrct the ector r defned below. r = Restrcton matr = 4 Applyng the restrcton matr: = ˆ ˆ ˆ 4 Notce that now, nstead of hang 7 ponts n or ector, we now hae a coarser 3 pont ector. Prolongaton: Prolongaton s essentally the opposte of restrcton; t brngs the coarse grd back to the fne grd. Snce the prolongaton matr and the restrcton matr are not nerses we do not get back the same grd pont ales, nstead what we are gettng back s the orgnal

13 3 grd ponts. The prolongaton process smply takes the transpose of the restrcton matr mltpled by a constant to get: I c ω If yo are crrently on the coarse grd c û, applyng the coarse grd wth the prolongaton matr gets: = c f c I c ˆ ω where c s a real constant whch s dependent on the restrcton scheme that has been sed. (Mathematcans hae already chosen, for many problems, optmal restrcton matrces and ther correspondng prolongaton matr.) Prolongaton essentally plls ot the fne-pont data from the coarse-grd data whle mantanng the essental ales of the coarse ponts. Eample: -D prolongaton of a length 3 ector Let s say we want to prolongate the ector c û defned below: c û = 3 ˆ ˆ ˆ Prolongaton matr = Applyng the prolongaton matr: = ˆ ˆ ˆ Notce that we started wth only 3 ponts n or ector bt after prolongaton, the reslt was a ector of 7 ponts.

14 The followng fgres ge a pctoral descrpton of the prolongaton and restrcton processes n both -D and -D: -D Fg..5 4

15 -D Fg..6 We hae now answered the qestons of the dfferent types of grds and how to get from one grd to another, bt what s the pont of all ths? The qeston that stll remans s how ths gets rd of the low freqency error. We wll eplan ths sng a erbal eplanaton and a sal eplanaton. Verbal eplanaton: By coarsenng the fne grd, the low freqency error on the fne grd has been strategcally nterpolated nto hgh freqency error on the coarse grd. The key to dealng wth hgh freqency error s tlzng the smoothng process. Snce hgh freqency error wll contnally arse n the followng coarse grd pon nterpolatng each grd, yo need to apply the smoothng process. After smoothng on the fnal coarse grd, prolongaton then takes place n order to get back to the orgnal grd sze. Vsal eplanaton: Referrng to Fgre.5, the top pctre represents some arbtrary error n a fne grd. The bottom pctre represents the correspondng coarse grd after nterpolaton. Notce that 5

16 there are now only half as many ponts n the coarse grd, so the error s now n a hgher freqency, whch can be dealt wth by tlzng the smoothng process. Varos Mltgrd Schemes: There est mltple schemes n whch Mltgrd can be mplemented. These combnatons nole aratons n the nmber of restrcton and prolongaton phases desred and the pattern they take on. What ths means s that yo can choose f yo wanted to restrct, restrct agan, restrct agan, and once yo are at the desred coarse grd, yo prolongate, prolongate agan, prolongate agan ntl yo get back to the orgnal grd sze. There are also other patterns of restrcton and prolongaton. Yo can choose to restrct once and prolongate once, then restrct twce and prolongate twce, (note: Remember that followng each restrcton, smoothng takes place n order to get rd of the hgh freqency error) There are three man types of Mltgrd cycles: V-cycle, W-cycle, and Fll Mltgrd scheme. Whle the nmber of leels s an nddal s preference, the essental patterns le as follows: V-cycle: Fg..8 6

17 W-cycle: Fg.9 FMG-cycle: Fg..0 Now that we e dscssed all the elements of Mltgrd, let s look at some problems. 7

18 3 Problem Upon nderstandng how Mltgrd works, I stded the followng problem: One-dmensonal Posson s eqaton wth Drchlet bondary condtons = f n Ω =(0,n); ( 0) = ( n) = 0 I took on ths problem n two dfferent fashons:. A drect approach eamnng conergence drectly (programmng the algorthm nto Matlab). An analytcal approach eamnng conergence by proof. Approach : Usng Matlab, I programmed a -leel V-cycle sng the fnte dfferences dscretzaton scheme. My ntal gess was a ector of length n made p only of zeros and eact solton s defned as follows (both ntal gess and eact soltons were chosen randoml: ( ) = ( ) π sn( ) n + where =,,n Snce, = f (the problem we are solng), the ale of f sed n the problem can be compted by takng the nd derate of wth respect to. I sed n=300, meanng the problem had 300 steps where each step was of sze. The smoother sed was defned as: B = 4I I dd 4 dfferent eperments. In the frst, I sed one smoothng step per cycle, n the second eperment, I sed two smoothng steps per cycle, n the thrd, three smoothng steps per cycle, and n the forth, for smoothng steps per cycle. I dd 5 cycles for each eperment. The chart on the followng page shows the reslts I receed. Snce I cannot wrte ot the resltng appromate solton for each step of each eperment, I took the -norm of the resltng ector of each cycle. The -norm s a sngle nmber representaton of the oerall error. Net to the error s the conergence rate ( the speed at whch a conergent seqence approaches ts lmt, n ths case, the lmt of the error shold always be 0) of each step. [6] 8

