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class="cmr-12">4 < / span > < / span > < a
id="x14-130004">< / a > < span
class="cmr-12">Getting Started< / span > < / h3 >
<!-- l. 5 --> < p class = "noindent" > < span
class="cmr-12">This section describes the basics for building and applying AMG4PSBLAS one-level< / span >
< span
class="cmr-12">and multilevel (i.e., AMG) preconditioners with the Krylov solvers included in< / span >
< span
class="cmr-12">PSBLAS< / span > < span
class="cmr-12">  < / span > < span class = "cite" > < span
class="cmr-12">[< / span > < a
href="userhtmlli5.html#XPSBLASGUIDE">< span
class="cmr-12">21< / span > < / a > < span
class="cmr-12">]< / span > < / span > < span
class="cmr-12">.< / span >
<!-- l. 9 --> < p class = "indent" > < span
class="cmr-12">The following steps are required:< / span >
< ol class = "enumerate1" >
< li
class="enumerate" id="x14-13002x1">
<!-- l. 11 --> < p class = "noindent" > < span
class="cmti-12">Declare the preconditioner data structure< / span > < span
class="cmr-12">. It is a derived data type,< / span >
< span class = "obeylines-h" > < code class = "verb" > amg_< / code > < / span > < span
class="cmti-12">x< / span > < span class = "obeylines-h" > < code class = "verb" > prec_< / code > < / span > < span class = "obeylines-h" > < code class = "verb" > type< / code > < / span > < span
class="cmr-12">, where < / span > < span
class="cmti-12">x < / span > < span
class="cmr-12">may be < / span > < span class = "obeylines-h" > < code class = "verb" > s< / code > < / span > < span
class="cmr-12">, < / span > < span class = "obeylines-h" > < code class = "verb" > d< / code > < / span > < span
class="cmr-12">, < / span > < span class = "obeylines-h" > < code class = "verb" > c< / code > < / span > < span
class="cmr-12">or < / span > < span class = "obeylines-h" > < code class = "verb" > z< / code > < / span > < span
class="cmr-12">, according to the basic data< / span >
< span
class="cmr-12">type of the sparse matrix (< / span > < span class = "obeylines-h" > < code class = "verb" > s< / code > < / span > < span
class="cmr-12">= real single precision; < / span > < span class = "obeylines-h" > < code class = "verb" > d< / code > < / span > < span
class="cmr-12">= real double precision;< / span >
< span class = "obeylines-h" > < code class = "verb" > c< / code > < / span > < span
class="cmr-12">= complex single precision; < / span > < span class = "obeylines-h" > < code class = "verb" > z< / code > < / span > < span
class="cmr-12">= complex double precision). This data< / span >
< span
class="cmr-12">structure is accessed by the user only through the AMG4PSBLAS routines,< / span >
< span
class="cmr-12">following an object-oriented approach.< / span >
< / li >
< li
class="enumerate" id="x14-13004x2">
<!-- l. 19 --> < p class = "noindent" > < span
class="cmti-12">Allocate and initialize the preconditioner data structure, according to a< / span >
< span
class="cmti-12">preconditioner type chosen by the user< / span > < span
class="cmr-12">. This is performed by the routine< / span >
< code class = "lstinline" > < span style = "color:#000000" > init< / span > < / code > < span
class="cmr-12">, which also sets defaults for each preconditioner type selected by< / span >
< span
class="cmr-12">the user. The preconditioner types and the defaults associated with them< / span >
< span
class="cmr-12">are given in Table< / span > < span
class="cmr-12">  < / span > < a
href="#x14-13015r1">< span
class="cmr-12">1< / span > <!-- tex4ht:ref: tab:precinit --> < / a > < span
class="cmr-12">, where the strings used by < / span > < code class = "lstinline" > < span style = "color:#000000" > init< / span > < / code > < span
