escript/py_src/pdetools.py


########################################################
#
# Copyright (c) 2003-2008 by University of Queensland
# Earth Systems Science Computational Center (ESSCC)
# http://www.uq.edu.au/esscc
#
# Primary Business: Queensland, Australia
# Licensed under the Open Software License version 3.0
# http://www.opensource.org/licenses/osl-3.0.php
#
########################################################

__copyright__="""Copyright (c) 2003-2008 by University of Queensland
Earth Systems Science Computational Center (ESSCC)
http://www.uq.edu.au/esscc
Primary Business: Queensland, Australia"""
__license__="""Licensed under the Open Software License version 3.0
http://www.opensource.org/licenses/osl-3.0.php"""
__url__="http://www.uq.edu.au/esscc/escript-finley"

"""
Provides some tools related to PDEs. 

Currently includes:
    - Projector - to project a discontinuous
    - Locator - to trace values in data objects at a certain location
    - TimeIntegrationManager - to handel extraplotion in time
    - SaddlePointProblem - solver for Saddle point problems using the inexact uszawa scheme

@var __author__: name of author
@var __copyright__: copyrights
@var __license__: licence agreement
@var __url__: url entry point on documentation
@var __version__: version
@var __date__: date of the version
"""

__author__="Lutz Gross, l.gross@uq.edu.au"


import escript
import linearPDEs
import numarray
import util
import math

##### Added by Artak
# from Numeric import zeros,Int,Float64
###################################


class TimeIntegrationManager:
  """
  a simple mechanism to manage time dependend values. 

  typical usage is::

     dt=0.1 # time increment
     tm=TimeIntegrationManager(inital_value,p=1)
     while t<1.
         v_guess=tm.extrapolate(dt) # extrapolate to t+dt
         v=...
         tm.checkin(dt,v)
         t+=dt

  @note: currently only p=1 is supported.
  """
  def __init__(self,*inital_values,**kwargs):
     """
     sets up the value manager where inital_value is the initial value and p is order used for extrapolation
     """
     if kwargs.has_key("p"):
            self.__p=kwargs["p"]
     else:
            self.__p=1
     if kwargs.has_key("time"):
            self.__t=kwargs["time"]
     else:
            self.__t=0.
     self.__v_mem=[inital_values]
     self.__order=0
     self.__dt_mem=[]
     self.__num_val=len(inital_values)

  def getTime(self):
      return self.__t
  def getValue(self):
      out=self.__v_mem[0]
      if len(out)==1:
          return out[0]
      else:
          return out

  def checkin(self,dt,*values):
      """
      adds new values to the manager. the p+1 last value get lost
      """
      o=min(self.__order+1,self.__p)
      self.__order=min(self.__order+1,self.__p)
      v_mem_new=[values]
      dt_mem_new=[dt]
      for i in range(o-1):
         v_mem_new.append(self.__v_mem[i])
         dt_mem_new.append(self.__dt_mem[i])
      v_mem_new.append(self.__v_mem[o-1])
      self.__order=o
      self.__v_mem=v_mem_new
      self.__dt_mem=dt_mem_new
      self.__t+=dt

  def extrapolate(self,dt):
      """
      extrapolates to dt forward in time.
      """
      if self.__order==0:
         out=self.__v_mem[0]
      else:
        out=[]
        for i in range(self.__num_val):
           out.append((1.+dt/self.__dt_mem[0])*self.__v_mem[0][i]-dt/self.__dt_mem[0]*self.__v_mem[1][i])

      if len(out)==0:
         return None
      elif len(out)==1:
         return out[0]
      else:
         return out
 

class Projector:
  """
  The Projector is a factory which projects a discontiuous function onto a
  continuous function on the a given domain.
  """
  def __init__(self, domain, reduce = True, fast=True):
    """
    Create a continuous function space projector for a domain.