19 Cycle -norm of the error Conergence factor (a) Cycle -norm of the error Conergence factor (b) Cycle -norm of the error Conergence factor (c) Cycle -norm of the error Conergence factor (d) a smoothng step b smoothng steps c 3 smoothng steps d 4 smoothng steps One nterestng obseraton s that for (a), the conergence rate was roghly arond a factor of 0.5. (b) had a conergence rate roghly arond 0.5. (c) had a conergence rate roghly arond 0.5. (d) had a conergence rate roghly arond Approach takes on a more analytcal approach to ths problem and wll proe that those ales are n fact the conergence rates for the correspondng nmber of smoothng steps. 9

20 Approach : Followng the reslts from Approach, I took a more analytcal approach n determnng the conergence of the same problem: One-dmensonal Posson s eqaton wth Drchlet bondary condtons = f n Ω =(0,n); ( 0) = ( n) = 0 (3.) The reason ths problem s sgnfcant s that t can be completely analyzed, meanng that t s able to completely represent how Mltgrd works (barrng comptatonal error). After dscretzaton, let In words, N = ( N + ) for some N and set h ( N + ). N refers to the sze of the dscretzed problem whereas = h represents the correspondng stepsze. We also se an arbtrary and N n order to show that ths problem can be completely analyzed meanng that the reslts that we obtan represent the solton for all (3.) regardless of sze, ntal gess, and tre solton. Net, we get that the matr A n (.) can be represented as: A = h [,, ] and the smoother beng sed s the damped Jacob scheme B = h 4 I We fnd that the egenales and egenectors of A are λ = h ( cos( πh )) and ( ψ ) = (h ) / sn( kπh ) respectely. k Wth respect to the egenector bass, we look for the transformed matrces  (transformed matr representng the dscretzaton), Bˆ (transformed smoothng matr), Rˆ (restrcton matr), Ŝ (error propagaton matr for the smoothng process), and Ĉ (proector for the coarse grd correcton). Here are the followng reslts:  = 4h 0 0 0

21 ˆ = 4h B 0 0 ( ) Rˆ = Sˆ = 0 0 Cˆ = ( ( )) / ( ( )) / S ˆ m / Cˆ Sˆ m / ( )) ( ( )) = m+ / m ( ( )) m ( ) m+ / where = cos( πh )) / ( / / After comptng ˆ m ˆ ˆ m S CS (the matr representng the oerall error of a -leel scheme), we eamne ts spectral rads ˆ m / ( ˆ ˆ m / ρ S CS ) to get an dea of ts conergence rate. m represents the nmber of desred smoothng steps. Nmber of smoothng steps ˆ m / ( ˆ ˆ m / ρ S CS ) ½ ¼ 3 /8 4 /6 4 Algebrac Mltgrd (AMG) 4. Introdcton to AMG One problem that consstently arses from geometrc Mltgrd (the type eplaned n secton ) s that t has certan restrctons when solng ellptc PDEs. The key problem to the geometrc case s that n order to mplement Mltgrd, the grd needs to be known. The qeston then arses, what f the grd s relately nknown? Algebrac Mltgrd (AMG) takes on the same essental concepts as the geometrc case descrbed aboe n regards to the need for ntalzaton, smoothng, prolongaton, and restrcton. The key

22 dfference comes wth knowledge of the grd. In the case of AMG, the grd s relately nknown. Instead of hang a set grd, the data ponts come n a random model as wll be shown later. The relatonshps among the nknowns are smlar to that of the geometrc case, bt the locatons are nknown. Becase of ths araton, there are essentally key components to AMG:. defnng the MG components (what s the restrcton matr, prolongaton matr, ). performng MG cycles (dong the actal Mltgrd) Let s pretend we want to sole an ellptc PDE wth the doman represented n fg. (4.). Notce that the grd s bascally random, bt the relatonshps between ponts are known. Fg. 4.

