Solver module

Detailed Description

This module is responsible for treating the systems of linear
equations arizing from a discretized a PDE.

This class provides iterative solvers, preconditioners, and related tools.

The iterative solvers are derived from the gsIterativeSolver, most notably there are the standard Krylov space solvers (cg, MINRES, GMRES) and the gradient method, which can use optimal step sizes based on line search or fixed step sizes. The latter allows methods, which G+Smo presents as preconditioners, to be used as solvers, this means that a standard multigrid solver would be realized as a gradient solver with multigrid preconditioner.

The preconditioners are based on the abstract class gsLinearOperator. Objects of this type realize linear operators \( A \). Their application to a vector is provided by the member function apply. The class gsPreconditionerOp is an abstract class derived from gsLinearOperator. An example would be the Jacobi iteration, which interpreted as a linear operator, would be \( P = (\mbox{diag } A)^{-1} \). Preconditioners additionally have an underlying operator (underlyingOp), which would be \( A \) in this case. It is also possible to directly iterate those preconditioners using the member function step, which takes parameters \( f \) and \( x \) and updates the latter as follows:

\[ x_{new} = x_{old} + P ( f - A * x_{old} ). \]

Linear operators and preconditioners can be combined dynamically with the respective classes (like gsSumOp, gsProductOp, gsScaledOp, gsBlockOp). The wrapper class gsMatrixOp allows to treat (sparse) matrices as linear operators, the wrapper class gsSolverOp allows to treat the direct solvers from Eigen as linear operators. Also the multigrid and IETI methods are presented as linear operators and preconditioners.

The files associated to this module are located in the folders

• src/gsSolver
• src/gsMultiGrid
• src/gsIeti

Classes
class	AndersonAcceleration< Scalar >
	Anderson acceleration solver and its (preconditioned) variants. More...
class	gsBiCgStab< T >
	Biconjugate gradient stabilized solver. More...
class	gsBlockOp< T >
	Simple class create a block operator structure. More...
class	gsCompositePrecOp< T >
	This class represents the composition of preconditioners of type gsPreconditionerOp. More...
class	gsConjugateGradient< T >
	The conjugate gradient method. More...
class	gsGaussSeidelOp< MatrixType, ordering >
	Gauss-Seidel preconditioner. More...
class	gsGMRes< T >
	The generalized minimal residual (GMRES) method. More...
class	gsGradientMethod< T >
	The gradient method. More...
class	gsGridHierarchy< T >
	Grid Hierarchy. More...
class	gsIdentityOp< T >
	Identity operator. More...
class	gsIetiMapper< T >
	Ieti Mapper. More...
class	gsIetiSystem< T >
	This class represents a IETI system. More...
class	gsIncompleteLUOp< MatrixType >
	Incomplete LU with thresholding preconditioner. More...
class	gsIterativeSolver< T >
	Abstract class for iterative solvers. More...
class	gsJacobiOp< MatrixType >
	Jacobi preconditioner. More...
class	gsKroneckerOp< T >
	Class for representing a Kronecker product of operators of type gsLinearOperator. More...
class	gsLanczosMatrix< T >
	Class for representing a Lanczos matrix and calculating its eigenvalues. More...
class	gsLinearLambdaOp< T, L >
	Wrapper that allows to use lambdas as a gsLinearOperator. More...
class	gsLinearOperator< T >
	Simple abstract class for discrete operators. More...
class	gsMatrixOp< MatrixType >
	Simple adapter class to use a matrix (or matrix-like object) as a linear operator. Needed for the iterative method classes. More...
class	gsMinimalResidual< T >
	The minimal residual (MinRes) method. More...
class	gsMinResQLP< T >
	The minimal residual (MinRes-QLP) method. More...
class	gsMultiGridOp< T >
	Multigrid preconditioner. More...
class	gsPatchPreconditionersCreator< T >
	Provides robust preconditioners for single patch geometries. More...
class	gsPreconditionerFromOp< T >
	Simple class allowing to construct a preconditioner from a linear operator. More...
class	gsPreconditionerOp< T >
	Simple abstract class for perconditioners. More...
class	gsPrimalSystem< T >
	This class represents the primal system for a IETI-DP algorithm. More...
class	gsProductOp< T >
	Class for representing the product of objects of type gsLinearOperator as gsLinearOperator. More...
class	gsRichardsonOp< MatrixType >
	Richardson preconditioner. More...
class	gsScaledDirichletPrec< T >
	This class represents the scaled Dirichlet preconditioner for a IETI problem. More...
class	gsScaledOp< T >
	Allows an operator to be multiplied with a scalar. More...
class	gsSolverOp< SolverType >
	Simple adapter class to use an Eigen solver (having a compute() and a solve() method) as a linear operator. More...
class	gsSolverUtils< T >
	Utility class for PDE's solver related utils. More...
class	gsSumOp< T >
	Class for representing the sum of objects of type gsLinearOperator as gsLinearOperator. More...

Enumerations
enum	ordering {   forward,   reverse,   symmetric }
	Specify the ordering of gsGaussSeidelOp preconditioner. More...

Enumeration Type Documentation

enum ordering

Specify the ordering of gsGaussSeidelOp preconditioner.

Enumerator
forward	standard forward ordering
reverse	reverse ordering
symmetric	one total step is = one forward + one backward