'AGGR_THRESH'
real(
kind_parameter )
Any real
@@ -451,7 +454,7 @@ number
0.01
The threshold in the aggregation algorithm,
see (3 ) in Section 4.2 .
@@ -468,7 +471,7 @@ number
Note. Different thresholds at different levels, such as
those used in [25 , Section 5.1], can be easily set by
+ HREF="node36.html#VANEK_MANDEL_BREZINA">26, Section 5.1], can be easily set by
invoking the rou-
tine set with
@@ -483,7 +486,7 @@ the parameter ilev .
-
+
Table 5:
Parameters defining the coarse-space correction at the coarsest
@@ -590,7 +593,7 @@ Note that UMF and SLU require the coarsest
-
+
Table 6:
Parameters defining the coarse-space correction at the coarsest
@@ -609,7 +612,7 @@ level (continued).
Any integer
number
10
@@ -621,7 +624,7 @@ number Any integer
number
0
@@ -635,7 +638,7 @@ number Any real
number
0
@@ -656,7 +659,7 @@ number
-
+
Table 7:
Parameters defining the smoother or the details of the one-level preconditioner.
@@ -748,7 +751,7 @@ Parameters defining the smoother or the details of the one-level preconditioner.
Any integer
number
@@ -767,7 +770,7 @@ Parameters defining the smoother or the details of the one-level preconditioner.
Any integer
number
@@ -783,7 +786,7 @@ Parameters defining the smoother or the details of the one-level preconditioner.
-
+
Table 8:
Parameters defining the smoother or the details of the one-level preconditioner
@@ -840,7 +843,7 @@ Parameters defining the smoother or the details of the one-level preconditioner
Any integer
number
@@ -856,7 +859,7 @@ Parameters defining the smoother or the details of the one-level preconditioner
real(
kind_parameter )
Any real number
@@ -870,8 +873,7 @@ Parameters defining the smoother or the details of the one-level preconditioner
SRC="img83.png"
ALT="$p,t$">) factorization.
-
-
+
diff --git a/docs/html/node21.html b/docs/html/node21.html
index 8a1d324e..be357628 100644
--- a/docs/html/node21.html
+++ b/docs/html/node21.html
@@ -1,6 +1,6 @@
-
+
Method hierarchy_build
@@ -9,7 +9,7 @@
-
+
diff --git a/docs/html/node22.html b/docs/html/node22.html
index 0bb9ab16..d9fa00f9 100644
--- a/docs/html/node22.html
+++ b/docs/html/node22.html
@@ -1,6 +1,6 @@
-
+
Method smoothers_build
@@ -9,7 +9,7 @@
-
+
diff --git a/docs/html/node23.html b/docs/html/node23.html
index 445ce6f3..54375c9a 100644
--- a/docs/html/node23.html
+++ b/docs/html/node23.html
@@ -1,6 +1,6 @@
-
+
Method build
@@ -9,7 +9,7 @@
-
+
diff --git a/docs/html/node24.html b/docs/html/node24.html
index be5f014a..447807ca 100644
--- a/docs/html/node24.html
+++ b/docs/html/node24.html
@@ -1,6 +1,6 @@
-
+
Method apply
@@ -9,7 +9,7 @@
-
+
@@ -62,9 +62,9 @@ This method computes
, where , where is a previously built
diff --git a/docs/html/node25.html b/docs/html/node25.html
index da626cdc..23211355 100644
--- a/docs/html/node25.html
+++ b/docs/html/node25.html
@@ -1,6 +1,6 @@
-
+
Method free
@@ -9,7 +9,7 @@
-
+
diff --git a/docs/html/node26.html b/docs/html/node26.html
index 494e7369..f07bcf9f 100644
--- a/docs/html/node26.html
+++ b/docs/html/node26.html
@@ -1,6 +1,6 @@
-
+
Method descr
@@ -9,7 +9,7 @@
-
+
diff --git a/docs/html/node27.html b/docs/html/node27.html
index b4165d36..64b9a5db 100644
--- a/docs/html/node27.html
+++ b/docs/html/node27.html
@@ -1,6 +1,6 @@
-
+
Auxiliary Methods
@@ -9,7 +9,7 @@
-
+
diff --git a/docs/html/node28.html b/docs/html/node28.html
index ffbacc9b..813e8e25 100644
--- a/docs/html/node28.html
+++ b/docs/html/node28.html
@@ -1,6 +1,6 @@
-
+
Method: dump
@@ -9,7 +9,7 @@
-
+
diff --git a/docs/html/node29.html b/docs/html/node29.html
index 888861cf..da717d1c 100644
--- a/docs/html/node29.html
+++ b/docs/html/node29.html
@@ -1,6 +1,6 @@
-
+
Method: clone
@@ -9,7 +9,7 @@
-
+
diff --git a/docs/html/node3.html b/docs/html/node3.html
index a5e870ba..b7f7bd1d 100644
--- a/docs/html/node3.html
+++ b/docs/html/node3.html
@@ -1,6 +1,6 @@
-
+
General Overview
@@ -9,7 +9,7 @@
-
+
@@ -57,8 +57,8 @@ General Overview
PSBLAS (MLD2P4) provides parallel Algebraic MultiGrid (AMG) and Domain
Decomposition preconditioners (see, e.g., [3 ,23 ,21 ]),
+ HREF="node36.html#Stuben_01">24,22 ]),
to be used in the iterative solution of linear systems,
@@ -72,7 +72,7 @@ Ax=b,
where is a square, real or complex, sparse matrix.
The name of the package comes from its original implementation, containing
@@ -95,7 +95,7 @@ multilevel cycles and smoothers widely used in multigrid methods.