23 The frst thng we need to do s translate ths grd nto a matr format that can be soled. Fgre 4. defnes the correspondng matr. The way to obtan the matr s to mark the ales tochng the pont n qeston. So for fgre 4., pont s tochng ponts and 0, so on lne (representng pont ) of the matr, ales at,, and 0 are marked. Pont s tochng, 3, and 0, so lne (whch representng pont ) of the matr, ales at,, 3, and 0 are marked. Contne workng on ths process ntl all ponts hae been marked. Correspondng matr: A= Fg. 4. (Note: when dong the real problem, each represents some nmber; are only sed for smplcty of eplanng AMG) There s lttle theory n the feld of Algebrac Mltgrd de to the complety of the grd. The key to Mltgrd as was descrbed n secton, s smoothng, determnng the coarse ponts/coarse grds, determnng how to get to the coarse grds, and then how to get back to the orgnal grd. Intal fne grds, sch as the ones n fgre (.5) & (.6) hae coarse grds that can easly be determned, bt snce the key to AMG s that the grd s relately free n desgn, there s no set way of fndng a proper coarse grd. Determnng the coarse-grd ponts s qte open-ended. One method of determnng the coarse-grd ponts tlzes a concept called strong connectons. A second method s called Mamal Independent Set. There are other methods, all based on the sers preference, bt only these two wll be dscssed. 3

24 Method : Strong Connectons Note: I dd not stdy ths method (I only sed method ). The prpose of pttng ths n s to show that there are mltple methods n determnng the coarse grd bt nether can be proed to be more adantageos than the other (renforcng the reasonng as to why there s lttle theory behnd AMG) A strong connecton s defned as: S = { : a > θ ma a} where 0 < θ When basng the coarse-grd correcton on these strong connectons, there are man crtera:. For each F, each pont S shold ether be n C or shold be strongly connected to at least one pont n C.. C shold be a mamal sbset wth the property that no two C -ponts are strongly connected to each other. C refers to the coarse-grd arables, F refers to the fne-grd arables, and C refers to the set of nterpolatory coarse-grd arables sed to nterpolate fne-grd ales. In ths case, sometmes both enforcng both crtera s mpossble so crtera # s more mportant than crtera #. Method : Mamal Independent Set Mamal Independent Set reqres the satsfacton of the followng crtera:. No two coarse ponts are adacent to each other. The set of coarse ponts are adacent to all ponts n the gen doman. We now se Method (Mamal Independent Set) to determne the coarse ponts on fgre (4.). 4

25 Eample: Usng Mamal Independent Set on fgre (4.) Fg. 4.3 Once coarse-grd ponts are determned the matr A s re-ordered wth permtaton matrces so that all coarse grd ponts are blocked together as shown: the restrcton matr and prolongaton matr are now: A ff A t fc P ˆ AP ˆ = (4.) Acf Acc I 0 ω I, t I ω 0 I respectely. ω acts mch lke ω n (.8). A reqrement s for ω and fc A to hae the same sparsty pattern (note: sparsty means a matr poplated prmarly wth zeros, sch as a dagonal matr, trdagonal matr, ). [6] 5

26 4. Implementng AMG Followng the determnaton of all the reqred components, AMG s essentally the same as the geometrc case of Mltgrd. The smoothng process takes on the same form as the geometrc Mltgrd. Bt, n ths case, A s the permtated matr defned n (4.). r k = b A k = k k + δ k Bδ = r (4.3) k k Agan, B s some knd of factorzaton of A. Snce A s generally sparse, the se of the ncomplete LU factorzaton s poplar. Bt other technqes may sed sch as symmetrc Gass-Sedel. Followng the smoothng step, AMG takes on the same concepts as the geometrc case of Mltgrd, bt nstead t s sng the AMG components dscssed n secton 4.. For most problems nolng AMG, the determnaton of all the components necessary n Mltgrd s what generally takes the most amont of work. Ths s largely n part becase there s no set formla/theory for determnng the coarse grd and the correspondng nterpolatng matrces. 4.3 Epermentaton Prof. Randolph Bank deeloped Mltgraph [], whch comptes the soltons for ellptc PDEs sng AMG. Mltgraph tlzes the Mamal Independent Set for determnng coarse grd ponts. At ths pont, I am crrently sng Mltgraph to stdy aros PDEs wth the goal of tryng to nderstand trends, rates of conergence, accracy, sng AMG combned wth a nmber of aros optons (choosng the factorzaton for the smoother B n (4.3), choosng the dscretzaton method, ) The followng are some eamples of the se of AMG on aros problems, wth bref commentary: 6

