Choosing a linear algebra backend

To select a backend in order to solve a linear system, we instantiate the Backend class associated :

#include <feel/feelalg/backend.hpp>
boost::shared_ptr<Backend<double> > backend =
     Backend<double>::build( BACKEND_PETSC );

The backend provides an interface to solve

\[ A x = b \]

where \(A\) is a \(n \times n \) sparse matrix and \(x,b\) vectors of size \(n\). The backend defines the C++ types for each of these, e.g :

Backend<double>::sparse_matrix_type A;

Backend<double>::vector_type x,b;

In practice, we use the boost::shared_ptr<> shared pointer to ensure that we won't get memory leaks. The backends provide a corresponding typedef

Backend<double>::sparse_matrix_ptrtype A( backend->newMatrix( Xh, Yh ) );
Backend<double>::vector_ptrtype x( backend->newVector( Yh ) );
Backend<double>::vector_ptrtype b( backend->newVector( Xh ) );

where \(X_h\) and \(Y_h\) are function spaces providing the number of degrees of freedom that will define the size of the matrix and vectors thanks to the helpers functions Backend::newMatrix() and Backend::newVector. In a parallel setting, the local/global processor mapping would be passed down by the function spaces.

Solving

To solve the linear problem \(Ax=b\), the backend provides a function solve.
Interface :

solve(_matrix=A, _solution=x, _rhs=b)

Required Parameters :

_matrix : the matrix \(A\) has a sparse_matrix_ptrtype type
_solution : the solution \(x\) has a type vector_type or vector_ptrtype
_rhs : the second member vector \(b\) has a type vector_ptrtype

Optional Parameters

_prec : preconditioner, instead of solving \(Ax=b\), we solve \(P^{-1}Ax= P^{-1}b\). This method can be applied in iterative methods and permits to decrease the number of iterations in the resolution system
_ maxit : maximum number of iterations, this option is used with an iterative solving method
_rtolerance : residual tolerance, the fraction \(\displaystyle{\frac{\mid\mid r^{(k)} \mid\mid }{\mid\mid r^{(0)} \mid\mid}}\) is inferior to the residual tolerance with \(r^{(k)}=b-Ax^{(k)}\) and \(x^{(k)}\) the solution at the \(k^{th}\) iteration
_atolerance : absolute tolerance, \(\mid\mid r^{(k)} \mid\mid \) is inferior to the absolute tolerance
_dtolerance : different tolerance, sometimes, the residue doesn’t decrease continuously during the iterations. The difference between two plots doesn’t have to exceed the parameter choosen for the difference tolerance.
_transpose : boolean to use transpose matrix, instead of solving \(Ax=b\), we solve \(A^{t}x=b\). If \(A\) is defined and positive, \(A^{t}=A\).

The library Boost::Parameters allows you to enter parameters in the order you want. It supports deduced parameters, that is to say parameters whose identity can be deduced from their types.

Table of Contents

Choosing a linear algebra backend

Solving