AMG cycles with smoothers and coarsest-level solvers. The V-, W-, and
K-cycles [3 ,19 ] are available, which allow to define
+ HREF="node36.html#Notay2008">20] are available, which allow to define
almost all the preconditioners in the package, including the multilevel hybrid
Schwarz ones; a specific cycle is implemented to obtain multilevel additive
Schwarz preconditioners. The Jacobi, hybrid
@@ -105,7 +105,7 @@ coarse-level matrices and operators, without explicitly using any information on
geometry of the original problem, e.g., the discretization of a PDE. To this end,
the smoothed aggregation technique [2 ,25 ]
+ HREF="node36.html#VANEK_MANDEL_BREZINA">26]
is applied. Either exact or approximate solvers can be used on the coarsest-level
system. Specifically, different sparse LU factorizations from external
packages, and native incomplete LU factorizations and Jacobi, hybrid Gauss-Seidel,
@@ -116,8 +116,7 @@ preconditioners.
MLD2P4 is written in Fortran 2003, following an
object-oriented design through the exploitation of features
such as abstract data type creation, type extension, functional overloading, and
-dynamic memory management.
-The parallel implementation is based on a Single Program Multiple Data
+dynamic memory management. The parallel implementation is based on a Single Program Multiple Data
(SPMD) paradigm. Single and
double precision implementations of MLD2P4 are available for both the
real and the complex case, which can be used through a single
diff --git a/docs/html/node30.html b/docs/html/node30.html
index e5cfa8fa..873b2304 100644
--- a/docs/html/node30.html
+++ b/docs/html/node30.html
@@ -1,6 +1,6 @@
-
+
Method: sizeof
@@ -9,7 +9,7 @@
-
+
diff --git a/docs/html/node31.html b/docs/html/node31.html
index a2044f73..1243f28c 100644
--- a/docs/html/node31.html
+++ b/docs/html/node31.html
@@ -1,6 +1,6 @@
-
+
Method: allocate_wrk
@@ -9,7 +9,7 @@
-
+
diff --git a/docs/html/node32.html b/docs/html/node32.html
index 66687d38..5805fd18 100644
--- a/docs/html/node32.html
+++ b/docs/html/node32.html
@@ -1,6 +1,6 @@
-
+
Method: free_wrk
@@ -9,7 +9,7 @@
-
+
diff --git a/docs/html/node33.html b/docs/html/node33.html
index a9e120a5..463a9879 100644
--- a/docs/html/node33.html
+++ b/docs/html/node33.html
@@ -1,6 +1,6 @@
-
+
Adding new smoother and solver objects to MLD2P4
@@ -9,7 +9,7 @@
-
+
diff --git a/docs/html/node34.html b/docs/html/node34.html
index 150cb8e1..a32c73cd 100644
--- a/docs/html/node34.html
+++ b/docs/html/node34.html
@@ -1,6 +1,6 @@
-
+
Error Handling
@@ -9,7 +9,7 @@
-
+
diff --git a/docs/html/node35.html b/docs/html/node35.html
index 5a7991e0..369bfe3a 100644
--- a/docs/html/node35.html
+++ b/docs/html/node35.html
@@ -1,6 +1,6 @@
-
+
License
@@ -9,7 +9,7 @@
-
+
diff --git a/docs/html/node36.html b/docs/html/node36.html
index bf1feef1..0504c1f0 100644
--- a/docs/html/node36.html
+++ b/docs/html/node36.html
@@ -1,6 +1,6 @@
-
+
Bibliography
@@ -9,7 +9,7 @@
-
+
@@ -138,53 +138,60 @@ S. Filippone, M. Colajanni,
PSBLAS: A Library for Parallel Linear Algebra
Computation on Sparse Matrices ,
ACM Transactions on Mathematical Software, 26 (4), 2000, 527-550.
-
16
+
16
+
+S. Gratton, P. Henon, P. Jiranek and X. Vasseur,
+ Reducing complexity of algebraic multigrid by aggregation ,
+Numerical Lin. Algebra with Applications, 2016, 23:501-518
+
+
+
17
W. Gropp, S. Huss-Lederman, A. Lumsdaine, E. Lusk, B. Nitzberg, W. Saphir, M. Snir,
MPI: The Complete Reference. Volume 2 - The MPI-2 Extensions ,
MIT Press, 1998.
-
17
+
18
C. L. Lawson, R. J. Hanson, D. Kincaid, F. T. Krogh,
Basic Linear Algebra Subprograms for FORTRAN usage ,
ACM Transactions on Mathematical Software, 5 (3), 1979, 308-323.
-
18
+
19
X. S. Li, J. W. Demmel,
SuperLU_DIST: A Scalable Distributed-memory
Sparse Direct Solver for Unsymmetric Linear Systems ,
ACM Transactions on Mathematical Software, 29 (2), 2003, 110-140.
-
19
+
20
Y. Notay, P. S. Vassilevski,
Recursive Krylov-based multigrid cycles ,
Numerical Linear Algebra with Applications, 15 (5), 2008, 473-487.
-
20
+
21
Y. Saad,
Iterative methods for sparse linear systems , 2nd edition, SIAM, 2003.
-
21
+
22
B. Smith, P. Bjorstad, W. Gropp,
Domain Decomposition: Parallel Multilevel Methods for Elliptic
Partial Differential Equations ,
Cambridge University Press, 1996.
-
22
+
23
M. Snir, S. Otto, S. Huss-Lederman, D. Walker, J. Dongarra,
MPI: The Complete Reference. Volume 1 - The MPI Core , second edition,
MIT Press, 1998.
-
23
+
24
K. Stüben,
An Introduction to Algebraic Multigrid ,
in A. Schüller, U. Trottenberg, C. Oosterlee, Multigrid,
Academic Press, 2001.
-
24
+
25
R. S. Tuminaro, C. Tong,
Parallel Smoothed Aggregation Multigrid: Aggregation Strategies on Massively Parallel Machines , in J. Donnelley, editor, Proceedings of SuperComputing 2000, Dallas, 2000.
-
25
+
26
P. Vanek, J. Mandel, M. Brezina,
Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems ,
diff --git a/docs/html/node37.html b/docs/html/node37.html
index 0eb0b5e6..657869fa 100644
--- a/docs/html/node37.html
+++ b/docs/html/node37.html
@@ -1,6 +1,6 @@
-
+
About this document ...
@@ -9,7 +9,7 @@
-
+
@@ -47,7 +47,7 @@ About this document ...
