Package esys :: Package escript :: Module pdetools :: Class SaddlePointProblem

Class SaddlePointProblem

object --+
         |
        SaddlePointProblem

This implements a solver for a saddle point problem

f(u,p)=0 g(u)=0

for u and p. The problem is solved with an inexact Uszawa scheme for p:

Q_f (u^{k+1}-u^{k}) = - f(u^{k},p^{k}) Q_g (p^{k+1}-p^{k}) = g(u^{k+1})

where Q_f is an approximation of the Jacobian A_f of f with respect to u and Q_f is an approximation of A_g A_f^{-1} A_g with A_g is the Jacobian of g with respect to p. As a the construction of a 'proper' Q_g can be difficult, non-linear conjugate gradient method is applied to solve for p, so Q_g plays in fact the role of a preconditioner.

Instance Methods

[hide private]

__init__(self, verbose=False, *args)
Initializes the problem.

float

inner(self, p0, p1)
Inner product of p0 and p1 approximating p.

solve(self, u0, p0, tolerance=1e-06, tolerance_u=None, iter_max=100, accepted_reduction=0.995, relaxation=None)
Runs the solver.

escript.Data

solve_f(self, u, p, tol=1e-08)
Solves

escript.Data

solve_g(self, u, tol=1e-08)
Solves

trace(self, text)
Prints text if verbose has been set.

Inherited from object: __delattr__, __getattribute__, __hash__, __new__, __reduce__, __reduce_ex__, __repr__, __setattr__, __str__

Class Variables

[hide private]

subiter_max = 3

Properties

[hide private]

Inherited from object: __class__

Method Details

[hide private]

init(self, verbose=False, args)
(Constructor)*

Initializes the problem.

Parameters:

verbose (bool) - if True, some information is printed in the process

Overrides: object.__init__

Note: this method may be overwritten by a particular saddle point problem

inner(self, p0, p1)

Inner product of p0 and p1 approximating p. Typically this returns integrate(p0*p1).

Returns: float: inner product of p0 and p1

solve(self, u0, p0, tolerance=1e-06, tolerance_u=None, iter_max=100, accepted_reduction=0.995, relaxation=None)

Runs the solver.

Parameters:

u0 (esys.escript.Data) - initial guess for u
p0 (esys.escript.Data) - initial guess for p
tolerance (positive float) - tolerance for relative error in u and p
tolerance_u (positive float) - tolerance for relative error in u if different from tolerance
iter_max (int) - maximum number of iteration steps
accepted_reduction (positive float or None) - if the norm g cannot be reduced by accepted_reduction backtracking to adapt the relaxation factor. If accepted_reduction=None no backtracking is used.
relaxation (float or None) - initial relaxation factor. If relaxation==None, the last relaxation factor is used.

solve_f(self, u, p, tol=1e-08)

Solves

A_f du = f(u,p)

with tolerance tol and returns du. A_f is Jacobian of f with respect to u.

Parameters:

u (escript.Data) - current approximation of u
p (escript.Data) - current approximation of p
tol (float) - tolerance expected for du

Returns: escript.Data

increment du

Note: this method has to be overwritten by a particular saddle point problem

solve_g(self, u, tol=1e-08)

Solves

Q_g dp = g(u)

where Q_g is a preconditioner for A_g A_f^{-1} A_g with A_g is the Jacobian of g with respect to p.

Parameters:

u (escript.Data) - current approximation of u
tol (float) - tolerance expected for dp

Returns: escript.Data

increment dp

Note: this method has to be overwritten by a particular saddle point problem

trace(self, text)

Prints text if verbose has been set.

Parameters:

text (str) - a text message

Class SaddlePointProblem

__init__(self, verbose=False, *args) (Constructor)

inner(self, p0, p1)

solve(self, u0, p0, tolerance=1e-06, tolerance_u=None, iter_max=100, accepted_reduction=0.995, relaxation=None)

solve_f(self, u, p, tol=1e-08)

solve_g(self, u, tol=1e-08)

trace(self, text)

init(self, verbose=False, args)
(Constructor)*