27 Fgre 4.4 Fgre 4.5 Fgre 4.4 and 4.5 represent the reslts of the problem:. 00 = wth a doman that looks lke the shape of Teas (bondary condtons). On the left sde, Fgre 4.4 shows the matr solton pror to beng redstrbted to the orgnal Teas grd (the doman of Teas as opposed to the matr solton s analogos to the relatonshp between fg. 4. & 4.). The top rght shows the aros nmercal ales (a legend) for the matr and the bottom rght shows the top-down ew of the matr on the left. Fgre 4.5 s what has more sgnfcance. The left hand fgre represents the norm of the error (the ales on the left of the graph are based on the logarthmc scale). Each teraton s represented by the nddal pont, so n ths case, the problem reqred 3 teratons. The top rght chart represents the sparsty of the restrcton/prolongaton matrces (yellow), the crrent grd (cyan), and ther rato (prple). Ths mplctly represents the nmber of leels beng sed; snce there are 5 sectons, ths means that 5 yy 7

28 leels were sed. (Note: becase there are 5 leels, only 4 restrcton/prolongaton matrces were reqred so notce that the ery left hand sde bar only has the sparsty of the crrent grd.) The bottom rght pe chart represents the amont of tme reqred to assemble all the necessary components (red) and the amont of tme actally needed to compte the solton (ble). Whle ths pe chart s dependent on the compter processor that s beng sed, t s sgnfcant when beng compared to other problems soled on the same compter. Ths problem was soled sng an ncomplete LU-factorzaton for B (the smoother) and sed a 7-pont star dscretzaton scheme (ths s a araton of the fnte dfferences scheme). Fgre 4.6 Fgre 4.7 Fgre 4.6 & 4.7 took on the same problem as 4.4 & 4.5 bt ths tme t was soled n a dfferent manner. The factorzaton for B was the symmetrc Gass-Sedel scheme and the dscretzaton scheme was an absolte ale appled onto the 5-pont star method (another araton of the fnte dfferences scheme). Notce that ths tme, 5 teratons were reqred for conergence bt more teratons were necessary largely n part becase only 3 leels were sed (as opposed to the 5 leels sed 8

29 n the preos eperment). Also notce that n Fgre 4.6, the solton appears to be mch less comple than the solton n Fgre 4.4. Whle they are essentally the same soltons, there are sbtle dfferences (that are magnfed de to the aros colors sed). At ths pont, ths s what I hae been stdyng; I hae been tryng to get acqanted wth all the aratons that can be sed n AMG and n the process, tryng to fgre ot what prodces the optmal solton. 9

30 Acknowledgements Many thanks to Professor Bank for takng the tme to adse me n ths year long proect. The knowledge and eperence that he has gen me wll not only last a lfetme bt has also properly prepared me for the ftre. I trly apprecate the kndness he has demonstrated to me n agreeng to work on ths proect. References [] Mltgraph.0, R.E. Bank, Department of Mathematcs, UC San Dego. [] Wllam Brggs, Van Emden Henson, and Stee McCormck, A Mltgrd Ttoral, SIAM, 3600 Unersty Cty Scence Center, Phladelpha, PA [3] R.E. Bank and R.K. Smth, An algebrac mltleel mltgraph algorthm, SIAM J. On Scentfc Comptng. [4] R.E. Bank and R.K. Smth, Mltgraph algorthms based on sparse gassan elmnaton, n Thrteenth Internatonal Symposm on Doman Decomposton Methods for Partal Dfferental Eqatons, Doman Decomposton Press, Bergen. [5] R.E. Bank and C.G. Doglas, Sharp estmates for mltgrd rates of conergence wth general smoothng and acceleraton, SIAM J., Nmercal Analyss, pg , 985. [6] Internet resorce terate method, dscretzaton, partal dfferental eqatons, rate of conergence, sparse. 30