    @param domain: Domain of the projection.
    @param reduce: Flag to reduce projection order (default is True)
    @param fast: Flag to use a fast method based on matrix lumping (default is true)
    """
    self.__pde = linearPDEs.LinearPDE(domain)
    if fast:
      self.__pde.setSolverMethod(linearPDEs.LinearPDE.LUMPING)
    self.__pde.setSymmetryOn()
    self.__pde.setReducedOrderTo(reduce)
    self.__pde.setValue(D = 1.)
    return

  def __call__(self, input_data):
    """
    Projects input_data onto a continuous function

    @param input_data: The input_data to be projected.
    """
    out=escript.Data(0.,input_data.getShape(),self.__pde.getFunctionSpaceForSolution())
    self.__pde.setValue(Y = escript.Data(), Y_reduced = escript.Data())
    if input_data.getRank()==0:
        self.__pde.setValue(Y = input_data)
        out=self.__pde.getSolution()
    elif input_data.getRank()==1:
        for i0 in range(input_data.getShape()[0]):
           self.__pde.setValue(Y = input_data[i0])
           out[i0]=self.__pde.getSolution()
    elif input_data.getRank()==2:
        for i0 in range(input_data.getShape()[0]):
           for i1 in range(input_data.getShape()[1]):
              self.__pde.setValue(Y = input_data[i0,i1])
              out[i0,i1]=self.__pde.getSolution()
    elif input_data.getRank()==3:
        for i0 in range(input_data.getShape()[0]):
           for i1 in range(input_data.getShape()[1]):
              for i2 in range(input_data.getShape()[2]):
                 self.__pde.setValue(Y = input_data[i0,i1,i2])
                 out[i0,i1,i2]=self.__pde.getSolution()
    else:
        for i0 in range(input_data.getShape()[0]):
           for i1 in range(input_data.getShape()[1]):
              for i2 in range(input_data.getShape()[2]):
                 for i3 in range(input_data.getShape()[3]):
                    self.__pde.setValue(Y = input_data[i0,i1,i2,i3])
                    out[i0,i1,i2,i3]=self.__pde.getSolution()
    return out

class NoPDE:
     """
     solves the following problem for u:

     M{kronecker[i,j]*D[j]*u[j]=Y[i]} 

     with constraint

     M{u[j]=r[j]}  where M{q[j]>0}

     where D, Y, r and q are given functions of rank 1.

     In the case of scalars this takes the form

     M{D*u=Y} 

     with constraint

     M{u=r}  where M{q>0}

     where D, Y, r and q are given scalar functions.

     The constraint is overwriting any other condition.

     @note: This class is similar to the L{linearPDEs.LinearPDE} class with A=B=C=X=0 but has the intention
            that all input parameter are given in L{Solution} or L{ReducedSolution}. The whole
            thing is a bit strange and I blame Robert.Woodcock@csiro.au for this.
     """
     def __init__(self,domain,D=None,Y=None,q=None,r=None):
         """
         initialize the problem

         @param domain: domain of the PDE.
         @type domain: L{Domain}
         @param D: coefficient of the solution. 
         @type D: C{float}, C{int}, L{numarray.NumArray}, L{Data}
         @param Y: right hand side
         @type Y: C{float}, C{int}, L{numarray.NumArray}, L{Data}
         @param q: location of constraints
         @type q: C{float}, C{int}, L{numarray.NumArray}, L{Data}
         @param r: value of solution at locations of constraints
         @type r: C{float}, C{int}, L{numarray.NumArray}, L{Data}
         """
         self.__domain=domain
         self.__D=D
         self.__Y=Y
         self.__q=q
         self.__r=r
         self.__u=None
         self.__function_space=escript.Solution(self.__domain)
     def setReducedOn(self):
         """
         sets the L{FunctionSpace} of the solution to L{ReducedSolution}
         """
         self.__function_space=escript.ReducedSolution(self.__domain)
         self.__u=None

     def setReducedOff(self):
         """
         sets the L{FunctionSpace} of the solution to L{Solution}
         """
         self.__function_space=escript.Solution(self.__domain)
         self.__u=None
         
     def setValue(self,D=None,Y=None,q=None,r=None):
         """
         assigns values to the parameters.