class="cmr-12">to identify the< / span >
< span
class="cmr-12">preconditioner types are also given. Note that these strings are valid also if< / span >
< span
class="cmr-12">uppercase letters are substituted by corresponding lowercase ones.< / span >
< / li >
< li
class="enumerate" id="x14-13006x3">
<!-- l. 29 --> < p class = "noindent" > < span
class="cmti-12">Modify the selected preconditioner type, by properly setting preconditioner< / span >
< span
class="cmti-12">parameters. < / span > < span
class="cmr-12">This is performed by the routine < / span > < code class = "lstinline" > < span style = "color:#000000" > set< / span > < / code > < span
class="cmr-12">. This routine must be< / span >
< span
class="cmr-12">called if the user wants to modify the default values of the parameters< / span >
< span
class="cmr-12">associated with the selected preconditioner type, to obtain a variant of that< / span >
< span
class="cmr-12">preconditioner. Examples of use of < / span > < code class = "lstinline" > < span style = "color:#000000" > set< / span > < / code > < span
class="cmr-12">are given in Section< / span > < span
class="cmr-12">  < / span > < a
href="userhtmlsu6.html#x15-140004.1">< span
class="cmr-12">4.1< / span > <!-- tex4ht:ref: sec:examples --> < / a > < span
class="cmr-12">; a complete< / span >
< span
class="cmr-12">list of all the preconditioner parameters and their allowed and default values< / span >
< span
class="cmr-12">is provided in Section< / span > < span
class="cmr-12">  < / span > < a
href="userhtmlse5.html#x17-160005">< span
class="cmr-12">5< / span > <!-- tex4ht:ref: sec:userinterface --> < / a > < span
class="cmr-12">, Tables< / span > < span
class="cmr-12">  < / span > < a
href="userhtmlsu9.html#x19-18009r2">< span
class="cmr-12">2< / span > <!-- tex4ht:ref: tab:p_cycle --> < / a > < span
class="cmr-12">-< / span > < a
href="userhtmlsu9.html#x19-18015r8">< span
class="cmr-12">8< / span > <!-- tex4ht:ref: tab:p_smoother_1 --> < / a > < span
class="cmr-12">.< / span >
< / li >
< li
class="enumerate" id="x14-13008x4">
<!-- l. 40 --> < p class = "noindent" > < span
class="cmti-12">Build the preconditioner for a given matrix< / span > < span
class="cmr-12">. If the selected preconditioner is< / span >
< span
class="cmr-12">multilevel, then two steps must be performed, as specified next.< / span >
< ol class = "enumerate2" >
< li
class="enumerate" id="x14-13009x0">
<!-- l. 43 --> < p class = "noindent" > < span
class="cmti-12">Build the AMG hierarchy for a given matrix. < / span > < span
class="cmr-12">This is performed by the< / span >
< span
class="cmr-12">routine < / span > < code class = "lstinline" > < span style = "color:#000000" > hierarchy_build< / span > < / code > < span
class="cmr-12">.< / span >
< / li >
< li
class="enumerate" id="x14-13010x0">
<!-- l. 45 --> < p class = "noindent" > < span
class="cmti-12">Build the preconditioner for a given matrix. < / span > < span
class="cmr-12">This is performed by the< / span >
< span
class="cmr-12">routine < / span > < code class = "lstinline" > < span style = "color:#000000" > smoothers_build< / span > < / code > < span
class="cmr-12">.< / span > < / li > < / ol >
<!-- l. 48 --> < p class = "noindent" > < span
class="cmr-12">If the selected preconditioner is one-level, it is built in a single step, performed by< / span >
< span
class="cmr-12">the routine < / span > < code class = "lstinline" > < span style = "color:#000000" > bld< / span > < / code > < span
class="cmr-12">.< / span >
< / li >
< li
class="enumerate" id="x14-13012x5">
<!-- l. 50 --> < p class = "noindent" > < span
class="cmti-12">Apply the preconditioner at each iteration of a Krylov solver. < / span > < span