This document was generated using the
-LaTeX 2HTML translator Version 2017.2 (Released Jan 23, 2017)
+LaTeX 2HTML translator Version 2018 (Released Feb 1, 2018)
Copyright © 1993, 1994, 1995, 1996,
Nikos Drakos,
@@ -60,7 +60,7 @@ Mathematics Department, Macquarie University, Sydney.
The command line arguments were:
latex2html -local_icons -noaddress -dir ../../html userhtml.tex
-The translation was initiated on 2018-01-27
+The translation was initiated on 2018-05-14
diff --git a/docs/html/node4.html b/docs/html/node4.html
index 7b2ac12c..c478e331 100644
--- a/docs/html/node4.html
+++ b/docs/html/node4.html
@@ -1,6 +1,6 @@
-
+
Code Distribution
@@ -9,7 +9,7 @@
-
+
@@ -80,7 +80,7 @@ constant
-->
diff --git a/docs/html/node5.html b/docs/html/node5.html
index 3e5ba8d6..a15ea834 100644
--- a/docs/html/node5.html
+++ b/docs/html/node5.html
@@ -1,6 +1,6 @@
-
+
Contributors
@@ -9,7 +9,7 @@
-
+
diff --git a/docs/html/node6.html b/docs/html/node6.html
index 8c177d8c..54be0f4b 100644
--- a/docs/html/node6.html
+++ b/docs/html/node6.html
@@ -1,6 +1,6 @@
-
+
Configuring and Building MLD2P4
@@ -9,7 +9,7 @@
-
+
diff --git a/docs/html/node7.html b/docs/html/node7.html
index 13445741..7eed7f21 100644
--- a/docs/html/node7.html
+++ b/docs/html/node7.html
@@ -1,6 +1,6 @@
-
+
Prerequisites
@@ -9,7 +9,7 @@
-
+
@@ -59,7 +59,7 @@ Prerequisites
[11 ,12 ,17 ] Many vendors provide optimized versions
+ HREF="node36.html#blas1">18] Many vendors provide optimized versions
of BLAS; if no vendor version is
available for a given platform, the ATLAS software
(
MPI
[16 ,22 ] A version of MPI is available on most
+ HREF="node36.html#MPI2">17 ,23 ] A version of MPI is available on most
high-performance computing systems.
diff --git a/docs/html/node8.html b/docs/html/node8.html
index f1b81928..b008d3ae 100644
--- a/docs/html/node8.html
+++ b/docs/html/node8.html
@@ -1,6 +1,6 @@
-
+
Optional third party libraries
@@ -9,7 +9,7 @@
-
+
@@ -92,7 +92,7 @@ for multilevel preconditioners may change to reflect their presence.
SuperLU_Dist
[18 ]
+ HREF="node36.html#SUPERLUDIST">19]
A sparse LU factorization package available
from the same site as SuperLU; it provides parallel factorization and
triangular system solution for double precision real and complex data.
diff --git a/docs/html/node9.html b/docs/html/node9.html
index df5e4663..212118d8 100644
--- a/docs/html/node9.html
+++ b/docs/html/node9.html
@@ -1,6 +1,6 @@
-
+
Configuration options
@@ -9,7 +9,7 @@
-
+
diff --git a/docs/html/userhtml.html b/docs/html/userhtml.html
index f026ce9d..1f0d2ee7 100644
--- a/docs/html/userhtml.html
+++ b/docs/html/userhtml.html
@@ -1,6 +1,6 @@
-
+
userhtml
@@ -9,7 +9,7 @@
-
+
@@ -69,17 +69,15 @@ based on PSBLAS
-Software version: 2.1
+Software version: 2.2
-July 31, 2017
+July 31, 2018
-
-
-
+
diff --git a/docs/mld2p4-2.1-guide.pdf b/docs/mld2p4-2.2-guide.pdf
similarity index 94%
rename from docs/mld2p4-2.1-guide.pdf
rename to docs/mld2p4-2.2-guide.pdf
index 859ae4c5..29db2a71 100644
--- a/docs/mld2p4-2.1-guide.pdf
+++ b/docs/mld2p4-2.2-guide.pdf
@@ -24,7 +24,7 @@ BT
/F17 11.9552 Tf 218.644 -79.389 Td [(P)31(asqua)-375(D'Am)31(bra)]TJ/F37 11.9552 Tf -22.655 -13.947 Td [(IA)27(C-CNR,)-326(Naples,)-326(Italy)]TJ/F17 11.9552 Tf 11.494 -29.39 Td [(Daniela)-375(di)-375(Sera\014no)]TJ/F37 11.9552 Tf -181.63 -13.948 Td [(Univ)27(ersit)27(y)-326(of)-326(Campania)-326(\134Luigi)-327(V)82(an)27(vitelli",)-326(Caserta,)-326(Italy)]TJ/F17 11.9552 Tf 179.561 -29.39 Td [(Salv)62(atore)-375(Filipp)-31(one)]TJ/F37 11.9552 Tf -134.787 -13.947 Td [(Cran\014eld)-326(Univ)27(ersit)27(y)82(,)-326(Cran\014eld,)-327(United)-326(Kingdom)]TJ
