closest to the -direction.

% (¦

& ¥

To each mesh point , associate the closed disk of radius centered at .

$ # &

$ $

S6

¦ # 6

What is the smallest such that the family is a -ply system? ) £ $

0

3 3

& Answer the same question for the case where the radius is reduced to . What is the overlap

& c@

d

graph (and associated mesh) for any such that R

3

4

X d

R

12d 1 d

What about when ?

¦R d

11 Determine the cost of a level set expansion algorithm starting from distinct centers.

6

12 Write a FORTRAN subroutine (or C function) which implements the Recursive Graph Partition-

ing algorithm.

13 Write recursive versions of the Recursive Graph Partitioning algorithm and Recursive Spectral

Bisection algorithm. [Hint: Recall that a recursive program unit is a subprogram or function,

say foo, which calls itself, so foo is allowed to make a subroutine call to foo within its body.

Recursivity is not allowed in FORTRAN but is possible in C or C++.] (a) Give a pseudo-code

for the RGB algorithm which processes the subgraphs in any order. (b) Give a pseudo-code for

the RGB algorithm case when the larger subgraph is to be processed before the smaller one in

any dissection. Is this second version equivalent to Algorithm 13.9?

NOTES AND REFERENCES. To start with, the original paper by Schwarz is the reference [193], but

an earlier note appeared in 1870. In recent years, research on Domain Decomposition techniques has

been very active and productive. This rebirth of an old technique has been in large part motivated

by parallel processing. However, the ¬rst practical use of Domain Decomposition ideas has been in

applications to very large structures; see [166, 29], and elasticity problems; see, e.g., [169, 205, 198,

51, 28] for references.

—‘

™

˜ ˜ ‚ —

{ ˜ wU

7p

¡

§

x"

!

Two recent monographs that describe the use of Domain Decomposition approaches in struc-

tural mechanics are [143] and [87]. Recent survey papers include those by Keyes and Gropp [135]

and another by Chan and Matthew [50]. The recent volume [136] discusses the various uses of

“domain-based” parallelism in computational sciences and engineering.

The bulk of recent work on Domain Decomposition methods has been geared toward a Partial

Differential Equations viewpoint. Often, there appears to be a dichotomy between this viewpoint

and that of “applied Domain Decomposition,” in that the good methods from a theoretical point of

view are hard to implement in practice. The Schwarz multiplicative procedure, with multicoloring,

represents a compromise between good intrinsic properties and ease of implementation. For example,

Venkatakrishnan concludes in [215] that although the use of global coarse meshes may accelerate

convergence of local, domain-based, ILU preconditioners, it does not necessarily reduce the overall

time to solve a practical aerodynamics problem.

Much is known about the convergence of the Schwarz procedure; refer to the work by Widlund

and co-authors [30, 72, 73, 74, 46]. The convergence results of Section 13.3.4 have been adapted

from Xu [230] as well as Hackbusch [116]. The result on the equivalence between Schwarz and

Schur complement iterations stated in Theorem 13.1 seems to have been originally proved by Chan

and Goovaerts [48]. The results on the equivalence between the full matrix techniques and the Schur

matrix techniques seen in Section 13.5 have been adapted from results by S. E. Eisenstat, reported

in [135]. These connections are rather interesting and useful in practice since they provide some

¬‚exibility on ways to implement a method. A number of preconditioners have also been derived

using these connections in the PDE framework [32, 31, 33, 34, 35].

Research on graph partitioning is currently very active. So far, variations of the Recursive Spec-

tral Bisection algorithm [165] seem to give the best results in terms of overall quality of the sub-

graphs. However, the algorithm is rather expensive, and less costly multilevel variations have been

developed [119]. Alternatives of the same class as those presented in Section 13.6.4 may be quite

attractive for a number of reasons, including cost, ease of implementation, and ¬‚exibility; see [107].