class="cmr-12">This is performed by< / span >
< span
class="cmr-12">the method < / span > < code class = "lstinline" > < span style = "color:#000000" > apply< / span > < / code > < span
class="cmr-12">. When using the PSBLAS Krylov solvers, this step is< / span >
< span
class="cmr-12">completely transparent to the user, since < / span > < code class = "lstinline" > < span style = "color:#000000" > apply< / span > < / code > < span
class="cmr-12">is called by the PSBLAS routine< / span >
< span
class="cmr-12">implementing the Krylov solver (< / span > < code class = "lstinline" > < span style = "color:#000000" > psb_krylov< / span > < / code > < span
class="cmr-12">).< / span >
< / li >
< li
class="enumerate" id="x14-13014x6">
<!-- l. 54 --> < p class = "noindent" > < span
class="cmti-12">Free the preconditioner data structure< / span > < span
class="cmr-12">. This is performed by the routine < / span > < code class = "lstinline" > < span style = "color:#000000" > free< / span > < / code > < span
class="cmr-12">.< / span >
< span
class="cmr-12">This step is complementary to step 1 and should be performed when the< / span >
< span
class="cmr-12">preconditioner is no more used.< / span > < / li > < / ol >
<!-- l. 59 --> < p class = "indent" > < span
class="cmr-12">All the previous routines are available as methods of the preconditioner object. A< / span >
< span
class="cmr-12">detailed description of them is given in Section< / span > < span
class="cmr-12">  < / span > < a
href="userhtmlse5.html#x17-160005">< span
class="cmr-12">5< / span > <!-- tex4ht:ref: sec:userinterface --> < / a > < span
class="cmr-12">. Examples showing the basic use of< / span >
< span
class="cmr-12">AMG4PSBLAS are reported in Section< / span > < span
class="cmr-12">  < / span > < a
href="userhtmlsu6.html#x15-140004.1">< span
class="cmr-12">4.1< / span > <!-- tex4ht:ref: sec:examples --> < / a > < span
class="cmr-12">.< / span >
< div class = "table" >
<!-- l. 63 --> < p class = "indent" > < a
id="x14-13015r1">< / a > < hr class = "float" > < div class = "float"
>
< div class = "center"
>
<!-- l. 64 --> < p class = "noindent" >
< div class = "tabular" > < table id = "TBL-1" class = "tabular"
>< colgroup id = "TBL-1-1g" > < col
id="TBL-1-1">< / colgroup > < colgroup id = "TBL-1-2g" > < col
id="TBL-1-2">< / colgroup > < colgroup id = "TBL-1-3g" > < col
id="TBL-1-3">< / colgroup > < tr
class="hline">< td > < hr > < / td > < td > < hr > < / td > < td > < hr > < / td > < / tr > < tr
style="vertical-align:baseline;" id="TBL-1-1-">< td style = "white-space:nowrap; text-align:left;" id = "TBL-1-1-1"
class="td11"> < span
class="cmcsc-10x-x-109">< span
class="small-caps">t< / span > < span
class="small-caps">y< / span > < span
class="small-caps">p< / span > < span
class="small-caps">e< / span > < / span > < / td > < td style = "white-space:normal; text-align:left;" id = "TBL-1-1-2"
class="td11"> <!-- l. 68 --> < p class = "noindent" > < span
class="cmcsc-10x-x-109">< span
class="small-caps">s< / span > < span
class="small-caps">t< / span > < span
class="small-caps">r< / span > < span
class="small-caps">i< / span > < span
class="small-caps">n< / span > < span
class="small-caps">g< / span > < / span > < / td > < td style = "white-space:normal; text-align:left;" id = "TBL-1-1-3"
class="td11"> <!-- l. 68 --> < p class = "noindent" > < span
class="cmcsc-10x-x-109">< span
class="small-caps">d< / span > < span
class="small-caps">e< / span > < span
class="small-caps">f< / span > < span
class="small-caps">a< / span > < span
class="small-caps">u< / span > < span
class="small-caps">l< / span > < span
class="small-caps">t< / span > < span