0 g 0 G
0 g 0 G
- 141.76 -78.924 Td [(Soft)27(w)28(are)-327(v)27(ersion)1(:)-436(2.1)]TJ 38.924 -13.948 Td [(July)-326(31,)-327(2017)]TJ
+ 141.76 -78.924 Td [(Soft)27(w)28(are)-327(v)27(ersion)1(:)-436(2.2)]TJ 38.924 -13.948 Td [(July)-326(31,)-327(2018)]TJ
0 g 0 G
0 g 0 G
ET
@@ -722,11 +722,11 @@ BT
0 g 0 G
[(,)]TJ
1 0 0 rg 1 0 0 RG
- [-269(23)]TJ
+ [-269(24)]TJ
0 g 0 G
[(,)]TJ
1 0 0 rg 1 0 0 RG
- [-268(21)]TJ
+ [-268(22)]TJ
0 g 0 G
[(]\051,)-281(to)-269(b)-27(e)-269(used)-268(in)-268(the)-269(iterativ)28(e)]TJ 0 -13.549 Td [(solution)-333(of)-334(lin)1(e)-1(ar)-333(systems,)]TJ/F22 10.9091 Tf 186.98 -16.123 Td [(Ax)]TJ/F15 10.9091 Tf 17.447 0 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(b;)]TJ
0 g 0 G
@@ -738,7 +738,7 @@ BT
0 g 0 G
[(,)]TJ
1 0 0 rg 1 0 0 RG
- [-263(19)]TJ
+ [-263(20)]TJ
0 g 0 G
[(])]TJ 0 -13.549 Td [(are)-392(a)28(v)56(ailable,)-407(whic)28(h)-392(allo)28(w)-392(to)-391(de\014ne)-392(almost)-392(all)-391(the)-392(preconditioners)-392(in)-392(th)1(e)-392(pac)28(k)55(age,)-406(in-)]TJ 0 -13.549 Td [(cluding)-394(the)-394(m)28(ultilev)28(el)-394(h)28(ybrid)-394(Sc)28(h)27(w)28(arz)-394(ones;)-424(a)-394(sp)-28(eci\014c)-394(cycle)-394(is)-394(implemen)28(ted)-394(to)-394(obtain)]TJ 0 -13.55 Td [(m)28(ultilev)28(el)-498(additiv)28(e)-498(Sc)28(h)28(w)28(arz)-498(preconditioners.)-937(The)-497(Jacobi,)-539(h)28(ybrid)-498(forw)28(ard/bac)28(kw)28(ard)]TJ 0 -13.549 Td [(Gauss-Seidel,)-366(blo)-27(c)27(k-Jacobi,)-365(and)-359(additiv)28(e)-360(Sc)28(h)28(w)28(arz)-359(m)-1(eth)1(o)-28(ds)-359(are)-360(a)28(v)56(ailable)-359(as)-360(smo)-27(others.)]TJ 0 -13.549 Td [(An)-279(algebraic)-279(appr)1(oac)27(h)-279(i)1(s)-279(used)-279(to)-279(generate)-279(a)-279(hierarc)28(h)28(y)-279(of)-279(coarse-lev)28(el)-279(matrices)-279(and)-278(op)-28(er-)]TJ 0 -13.549 Td [(ators,)-283(without)-270(explicitly)-270(using)-270(an)28(y)-271(inf)1(ormation)-271(on)-270(the)-270(geometry)-270(of)-270(the)-271(original)-270(problem,)]TJ 0 -13.549 Td [(e.g.,)-256(the)-237(discretization)-237(of)-237(a)-237(PDE.)-237(T)84(o)-237(this)-237(end,)-256(the)-237(smo)-28(othed)-237(aggregation)-237(tec)28(hnique)-237([)]TJ
1 0 0 rg 1 0 0 RG
@@ -746,7 +746,7 @@ BT
0 g 0 G
[(,)]TJ
1 0 0 rg 1 0 0 RG
- [-236(25)]TJ
+ [-236(26)]TJ
0 g 0 G
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1 0 0 rg 1 0 0 RG
@@ -908,7 +908,7 @@ BT
0 g 0 G
[(,)]TJ
1 0 0 rg 1 0 0 RG
- [-424(17)]TJ
+ [-424(18)]TJ
0 g 0 G
[(])-425(Man)28(y)-425(v)28(endors)-424(pro)27(v)1(ide)-425(optimized)-425(v)28(ersions)-425(of)-424(BLAS;)-425(if)-424(no)-425(v)28(endor)]TJ -11.105 -13.55 Td [(v)28(ersion)-523(is)-524(a)28(v)56(ailable)-523(for)-523(a)-524(giv)28(en)-523(platform,)-570(the)-524(A)84(TLAS)-523(soft)28(w)27(are)-523(\050)]TJ
0 1 0 0 k 0 1 0 0 K
@@ -928,11 +928,11 @@ BT
0 g 0 G
/F15 10.9091 Tf 30.697 0 Td [([)]TJ
1 0 0 rg 1 0 0 RG
- [(16)]TJ
+ [(17)]TJ
0 g 0 G
[(,)]TJ
1 0 0 rg 1 0 0 RG
- [-340(22)]TJ
+ [-340(23)]TJ
0 g 0 G
[(])-340(A)-341(v)28(ersion)-340(of)-340(MPI)-340(is)-341(a)28(v)56(ailable)-341(on)-340(most)-340(high-p)-28(erformance)-340(computing)-340(sys-)]TJ -3.424 -13.549 Td [(tems.)]TJ
0 g 0 G
@@ -1018,7 +1018,7 @@ BT
0 g 0 G
/F15 10.9091 Tf 28.388 0 Td [([)]TJ
1 0 0 rg 1 0 0 RG
- [(18)]TJ
+ [(19)]TJ
0 g 0 G
