Linear Solvers#

Base class for providing ApplyInverse() methods on operators.

Matrix provides existing Operators with an inverse. Note this may be iterative or non-iterative, assembled or non-assembled, approximate or exact to machine precision.

inverse-typed-spec

  • “iterative method” [string] optional

  • “direct method” [string] optional

  • “preconditioning method” [string] optional

DOCUMENT ME!

Linear Solver: PCG#

Preconditioned conjugate gradient method for a linear solver.

iterative-method-pcg-spec

  • “error tolerance” [double] 1.e-6 Tolerance on which to declare success.

  • “maximum number of iterations” [int] 100 Maximum iterations before declaring failure.

  • “overflow tolerance” [double] 3.e50 Error above this value results in failure.

  • “convergence criterial” [Array(string)] {relative rhs} A list of criteria, any of which can be applied. Valid include:

    • “relative rhs” : measure error relative to the norm of the RHS vector

    • “relative residual” : measure error relative to the norm of the residual

    • “absolute residual” : measure error directly, norm of error

    • “make one iteration” : require at least one iteration to be performed before declaring success

<ParameterList name="_PCG with HYPRE AMG">  <!-- parent list -->
<ParameterList name="pcg parameters">
  <Parameter name="error tolerance" type="double" value="1e-12"/>
  <Parameter name="maximum number of iterations" type="int" value="400"/>
  <Parameter name="convergence criteria" type="Array(string)" value="{relative residual,make one iteration}"/>
  <Parameter name="overflow tolerance" type="double" value="3.0e+50"/>

  <ParameterList name="verbose object">
    <Parameter name="verbosity level" type="string" value="high"/>
  </ParameterList>
</ParameterList>
</ParameterList>

Linear Solver: GMRES#

Generalized minimum residual method for a linear solver.

Based on the methods of Yu. Kuznetsov, 1968; Y.Saad, 1986. Deflated version of GMRES is due to R.Morgan, GMRES with deflated restarting, 2002 SISC; S.Rollin, W.Fichtner, Improving accuracy of GMRES with deflated restarting, 2007 SISC.

iterative-method-gmres-spec

  • “error tolerance” [double] 1.e-6 Tolerance on which to declare success.

  • “maximum number of iterations” [int] 100 Maximum iterations before declaring failure.

  • “overflow tolerance” [double] 3.e50 Error above this value results in failure.

  • “convergence criterial” [Array(string)] {relative rhs} A list of criteria, any of which can be applied. Valid include:

    • “relative rhs” : measure error relative to the norm of the RHS vector

    • “relative residual” : measure error relative to the norm of the residual

    • “absolute residual” : measure error directly, norm of error

    • “make one iteration” : require at least one iteration to be performed before declaring success

  • “size of Krylov space” [int] 10 Size of the Krylov space used to span the residual.

  • “controller training start” [int] 0 Start iteration for determining convergence rates. (Add more please!)

  • “controller training end” [int] 3 Start iteration for determining convergence rates. (Add more please!)

  • “preconditioning strategy” [string] left Valid are “left” and “right”-type preconditioning (see Saad 1986)

  • “maximum size of deflation space” [int] 0 Size of the deflation space, see Rollin et al.

<ParameterList name="_GMRES with HYPRE AMG">  <!-- parent list -->
<ParameterList name="gmres parameters">
  <Parameter name="error tolerance" type="double" value="1e-12"/>
  <Parameter name="maximum number of iterations" type="int" value="400"/>
  <Parameter name="convergence criteria" type="Array(string)" value="{relative residual}"/>
  <Parameter name="size of Krylov space" type="int" value="10"/>
  <Parameter name="overflow tolerance" type="double" value="3.0e+50"/>
  <Parameter name="maximum size of deflation space" type="int" value="0"/>
  <Parameter name="preconditioning strategy`" type="string" value="left"/>
  <Parameter name="release Krylov vectors" type="bool" value="false"/>

  <ParameterList name="verbose object">
    <Parameter name="verbosity level" type="string" value="high"/>
  </ParameterList>
</ParameterList>

<!-- Alternative implementation
<ParameterList name="belos gmres parameters">
  <Parameter name="error tolerance" type="double" value="1e-12"/>
  <Parameter name="maximum number of iterations" type="int" value="400"/>
  <Parameter name="convergence criteria" type="Array(string)" value="{relative residual}"/>
  <Parameter name="size of Krylov space" type="int" value="10"/>
  <Parameter name="overflow tolerance" type="double" value="3.0e+50"/>
</ParameterList-->
</ParameterList>

Linear Solver: NKA#

Uses NKA method as a linear solver.

This is effectively equivalent to GMRES with a rolling restart, where vectors fall off the end of the space.

iterative-method-nka-spec

  • “error tolerance” [double] 1.e-6 Tolerance on which to declare success.

  • “maximum number of iterations” [int] 100 Maximum iterations before declaring failure.

  • “overflow tolerance” [double] 3.e50 Error above this value results in failure.

  • “convergence criterial” [Array(string)] {relative rhs} A list of criteria, any of which can be applied. Valid include:

    • “relative rhs” : measure error relative to the norm of the RHS vector

    • “relative residual” : measure error relative to the norm of the residual

    • “absolute residual” : measure error directly, norm of error

    • “make one iteration” : require at least one iteration to be performed before declaring success

  • “max nka vectors” [int] 10 Size of the NKA space used to span the residual, conceptually equivalent to the size of the Krylov space.

  • “nka vector tolerance” [double] 0.05 Vectors whose dot product are within this tolerance are considered parallel, and therefore the old vector is thrown out.

<ParameterList name="_NKA">  <!-- parent list -->
<ParameterList name="nka parameters">
  <Parameter name="error tolerance" type="double" value="1e-12"/>
  <Parameter name="maximum number of iterations" type="int" value="400"/>
  <Parameter name="convergence criteria" type="Array(string)" value="{relative residual}"/>
  <Parameter name="overflow tolerance" type="double" value="3.0e+50"/>
  <Parameter name="max nka vectors" type="int" value="10"/>
  <Parameter name="nka vector tolerance" type="double" value="0.05"/>

  <ParameterList name="verbose object">
    <Parameter name="verbosity level" type="string" value="high"/>
  </ParameterList>
</ParameterList>
</ParameterList>

Linear Solver: Amesos#

Direct solvers via Trilinos.

Amesos library of Trilinos package provides interfaces to a few direct solvers. List “amesos parameters” contains parameters that understood by this library. These parameters may violate the camel-case convention employed by this spec. Additional parameters are:

  • “solver name” [string] declares name of one of the supported direct solvers. Available options are “klu”, “superludist”, “basker”, etc, see Amesos and Amesos2 manuals for details. The default value is serial solver “klu”.

  • “amesos version” [int] specifies version of Amesos. Available options are 1 and 2. The default value is 1.

<ParameterList name="_AMESOS KLU">  <!-- parent list -->
<Parameter name="direct method" type="string" value="amesos"/>
<ParameterList name="amesos parameters">
  <Parameter name="solver name" type="string" value="klu"/>
  <Parameter name="amesos version" type="int" value="1"/>
</ParameterList>
</ParameterList>

Linear Solver: Amesos#

Direct solvers via Trilinos.

Warning

undocumented

Linear Solver: Belos (GMRES)#

Trilinos/Belos implementations of iterative methods.

Includes GMRES and other Belos methods.

Warning

undocumented