class="small-caps">p< / span > < span
class="small-caps">r< / span > < span
class="small-caps">e< / span > < span
class="small-caps">c< / span > < span
class="small-caps">o< / span > < span
class="small-caps">n< / span > < span
class="small-caps">d< / span > < span
class="small-caps">i< / span > < span
class="small-caps">t< / span > < span
class="small-caps">i< / span > < span
class="small-caps">o< / span > < span
class="small-caps">n< / span > < span
class="small-caps">e< / span > < span
class="small-caps">r< / span > < / span > < / td > < / tr > < tr
class="hline">< td > < hr > < / td > < td > < hr > < / td > < td > < hr > < / td > < / tr > < tr
style="vertical-align:baseline;" id="TBL-1-2-">< td style = "white-space:nowrap; text-align:left;" id = "TBL-1-2-1"
class="td11">
No preconditioner < / td > < td style = "white-space:normal; text-align:left;" id = "TBL-1-2-2"
class="td11"> <!-- l. 69 --> < p class = "noindent" > < code class = "lstinline" > < span style = "color:#000000" > ’ < / span > < span style = "color:#000000" > NONE< / span > < span style = "color:#000000" > ’ < / span > < / code >
< / td > < td style = "white-space:normal; text-align:left;" id = "TBL-1-2-3"
class="td11"> <!-- l. 69 --> < p class = "noindent" > Considered to use the PSBLAS Krylov
solvers with no preconditioner.
< / td >
< / tr > < tr
class="hline">< td > < hr > < / td > < td > < hr > < / td > < td > < hr > < / td > < / tr > < tr
style="vertical-align:baseline;" id="TBL-1-3-">< td style = "white-space:nowrap; text-align:left;" id = "TBL-1-3-1"
class="td11">
Diagonal < / td > < td style = "white-space:normal; text-align:left;" id = "TBL-1-3-2"
class="td11"> <!-- l. 71 --> < p class = "noindent" > < code class = "lstinline" > < span style = "color:#000000" > ’ < / span > < span style = "color:#000000" > DIAG< / span > < span style = "color:#000000" > ’ < / span > < / code > ,
< code class = "lstinline" > < span style = "color:#000000" > ’ < / span > < span style = "color:#000000" > JACOBI< / span > < span style = "color:#000000" > ’ < / span > < / code > ,
< code class = "lstinline" > < span style = "color:#000000" > ’ < / span > < span style = "color:#000000" > L1< / span > < span style = "color:#000000" > -< / span > < span style = "color:#000000" > JACOBI< / span > < span style = "color:#000000" > ’ < / span > < / code >
< / td > < td style = "white-space:normal; text-align:left;" id = "TBL-1-3-3"
class="td11"> <!-- l. 71 --> < p class = "noindent" > Diagonal preconditioner. For any zero
diagonal entry of the matrix to be
preconditioned, the corresponding entry
of the preconditioner is set to  1.
< / td >
< / tr > < tr
class="hline">< td > < hr > < / td > < td > < hr > < / td > < td > < hr > < / td > < / tr > < tr
style="vertical-align:baseline;" id="TBL-1-4-">< td style = "white-space:nowrap; text-align:left;" id = "TBL-1-4-1"
class="td11">
Gauss-Seidel < / td > < td style = "white-space:normal; text-align:left;" id = "TBL-1-4-2"
class="td11"> <!-- l. 74 --> < p class = "noindent" > < code class = "lstinline" > < span style = "color:#000000" > ’ < / span > < span style = "color:#000000" > GS< / span > < span style = "color:#000000" > ’ < / span > < / code > ,
< code class = "lstinline" > < span style = "color:#000000" > ’ < / span > < span style = "color:#000000" > L1< / span > < span style = "color:#000000" > -< / span > < span style = "color:#000000" > GS< / span > < span style = "color:#000000" > ’ < / span > < / code >
< / td > < td style = "white-space:normal; text-align:left;" id = "TBL-1-4-3"
class="td11"> <!-- l. 74 --> < p class = "noindent" > Hybrid Gauss-Seidel (forward), that is,
global block Jacobi with Gauss-Seidel as
local solver.