[(])-417(A)-416(sparse)-417(LU)-416(factorization)-417(pac)28(k)55(age)-416(a)28(v)55(ailable)-416(from)-417(the)-417(same)-416(site)]TJ -55.002 -13.549 Td [(as)-366(Sup)-28(erLU;)-366(it)-367(pro)28(vides)-366(parallel)-366(factorization)-367(and)-366(triangular)-366(system)-367(solution)-366(for)]TJ 0 -13.549 Td [(double)-384(precision)-384(real)-384(and)-384(complex)-384(data.)-596(W)83(e)-384(tested)-384(v)28(ersions)-384(3.3)-384(and)-384(4.2.)-596(If)-384(y)28(ou)]TJ 0 -13.549 Td [(installed)-384(BLAS)-384(f)1(rom)-384(A)83(TLAS,)-384(remem)28(b)-28(er)-384(to)-383(de\014ne)-384(the)-384(BLASLIB)-384(v)56(ariable)-384(in)-384(the)]TJ 0 -13.549 Td [(mak)28(e.inc)-401(\014le)-401(and)-401(to)-401(add)-402(t)1(he)]TJ/F45 10.9091 Tf 142.011 0 Td [(-std=c99)]TJ/F15 10.9091 Tf 50.193 0 Td [(option)-401(to)-401(the)-401(C)-401(compiler)-401(options.)-648(Note)]TJ -192.204 -13.55 Td [(that)-432(this)-432(library)-432(requires)-433(t)1(he)-433(P)28(arMETIS)-432(library)-432(for)-432(parallel)-432(graph)-432(partitioning)]TJ 0 -13.549 Td [(and)-374(\014ll-reducing)-374(matrix)-374(ordering,)-385(a)28(v)56(ailable)-374(from)]TJ
0 1 0 0 k 0 1 0 0 K
@@ -1891,11 +1891,11 @@ BT
0 g 0 G
[(,)]TJ
1 0 0 rg 1 0 0 RG
- [-313(23)]TJ
+ [-313(24)]TJ
0 g 0 G
[(,)]TJ
1 0 0 rg 1 0 0 RG
- [-313(21)]TJ
+ [-313(22)]TJ
0 g 0 G
[(])]TJ 0 -13.549 Td [(for)-333(details.\051)]TJ 16.937 -14.105 Td [(MLD2P4)-329(uses)-330(a)-329(pure)-330(algebraic)-329(approac)28(h,)-331(b)1(as)-1(ed)-329(on)-329(the)-330(smo)-28(othed)-329(aggregation)-330(algo-)]TJ -16.937 -13.549 Td [(rithm)-298([)]TJ
1 0 0 rg 1 0 0 RG
@@ -1903,11 +1903,11 @@ BT
0 g 0 G
[(,)]TJ
1 0 0 rg 1 0 0 RG
- [-298(25)]TJ
+ [-298(26)]TJ
0 g 0 G
[(],)-305(for)-298(building)-298(the)-298(sequence)-298(of)-298(coarse)-298(matrices)-298(and)-298(transfer)-298(op)-28(erators,)-305(start-)]TJ 0 -13.549 Td [(ing)-306(from)-306(the)-305(original)-306(one.)-435(A)-306(decoupled)-306(v)28(ersion)-306(of)-306(this)-306(algori)1(thm)-306(is)-306(implemen)28(ted,)-312(where)]TJ 0 -13.55 Td [(the)-316(smo)-28(othed)-316(aggregation)-315(is)-316(applied)-316(lo)-28(cally)-316(to)-316(eac)28(h)-316(submatrix)-316([)]TJ
1 0 0 rg 1 0 0 RG
- [(24)]TJ
+ [(25)]TJ
0 g 0 G
[(].)-438(A)-316(brief)-316(description)]TJ 0 -13.549 Td [(of)-333(the)-333(AMG)-333(prec)-1(on)1(ditioners)-334(implemen)28(ted)-333(in)-333(MLD2P4)-333(is)-333(giv)27(en)-333(in)-333(Sections)]TJ
0 0 1 rg 0 0 1 RG
@@ -1955,7 +1955,7 @@ BT
0 g 0 G
-398.511 -19.516 Td [(where)]TJ/F22 10.9091 Tf 32.496 0 Td [(A)]TJ/F15 10.9091 Tf 12.796 0 Td [(=)-423(\050)]TJ/F22 10.9091 Tf 17.342 0 Td [(a)]TJ/F23 7.9701 Tf 5.767 -1.636 Td [(ij)]TJ/F15 10.9091 Tf 7.265 1.636 Td [(\051)]TJ/F25 10.9091 Tf 8.857 0 Td [(2)]TJ/F34 10.9091 Tf 11.888 0 Td [(R)]TJ/F23 7.9701 Tf 7.878 3.959 Td [(n)]TJ/F26 7.9701 Tf 5.139 0 Td [(\002)]TJ/F23 7.9701 Tf 6.586 0 Td [(n)]TJ/F15 10.9091 Tf 10.223 -3.959 Td [(is)-421(a)-420(nonsingular)-420(sparse)-421(matrix;)-464(for)-420(ease)-421(of)-420(presen)28(tation)-421(w)28(e)]TJ -126.237 -13.549 Td [(assume)]TJ/F22 10.9091 Tf 37.697 0 Td [(A)]TJ/F15 10.9091 Tf 11.818 0 Td [(has)-333(a)-334(symmetric)-333(sparsit)28(y)-334(pattern.)]TJ -32.579 -13.549 Td [(Let)-468(us)-469(consider)-468(as)-468(\014nest)-469(ind)1(e)-1(x)-468(space)-468(the)-468(s)-1(et)-468(of)-468(ro)28(w)-469(\050column\051)-468(indices)-468(of)]TJ/F22 10.9091 Tf 361.856 0 Td [(A)]TJ/F15 10.9091 Tf 8.182 0 Td [(,)-502(i.e.,)]TJ -386.974 -13.55 Td [(\012)-377(=)]TJ/F25 10.9091 Tf 24.593 0 Td [(f)]TJ/F15 10.9091 Tf 5.454 0 Td [(1)]TJ/F22 10.9091 Tf 5.455 0 Td [(;)]TJ/F15 10.9091 Tf 4.848 0 Td [(2)]TJ/F22 10.9091 Tf 5.455 0 Td [(;)-167(:)-166(:)-167(:)-167(;)-166(n)]TJ/F25 10.9091 Tf 30.79 0 Td [(g)]TJ/F15 10.9091 Tf 5.455 0 Td [(.)