There is a parallel between the techniques based on level set expansions and the ideas behind Voronoi

diagrams known in computational geometry. The description of the geometric partitioning techniques

in Section 13.6.2 is based on the recent papers [105] and [150]. Earlier approaches have been devel-

oped in [55, 56, 57].

‘

g …e

@ @ ˜ « ¢ $ ! r a `

guTSB‚XSES4s ’’ &%d s1da%!!%%¡i'a #!gWe d#! 1%!! %kI£¢ D$£df X d!#”e I£w` cd W!d &W„d££cd I)Qe¬IFE 7$7aC sd`v #a S¦guT¤@! UIg™Py&'fX w" £ f ! A X § ’¢ C§ v „ ¢ ¢

g X

¢ ! A IWfa X e $ Wv" u„!X¦If d %egX IH 6GFE! Cuvr FBE r ¨y a ‚FX ‚Cs¨aV I3d v7D4dªr@ { !y iI4 v7‘UH ¥(d¢&£#$! !% ¡W u! &—¢ ¥$m a ` sdr¤ %!§ '¦–ya ¢ e

' E$ m H v $ ' @ E © a ! ! ` $ a ! a

d%'#a&£d&— I #" ss¦–g&!‚‚¦ g% !s|D¨¦9¡ d#! 1%8 %kII$ p eI77#ad¦ug¢„¦"X

! §e

d%'#a&£d&— I #" ss¦–gp &!‚‚¦ g% !s|D¨¦9¡ d#! 1%8 %kII$ eI77#ad¦ug„y¢ d X

! a ¦¢ f $ "w dfX e Dj gd!i q!„ru a ¦e 'A%X{$ m f !$ ¨de{tr I!` #" gud"d ˜Qd¢&£#$! !% ¡W u! &—¢ ¥` $m a ` sdr¤ %!§ '¦¢a –a

r tr aw $ d ! ! — a ! ! $ a !

¢ d £WfX e £C„ACcf d X oI5Q’ $&%d !szig&( d!#” I !&d£ pQeI7$7a#ad`¦ug¢„¢£X X

w wvf ef e ¨t ) ¢ r t y ` ! § ¢ !¤ § y

¢ ££dfX £)W…Xd„d d !X f

gg eX

I6

e C0

‚HEFC ˜qi ‚ £$y %vBG" FE%Cm ‚XFE! I|v&`C •Vqr I$H vx FEgSH|@ DE Bq$ (" I!H o£y qSH4 Ia˜¢ 6 ˜ u7'§ ¦¢t pd¢¢ %a t gI —$ £D$"d"%!d a ig& ¢ A

" B C 0 6j v } P 0 ! t y

£$#! #”!% I ’% #!gs£Dg# %#(#”!% d3iy(dd!s#r q„§qc%I mA g7u§q¢ p ¢ p ¦"X

p ¨

vy

¤! r’&%d IQI$##£7$a’&%d £¡#X Wf£X #$e !If )d W„%!Aw !X fqd!e—#”¦& UTI#rC %# P’¦—rqm#!'˜g9¦` #Cu(Ig&#v"4 5˜ g5drd QF¢I© %$ I%g! #q§` dqI•! o¦f

! m$ ` ! ¢ … $ ‚ ¥ ! — #!g ` g ¢ j ! ¢ ¢

¢ ` $

vC 4

d¦#!¢` £`f #IIg` Wf8X e cf IW„ csXdr%!I'e ¦UHa –FEig˜ 8igI£(%V%@§I7 ! %a%T!0 A—7i…–#o#j6 I£¨$v`y uH(˜’6¢ o£¨H„G6¢ FEl¨uHS4 @9gIuHC 7$tQ%a d¢&%a! % %! gg¢§ s£$¢r D¥¦g

£ $ e a w £ a ¦ ` 6 I @& E x£ % ¦ } y v¨ ¢ Iv

a C " ¢

!d — i ¢gg e x h $ x $ ! he fp

… g $ ! t d¢#adag5lQp¦£IWfX khge %© gde#a g# #`k`¥m•k %a #j%g„t p

ecA u ¨d! ue` #" %$ &" Sd"de!! Q¢d gI#$ !$ " !W¡m !$ (g! # `q Iasdr%!'¦a –7!%a` !%7s%b # ¢ #j£` 7$$ ! $U5& § g%d !sr