< / td >
< / tr > < tr
class="hline">< td > < hr > < / td > < td > < hr > < / td > < td > < hr > < / td > < / tr > < tr
style="vertical-align:baseline;" id="TBL-1-5-">< td style = "white-space:nowrap; text-align:left;" id = "TBL-1-5-1"
class="td11">
Symmetrized Gauss-Seidel < / td > < td style = "white-space:normal; text-align:left;" id = "TBL-1-5-2"
class="td11"> <!-- l. 77 --> < p class = "noindent" > < code class = "lstinline" > < span style = "color:#000000" > ’ < / span > < span style = "color:#000000" > FBGS< / span > < span style = "color:#000000" > ’ < / span > < / code > ,
< code class = "lstinline" > < span style = "color:#000000" > ’ < / span > < span style = "color:#000000" > L1< / span > < span style = "color:#000000" > -< / span > < span style = "color:#000000" > FBGS< / span > < span style = "color:#000000" > ’ < / span > < / code >
< / td > < td style = "white-space:normal; text-align:left;" id = "TBL-1-5-3"
class="td11"> <!-- l. 77 --> < p class = "noindent" > Symmetrized hybrid Gauss-Seidel, that
is, forward Gauss-Seidel followed by
backward Gauss-Seidel.
< / td >
< / tr > < tr
class="hline">< td > < hr > < / td > < td > < hr > < / td > < td > < hr > < / td > < / tr > < tr
style="vertical-align:baseline;" id="TBL-1-6-">< td style = "white-space:nowrap; text-align:left;" id = "TBL-1-6-1"
class="td11">
Block Jacobi < / td > < td style = "white-space:normal; text-align:left;" id = "TBL-1-6-2"
class="td11"> <!-- l. 80 --> < p class = "noindent" > < code class = "lstinline" > < span style = "color:#000000" > ’ < / span > < span style = "color:#000000" > BJAC< / span > < span style = "color:#000000" > ’ < / span > < / code > ,
< code class = "lstinline" > < span style = "color:#000000" > ’ < / span > < span style = "color:#000000" > L1< / span > < span style = "color:#000000" > -< / span > < span style = "color:#000000" > BJAC< / span > < span style = "color:#000000" > ’ < / span > < / code >
< / td > < td style = "white-space:normal; text-align:left;" id = "TBL-1-6-3"
class="td11"> <!-- l. 80 --> < p class = "noindent" > Block-Jacobi with ILU(0) on the local
blocks.
< / td >
< / tr > < tr
class="hline">< td > < hr > < / td > < td > < hr > < / td > < td > < hr > < / td > < / tr > < tr
style="vertical-align:baseline;" id="TBL-1-7-">< td style = "white-space:nowrap; text-align:left;" id = "TBL-1-7-1"
class="td11">
Additive Schwarz < / td > < td style = "white-space:normal; text-align:left;" id = "TBL-1-7-2"
class="td11"> <!-- l. 81 --> < p class = "noindent" > < code class = "lstinline" > < span style = "color:#000000" > ’ < / span > < span style = "color:#000000" > AS< / span > < span style = "color:#000000" > ’ < / span > < / code >
< / td > < td style = "white-space:normal; text-align:left;" id = "TBL-1-7-3"
class="td11"> <!-- l. 81 --> < p class = "noindent" > Additive Schwarz (AS), with overlap  1
and ILU(0) on the local blocks.