-623(An)28(y)-393(algebraic)-393(m)27(ultilev)28(el)-393(preconditioners)-393(implemen)28(ted)-393(in)-393(MLD2P4)]TJ -82.05 -13.549 Td [(generates)-333(a)-334(hierarc)28(h)28(y)-333(of)-334(index)-333(spaces)-333(and)-334(a)-333(corresp)-28(onding)-333(hierarc)28(h)28(y)-334(of)-333(matrices,)]TJ 81.377 -23.491 Td [(\012)]TJ/F20 7.9701 Tf 7.879 4.505 Td [(1)]TJ/F25 10.9091 Tf 7.762 -4.505 Td [(\021)]TJ/F15 10.9091 Tf 11.515 0 Td [(\012)]TJ/F25 10.9091 Tf 10.909 0 Td [(\033)]TJ/F15 10.9091 Tf 11.515 0 Td [(\012)]TJ/F20 7.9701 Tf 7.879 4.505 Td [(2)]TJ/F25 10.9091 Tf 7.763 -4.505 Td [(\033)]TJ/F22 10.9091 Tf 11.515 0 Td [(:)-167(:)-166(:)]TJ/F25 10.9091 Tf 15.757 0 Td [(\033)]TJ/F15 10.9091 Tf 11.516 0 Td [(\012)]TJ/F23 7.9701 Tf 7.878 4.505 Td [(nl)-12(ev)]TJ/F22 10.9091 Tf 16.597 -4.505 Td [(;)-1167(A)]TJ/F20 7.9701 Tf 23.939 4.505 Td [(1)]TJ/F25 10.9091 Tf 7.763 -4.505 Td [(\021)]TJ/F22 10.9091 Tf 11.515 0 Td [(A;)-167(A)]TJ/F20 7.9701 Tf 21.212 4.505 Td [(2)]TJ/F22 10.9091 Tf 4.732 -4.505 Td [(;)-167(:)-166(:)-167(:)-167(;)-166(A)]TJ/F23 7.9701 Tf 32.424 4.505 Td [(nl)-12(ev)]TJ/F22 10.9091 Tf 16.597 -4.505 Td [(;)]TJ/F15 10.9091 Tf -328.044 -23.49 Td [(b)28(y)-282(using)-282(the)-282(information)-282(con)28(tained)-282(in)]TJ/F22 10.9091 Tf 180.373 0 Td [(A)]TJ/F15 10.9091 Tf 8.182 0 Td [(,)-292(without)-282(assuming)-282(an)27(y)-282(kno)28(wledge)-282(of)-282(the)-282(geom-)]TJ -188.555 -13.55 Td [(etry)-295(of)-294(the)-295(problem)-294(from)-295(whic)28(h)]TJ/F22 10.9091 Tf 150.888 0 Td [(A)]TJ/F15 10.9091 Tf 11.395 0 Td [(originates.)-431(A)-295(v)28(ector)-295(space)]TJ/F34 10.9091 Tf 126.348 0 Td [(R)]TJ/F23 7.9701 Tf 7.879 3.959 Td [(n)]TJ/F24 5.9776 Tf 5.138 -1.406 Td [(k)]TJ/F15 10.9091 Tf 8.285 -2.553 Td [(is)-295(asso)-27(ciate)-1(d)-294(with)-295(\012)]TJ/F23 7.9701 Tf 94.368 3.959 Td [(k)]TJ/F15 10.9091 Tf 5.12 -3.959 Td [(,)]TJ -409.421 -13.549 Td [(where)]TJ/F22 10.9091 Tf 31.714 0 Td [(n)]TJ/F23 7.9701 Tf 6.548 -1.777 Td [(k)]TJ/F15 10.9091 Tf 8.925 1.777 Td [(is)-349(the)-349(size)-348(of)-349(\012)]TJ/F23 7.9701 Tf 71.401 3.959 Td [(k)]TJ/F15 10.9091 Tf 5.12 -3.959 Td [(.)-491(F)84(or)-349(all)]TJ/F22 10.9091 Tf 43.449 0 Td [(k)-335(<)-304(nl)-19(ev)]TJ/F15 10.9091 Tf 41.907 0 Td [(,)-353(a)-348(restriction)-349(op)-28(erator)-349(and)-348(a)-349(prolongation)]TJ -209.064 -13.549 Td [(one)-333(are)-334(built,)-333(whic)28(h)-333(connect)-334(t)28(w)28(o)-334(l)1(e)-1(v)28(els)]TJ/F22 10.9091 Tf 188.273 0 Td [(k)]TJ/F15 10.9091 Tf 9.659 0 Td [(and)]TJ/F22 10.9091 Tf 21.212 0 Td [(k)]TJ/F15 10.9091 Tf 8.447 0 Td [(+)-222(1:)]TJ/F22 10.9091 Tf -101.285 -23.491 Td [(P)]TJ/F23 7.9701 Tf 8.519 4.504 Td [(k)]TJ/F25 10.9091 Tf 8.15 -4.504 Td [(2)]TJ/F34 10.9091 Tf 10.303 0 Td [(R)]TJ/F23 7.9701 Tf 7.879 4.504 Td [(n)]TJ/F24 5.9776 Tf 5.138 -1.405 Td [(k)]TJ/F26 7.9701 Tf 4.573 1.405 Td [(\002)]TJ/F23 7.9701 Tf 6.587 0 Td [(n)]TJ/F24 5.9776 Tf 5.138 -1.405 Td [(k)]TJ/F21 5.9776 Tf 4.075 0 Td [(+1)]TJ/F22 10.9091 Tf 10.239 -3.099 Td [(;)-1167(R)]TJ/F23 7.9701 Tf 24.125 4.504 Td [(k)]TJ/F25 10.9091 Tf 8.15 -4.504 Td [(2)]TJ/F34 10.9091 Tf 10.303 0 Td [(R)]TJ/F23 7.9701 Tf 7.879 4.504 Td [(n)]TJ/F24 5.9776 Tf 5.138 -1.405 Td [(k)]TJ/F21 5.9776 Tf 4.075 0 Td [(+1)]TJ/F26 7.9701 Tf 9.741 1.405 Td [(\002)]TJ/F23 7.9701 Tf 6.587 0 Td [(n)]TJ/F24 5.9776 Tf 5.138 -1.405 Td [(k)]TJ/F15 10.9091 Tf 5.071 -3.099 Td [(;)]TJ -283.114 -23.491 Td [(the)-282(matrix)]TJ/F22 10.9091 Tf 53.146 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.959 Td [(k)]TJ/F20 7.9701 Tf 4.621 0 Td [(+1)]TJ/F15 10.9091 Tf 14.392 -3.959 Td [(is)-282(computed)-281(b)27(y)-281(using)-282(the)-282(previou)1(s)-282(op)-28(erators)-282(according)-281(to)-282(the)-282(Galerkin)]TJ -80.341 -13.549 Td [(approac)28(h,)-333(i.e.,)]TJ/F22 10.9091 Tf 165.163 -13.549 Td [(A)]TJ/F23 7.9701 Tf 8.182 4.504 Td [(k)]TJ/F20 7.9701 Tf 4.621 0 Td [(+1)]TJ/F15 10.9091 Tf 14.349 -4.504 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(R)]TJ/F23 7.9701 Tf 8.368 4.504 Td [(k)]TJ/F22 10.9091 Tf 5.12 -4.504 Td [(A)]TJ/F23 7.9701 Tf 8.181 4.504 Td [(k)]TJ/F22 10.9091 Tf 5.12 -4.504 Td [(P)]TJ/F23 7.9701 Tf 8.519 4.504 Td [(k)]TJ/F22 10.9091 Tf 5.12 -4.504 Td [(:)]TJ/F15 10.9091 Tf -244.258 -19.516 Td [(In)-476(the)-477(curren)28(t)-476(implemen)28(tation)-477(of)-476(MLD2P4)-476(w)28(e)-477(ha)28(v)28(e)]TJ/F22 10.9091 Tf 261.479 0 Td [(R)]TJ/F23 7.9701 Tf 8.368 3.958 Td [(k)]TJ/F15 10.9091 Tf 10.749 -3.958 Td [(=)-516(\050)]TJ/F22 10.9091 Tf 18.358 0 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.958 Td [(k)]TJ/F15 10.9091 Tf 5.119 -3.958 Td [(\051)]TJ/F23 7.9701 Tf 4.243 3.958 Td [(T)]TJ/F15 10.9091 Tf 11.801 -3.958 Td [(A)-476(smo)-28(other)-476(with)]TJ -328.636 -13.55 Td [(iteration)-441(matrix)]TJ/F22 10.9091 Tf 82.101 0 Td [(M)]TJ/F23 7.9701 Tf 11.773 3.959 Td [(k)]TJ/F15 10.9091 Tf 9.928 -3.959 Td [(is)-441(set)-441(up)-440(at)-441(eac)28(h)-441(lev)28(el)]TJ/F22 10.9091 Tf 113.517 0 Td [(k)-488(<)-457(nl)-20(ev)]TJ/F15 10.9091 Tf 45.251 0 Td [(,)-468(and)-440(a)-441(solv)28(er)-441(is)-441(set)-441(up)-440(at)-441(the)]TJ -262.57 -13.549 Td [(coarsest)-369(lev)28(el,)-378(so)-368(that)-369(they)-369(are)-369(ready)-368(for)-369(application)-369(\050for)-368(example,)-378(setting)-369(up)-368(a)-369(solv)28(er)]TJ 0 -13.549 Td [(based)-292(on)-291(the)]TJ/F22 10.9091 Tf 62.938 0 Td [(LU)]TJ/F15 10.9091 Tf 19.243 0 Td [(factorization)-292(means)-291(computing)-292(and)-291(storing)-292(the)]TJ/F22 10.9091 Tf 224.422 0 Td [(L)]TJ/F15 10.9091 Tf 10.605 0 Td [(and)]TJ/F22 10.9091 Tf 20.757 0 Td [(U)]TJ/F15 10.9091 Tf 11.819 0 Td [(factors\051.)-431(The)]TJ -349.784 -13.549 Td [(construction)-345(of)-344(the)-345(hierarc)28(h)27(y)-344(of)-345(AMG)-345(comp)-28(on)1(e)-1(n)28(ts)-345(d)1(e)-1(scrib)-27(ed)-345(so)-345(far)-345(corresp)-27(onds)-345(to)-345(the)]TJ 0 -13.549 Td [(so-called)-333(build)-334(ph)1(as)-1(e)-333(of)-333(the)-334(p)1(rec)-1(on)1(ditioner.)]TJ 16.936 -13.55 Td [(The)-359(com)-1(p)-27(onen)28(ts)-360(pro)-28(du)1(c)-1(ed)-359(in)-359(the)-360(build)-359(phase)-359(ma)27(y)-359(b)-28(e)-359(com)28(bined)-360(in)-359(sev)28(eral)-360(w)28(a)28(ys)-360(to)]TJ -16.936 -13.549 Td [(obtain)-306(di\013eren)28(t)-306(m)27(ultilev)28(el)-306(preconditioners;)-315(this)-306(is)-306(done)-307(in)-306(the)-306(application)-306(phase,)-312(i)1(.e)-1(.)1(,)-312(in)]TJ 0 -13.549 Td [(the)-391(computation)-391(of)-391(a)-391(v)27(ector)-391(of)-391(t)28(yp)-28(e)]TJ/F22 10.9091 Tf 178.081 0 Td [(w)]TJ/F15 10.9091 Tf 12.185 0 Td [(=)]TJ/F22 10.9091 Tf 12.566 0 Td [(B)]TJ/F26 7.9701 Tf 8.822 3.959 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ/F22 10.9091 Tf 4.732 -3.959 Td [(v)]TJ/F15 10.9091 Tf 5.68 0 Td [(,)-406(where)]TJ/F22 10.9091 Tf 39.631 0 Td [(B)]TJ/F15 10.9091 Tf 13.089 0 Td [(denotes)-391(the)-391(preconditioner,)]TJ -281.372 -13.549 Td [(usually)-334(with)1(in)-334(an)-334(iteration)-333(of)-334(a)-334(Kr)1(ylo)27(v)-333(solv)27(er)-333([)]TJ
1 0 0 rg 1 0 0 RG
- [(20)]TJ
+ [(21)]TJ
0 g 0 G
[(].)-446(An)-333(example)-334(of)-333(s)-1(u)1(c)27(h)-333(a)-334(com)28(bination,)]TJ 0 -13.549 Td [(kno)28(wn)-448(as)-448(V-cycle,)-476(is)-448(giv)28(en)-448(in)-447(Figure)]TJ
0 0 1 rg 0 0 1 RG
@@ -1967,7 +1967,7 @@ BT
0 g 0 G
[(],)-334(and)-333(a)-333(v)28(e)-1(r)1(s)-1(ion)-333(of)-333(the)-333(K-cyc)-1(l)1(e)-334(describ)-28(ed)-333(in)-333([)]TJ
1 0 0 rg 1 0 0 RG
- [(19)]TJ