— a ! m $ ! a a a a m a $ ¢

d!# 1%! %kI£Dc gI#$ Idd%oIl£#$! dsrd#!`&sr— d#!—`` ! ceI7% &%g¢ p ¥¢ w

yI¨uH¦˜6 ¦d¢dad!"#! !7%a8 7a¥IQI#$! #” I%7a— l I¢ A IWf#$X !¦gI&cAv`e —)„d AI£v dg!XQd &I&$e¢ H‚FEC !q#jig‚6¥¦v

v 4 } '0 r m $ ` j fI i ˜ § 6 ¢ g¢ ‘ ¢

` hH

…f e¤ i …

a ¢ ££dfX &!irgFe #``# Y3d c` ’ed da#!!%$! ¨© ˜`—D`‚—q£c%XW{&fsu# 7"g ` i $ h ! — ™ § e X ` $ r a

!a— ¦

¨¦I` k% {–£’d”D$§a Ig("` —" •gc£r¢ u”d¦!! ¨$ 1%!“d ’g"Y(—”gud"d˜! ’d¢e %a&%$d%!d"£gs% #”¢ !gt 'e Di#!a £—` ! `&‚#$ g ¢xxqf ¦Ue $ m–&r u`%d &§ ¢S£q¦¥¢ …

hsr § i t ! I ¢ n3 — ! ‘ j ¢ &¢ ! $ x ¢¤

h r

¢ d £WfX ¦v

ge

Da! yv H C 4I… „Ae …

m4 12 I I ` § g yq@&¤ C’ 6 X a ¢ 52¨ hx ¢

¢ i

¢£d%a% %!¨I&e suE UH %X£SXr $ u@pWB76D`5—!4 #`g`2 Dy!@—‚y £bi47v 7v‚$IlcgI%dH$ 6xc¨7@SHiBu@! iW@S4t gW…y i6$!—$d£|"y ¨{v IH F6&FE` ICx¨S4Hrv u @9QuH¦t¢ s(£%! £ £ xgUT –…„ IsH‚II%@Er 7V @uWB§q76pQ¥¢ 9¥&f

V H C 0 H 6 y ˜ P t ` ` ¢z ¦v x c vI¤I”YFEC a I£ ¢&h ¢

qIvsB FEgSH|@DEBqQIq£vqSH4 I6 }˜u7'¦0 # g|©7{!$—I$dIiWftg gXy kaIc… eA„¢ A %£f f ¦ ¨g‚Hc˜qi s‚%v GBFEq%C§ pIiCE¢ ‚XF ¨|v¥0¥¦e "… e A $e¢ 6 ry C

£d A £vdX ¨du4t %$ " I$ § § r § ¢ d g e § n e

¢w v $ !

p ¦’is7$#a`(C£dfX q¢ ¨£#!b #`#` 7j%a #$%`!%d p

Dy—@‚y£–vI¦ev •%I#H ”Xx So%Cn 9@ IXm l’kX%“$ I4 v7%a‘‚!@ H E#jyxiXD@ uhqIde4 ¦Dav G"—X 8@ FIdr4 #vg%dy! I84 6d #!UT ‚CgE` u&I gx#adev Ida@ UwgI%dC!FE 7v S4¢xC9@uf ¥sI%¢r 4XSS 9@C§q˜q¨I¥%6¢i B¥¢ d

y 4 6 … i „ 4 §£

X 6y t ed d¢a™ p h ¢

¦£IWfX dea7da%! c %!'a#I Q%¢IWV6&UTSCRQI6GFEDCB7A@98675231)(&%$8 #" ¨ ¨ ¨%¨¦¥¢£X

¢fg b ` Y X @ P H 4 0 ' ¢ ! ¢ © §¢¤

6 # &5" ¡"