< / td >
< / tr > < tr
class="hline">< td > < hr > < / td > < td > < hr > < / td > < td > < hr > < / td > < / tr > < tr
style="vertical-align:baseline;" id="TBL-1-8-">< td style = "white-space:nowrap; text-align:left;" id = "TBL-1-8-1"
class="td11">
Multilevel < / td > < td style = "white-space:normal; text-align:left;" id = "TBL-1-8-2"
class="td11"> <!-- l. 83 --> < p class = "noindent" > < code class = "lstinline" > < span style = "color:#000000" > ’ < / span > < span style = "color:#000000" > ML< / span > < span style = "color:#000000" > ’ < / span > < / code >
< / td > < td style = "white-space:normal; text-align:left;" id = "TBL-1-8-3"
class="td11"> <!-- l. 83 --> < p class = "noindent" > V-cycle with one hybrid
forward Gauss-Seidel (GS) sweep as
pre-smoother and one hybrid backward
GS sweep as post-smoother, decoupled
smoothed aggregation as coarsening
algorithm, and LU (plus triangular solve)
as coarsest-level solver. See the default
values in Tables  < a
href="userhtmlsu9.html#x19-18009r2">2<!-- tex4ht:ref: tab:p_cycle --> < / a > -< a
href="userhtmlsu9.html#x19-18015r8">8<!-- tex4ht:ref: tab:p_smoother_1 --> < / a > for further details of
the preconditioner.
< / td >
< / tr > < tr
class="hline">< td > < hr > < / td > < td > < hr > < / td > < td > < hr > < / td > < / tr > < tr
style="vertical-align:baseline;" id="TBL-1-9-">< td style = "white-space:nowrap; text-align:left;" id = "TBL-1-9-1"
class="td11"> < / td > < / tr > < / table > < / div >
< br / > < div class = "caption"
>< span class = "id" > Table  1: < / span > < span
class="content">Preconditioner types, corresponding strings and default choices. < / span > < / div > <!-- tex4ht:label?: x14 - 13015r1 -->
< / div >
< / div > < hr class = "endfloat" / >
< / div >
<!-- l. 98 --> < p class = "indent" > < span
class="cmr-12">Note that the module < / span > < code class = "lstinline" > < span style = "color:#000000" > amg_prec_mod< / span > < / code > < span
class="cmr-12">, containing the definition of the preconditioner< / span >
< span
class="cmr-12">data type and the interfaces to the routines of AMG4PSBLAS, must be used< / span >
< span
class="cmr-12">in any program calling such routines. The modules < / span > < code class = "lstinline" > < span style = "color:#000000" > psb_base_mod< / span > < / code > < span
class="cmr-12">, for the< / span >
< span
class="cmr-12">sparse matrix and communication descriptor data types, and < / span > < code class = "lstinline" > < span style = "color:#000000" > psb_krylov_mod< / span > < / code > < span
class="cmr-12">,< / span >
< span
class="cmr-12">for interfacing with the Krylov solvers, must be also used (see Section< / span > < span
class="cmr-12">  < / span > < a
href="userhtmlsu6.html#x15-140004.1">< span
class="cmr-12">4.1< / span > <!-- tex4ht:ref: sec:examples --> < / a > < span
class="cmr-12">).< / span >
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<!-- l. 105 --> < p class = "indent" > < span
class="cmbx-12">Remark 1. < / span > < span
class="cmr-12">Coarsest-level solvers based on the LU factorization, such as those< / span >
< span
class="cmr-12">implemented in UMFPACK, MUMPS, SuperLU, and SuperLU< / span > < span
class="cmr-12">_Dist, usually lead to< / span >
< span
class="cmr-12">smaller numbers of preconditioned Krylov iterations than inexact solvers, when the< / span >
< span
class="cmr-12">linear system comes from a standard discretization of basic scalar elliptic PDE< / span >
< span
class="cmr-12">problems. However, this does not necessarily correspond to the shortest execution time< / span >
< span
class="cmr-12">on parallel< / span > < span
class="cmr-12">  computers.< / span >
< div class = "subsectionTOCS" >
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class="cmr-12">4.1 < / span > < a
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class="cmr-12">Examples< / span > < / a > < / span >
< br / > < span class = "subsectionToc" > < span
class="cmr-12">4.2 < / span > < a
href="userhtmlsu7.html#x16-150004.2">< span
class="cmr-12">GPU example< / span > < / a > < / span >
< / div >
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