+ [(20)]TJ
0 g 0 G
[(].)]TJ/F17 11.9552 Tf 0 -29.053 Td [(4.2)-1125(Smo)-31(othed)-375(Aggregation)]TJ/F15 10.9091 Tf 0 -20.595 Td [(In)-374(order)-374(to)-374(de\014ne)-374(the)-375(prolongator)]TJ/F22 10.9091 Tf 167.339 0 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.959 Td [(k)]TJ/F15 10.9091 Tf 5.12 -3.959 Td [(,)-384(used)-375(to)-374(compute)-374(the)-374(coarse-lev)28(el)-375(matrix)]TJ/F22 10.9091 Tf 204.32 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.959 Td [(k)]TJ/F20 7.9701 Tf 4.622 0 Td [(+1)]TJ/F15 10.9091 Tf 11.319 -3.959 Td [(,)]TJ -409.421 -13.549 Td [(MLD2P4)-319(uses)-320(the)-319(smo)-28(othed)-319(aggregation)-319(algorithm)-320(describ)-27(ed)-320(in)-319([)]TJ
1 0 0 rg 1 0 0 RG
@@ -1975,7 +1975,7 @@ BT
0 g 0 G
[(,)]TJ
1 0 0 rg 1 0 0 RG
- [-319(25)]TJ
+ [-319(26)]TJ
0 g 0 G
[(].)-440(The)-319(basic)-320(idea)]TJ 0 -13.549 Td [(of)-270(this)-271(algorithm)-270(is)-271(to)-270(build)-270(a)-271(coarse)-270(se)-1(t)-270(of)-270(indices)-271(\012)]TJ/F23 7.9701 Tf 243.122 3.959 Td [(k)]TJ/F20 7.9701 Tf 4.621 0 Td [(+1)]TJ/F15 10.9091 Tf 14.27 -3.959 Td [(b)28(y)-271(suitabl)1(y)-271(grouping)-270(the)-271(indices)]TJ -262.013 -13.549 Td [(of)-350(\012)]TJ/F23 7.9701 Tf 20.481 3.959 Td [(k)]TJ/F15 10.9091 Tf 8.934 -3.959 Td [(in)28(to)-350(disjoin)28(t)-350(sub)1(s)-1(ets)-349(\050aggregates\051,)-354(and)-350(to)-349(de\014ne)-350(the)-349(coarse)-1(-to-\014n)1(e)-350(space)-350(transfer)]TJ
0 g 0 G
@@ -2038,7 +2038,7 @@ BT
0 g 0 G
[-500(application)-333(of)]TJ/F22 10.9091 Tf 82.727 0 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.958 Td [(k)]TJ/F15 10.9091 Tf 8.756 -3.958 Td [(and)]TJ/F22 10.9091 Tf 21.212 0 Td [(R)]TJ/F23 7.9701 Tf 8.368 3.958 Td [(k)]TJ/F15 10.9091 Tf 8.15 -3.958 Td [(=)-278(\050)]TJ/F22 10.9091 Tf 15.757 0 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.958 Td [(k)]TJ/F15 10.9091 Tf 5.12 -3.958 Td [(\051)]TJ/F23 7.9701 Tf 4.242 3.958 Td [(T)]TJ/F15 10.9091 Tf 10.241 -3.958 Td [(to)-333(build)]TJ/F22 10.9091 Tf 41.212 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.958 Td [(k)]TJ/F20 7.9701 Tf 4.622 0 Td [(+1)]TJ/F15 10.9091 Tf 11.319 -3.958 Td [(.)]TJ -243.343 -23.467 Td [(In)-407(order)-407(to)-407(p)-28(erform)-407(the)-407(coarsening)-407(s)-1(tep,)-425(the)-407(smo)-28(othed)-407(aggregation)-407(algorithm)-407(de-)]TJ -16.937 -13.549 Td [(scrib)-28(ed)-478(in)-478([)]TJ
1 0 0 rg 1 0 0 RG
- [(25)]TJ
+ [(26)]TJ
0 g 0 G
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ET
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5.138 -1.406 Td [(k)-21(;i)]TJ/F26 7.9701 Tf 9.506 1.406 Td [(\002)]TJ/F23 7.9701 Tf 6.587 0 Td [(n)]TJ/F24 5.9776 Tf 5.138 -1.406 Td [(k)]TJ/F15 10.9091 Tf -407.38 -16.102 Td [(that)-366(maps)-366(a)-365(v)27(ector)]TJ/F22 10.9091 Tf 95.448 0 Td [(x)]TJ/F23 7.9701 Tf 6.235 3.959 Td [(k)]TJ/F15 10.9091 Tf 9.11 -3.959 Td [(to)-366(the)-366(v)28(ector)]TJ/F22 10.9091 Tf 65.942 0 Td [(x)]TJ/F23 7.9701 Tf 6.235 3.959 Td [(k)]TJ 0 -7.015 Td [(i)]TJ/F15 10.9091 Tf 9.11 3.056 Td [(made)-366(of)-366(the)-365(com)-1(p)-27(onen)28(ts)-366(of)]TJ/F22 10.9091 Tf 134.56 0 Td [(x)]TJ/F23 7.9701 Tf 6.234 3.959 Td [(k)]TJ/F15 10.9091 Tf 9.111 -3.959 Td [(with)-366(indices)-366(in)]TJ -341.985 -13.549 Td [(\012)]TJ/F23 7.9701 Tf 7.879 3.959 Td [(k)]TJ 0 -7.015 Td [(i)]TJ/F15 10.9091 Tf 5.12 3.056 Td [(,)-301(and)-292(the)-293(prolongation)-292(op)-28(erator)]TJ/F22 10.9091 Tf 152.201 0 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.959 Td [(k)]TJ -1.515 -7.015 Td [(i)]TJ/F15 10.9091 Tf 9.665 3.056 Td 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