         @param D: coefficient of the solution. 
         @type D: C{float}, C{int}, L{numarray.NumArray}, L{Data}
         @param Y: right hand side
         @type Y: C{float}, C{int}, L{numarray.NumArray}, L{Data}
         @param q: location of constraints
         @type q: C{float}, C{int}, L{numarray.NumArray}, L{Data}
         @param r: value of solution at locations of constraints
         @type r: C{float}, C{int}, L{numarray.NumArray}, L{Data}
         """
         if not D==None:
            self.__D=D
            self.__u=None
         if not Y==None:
            self.__Y=Y
            self.__u=None
         if not q==None:
            self.__q=q
            self.__u=None
         if not r==None:
            self.__r=r
            self.__u=None

     def getSolution(self):
         """
         returns the solution
        
         @return: the solution of the problem
         @rtype: L{Data} object in the L{FunctionSpace} L{Solution} or L{ReducedSolution}.
         """
         if self.__u==None:
            if self.__D==None:
               raise ValueError,"coefficient D is undefined"
            D=escript.Data(self.__D,self.__function_space)
            if D.getRank()>1:
               raise ValueError,"coefficient D must have rank 0 or 1"
            if self.__Y==None:
               self.__u=escript.Data(0.,D.getShape(),self.__function_space)
            else:
               self.__u=util.quotient(self.__Y,D)
            if not self.__q==None:
                q=util.wherePositive(escript.Data(self.__q,self.__function_space))
                self.__u*=(1.-q)
                if not self.__r==None: self.__u+=q*self.__r
         return self.__u
             
class Locator:
     """
     Locator provides access to the values of data objects at a given
     spatial coordinate x.  
     
     In fact, a Locator object finds the sample in the set of samples of a
     given function space or domain where which is closest to the given
     point x. 
     """

     def __init__(self,where,x=numarray.zeros((3,))):
       """
       Initializes a Locator to access values in Data objects on the Doamin 
       or FunctionSpace where for the sample point which
       closest to the given point x.

       @param where: function space
       @type where: L{escript.FunctionSpace} 
       @param x: coefficient of the solution. 
       @type x: L{numarray.NumArray} or C{list} of L{numarray.NumArray}
       """
       if isinstance(where,escript.FunctionSpace):
          self.__function_space=where
       else:
          self.__function_space=escript.ContinuousFunction(where)
       if isinstance(x, list):
           self.__id=[]
           for p in x:
              self.__id.append(util.length(self.__function_space.getX()-p[:self.__function_space.getDim()]).minGlobalDataPoint())
       else:
           self.__id=util.length(self.__function_space.getX()-x[:self.__function_space.getDim()]).minGlobalDataPoint()

     def __str__(self):
       """
       Returns the coordinates of the Locator as a string.
       """
       x=self.getX()
       if instance(x,list):
          out="["
          first=True
          for xx in x:
            if not first:
                out+=","
            else:
                first=False
            out+=str(xx)
          out+="]>"
       else:
          out=str(x)
       return out

     def getX(self):
        """
    Returns the exact coordinates of the Locator.
    """
        return self(self.getFunctionSpace().getX())

     def getFunctionSpace(self):
        """
    Returns the function space of the Locator.
    """
        return self.__function_space

     def getId(self,item=None):
        """
    Returns the identifier of the location.
    """
        if item == None:
           return self.__id
        else:
           if isinstance(self.__id,list):
              return self.__id[item]
           else:
              return self.__id


     def __call__(self,data):
        """
    Returns the value of data at the Locator of a Data object otherwise 
    the object is returned.
    """
        return self.getValue(data)

     def getValue(self,data):
        """
    Returns the value of data at the Locator if data is a Data object 
    otherwise the object is returned.
    """
        if isinstance(data,escript.Data):
           if data.getFunctionSpace()==self.getFunctionSpace():
             dat=data
           else:
             dat=data.interpolate(self.getFunctionSpace())
           id=self.getId()
           r=data.getRank()
           if isinstance(id,list):
               out=[]
               for i in id:
                  o=data.getValueOfGlobalDataPoint(*i)
                  if data.getRank()==0:
                     out.append(o[0])
                  else:
                     out.append(o)
               return out
           else:
             out=data.getValueOfGlobalDataPoint(*id)
             if data.getRank()==0:
                return out[0]
             else:
                return out
        else:
           return data

class SolverSchemeException(Exception):
   """
   exceptions thrown by solvers 
   """
   pass

class IndefinitePreconditioner(SolverSchemeException):
   """
   the preconditioner is not positive definite.
   """
   pass
class MaxIterReached(SolverSchemeException):
   """
   maxium number of iteration steps is reached.
   """
   pass
class CorrectionFailed(SolverSchemeException):
   """
   no convergence has been achieved in the solution correction scheme.
   """
   pass
class IterationBreakDown(SolverSchemeException):
   """
   iteration scheme econouters an incurable breakdown.
   """
   pass
class NegativeNorm(SolverSchemeException):
   """
   a norm calculation returns a negative norm.
   """
   pass

def PCG(b, Aprod, Msolve, bilinearform, atol=0, rtol=1.e-8, x=None, iter_max=100, initial_guess=True, verbose=False):
   """
   Solver for 

   M{Ax=b}

   with a symmetric and positive definite operator A (more details required!). 
   It uses the conjugate gradient method with preconditioner M providing an approximation of A. 

   The iteration is terminated if the C{stoppingcriterium} function return C{True}.

   For details on the preconditioned conjugate gradient method see the book:

   Templates for the Solution of Linear Systems by R. Barrett, M. Berry, 
   T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, 
   C. Romine, and H. van der Vorst.

   @param b: the right hand side of the liner system. C{b} is altered.
   @type b: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
   @param Aprod: returns the value Ax
   @type Aprod: function C{Aprod(x)} where C{x} is of the same object like argument C{x}. 
       The returned object needs to be of the same type like argument C{b}.
   @param Msolve: solves Mx=r
   @type Msolve: function C{Msolve(r)} where C{r} is of the same type like argument C{b}.
      The returned object needs to be of the same type like argument C{x}.
   @param bilinearform: inner product C{<x,r>}
   @type bilinearform: function C{bilinearform(x,r)} where C{x} is of the same type like argument C{x} and C{r} is.
       The returned value is a C{float}.
   @param stoppingcriterium: function which returns True if a stopping criterium is meet.
       C{stoppingcriterium} has the arguments C{norm_r}, C{r} and C{x} giving the current 
       norm of the residual (=C{sqrt(bilinearform(Msolve(r),r)}), the current residual and
       the current solution approximation. C{stoppingcriterium} is called in each iteration step.
   @type stoppingcriterium: function that returns C{True} or C{False}
   @param x: an initial guess for the solution. If no C{x} is given 0*b is used.
   @type x: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
   @param iter_max: maximum number of iteration steps.
   @type iter_max: C{int}
   @return: the solution approximation and the corresponding residual
   @rtype: C{tuple} 
   @warning: C{b} and C{x} are altered.
   """
   iter=0
   if x==None:
      x=0*b
      r=b*1.
      initial_guess=False
   if initial_guess:
      r=b+(-1)*Aprod(x) 
      x=x*1.
   else:
      x=x*0
      r=b*1
   rhat=Msolve(r)
   d = rhat 
   rhat_dot_r = bilinearform(rhat, r)
   if rhat_dot_r<0: raise NegativeNorm,"negative norm."
   norm_b=math.sqrt(rhat_dot_r)
   if verbose: print "PCG: initial residual norm = %e (absolute tolerance = %e)"%(norm_b, atol+rtol*norm_b)

   while not math.sqrt(rhat_dot_r) <= atol+rtol*norm_b:
       iter+=1
       if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max

       q=Aprod(d)
       alpha = rhat_dot_r / bilinearform(d, q)
       x += alpha * d
       r += (-alpha) * q

       rhat=Msolve(r)
       rhat_dot_r_new = bilinearform(rhat, r)
       beta = rhat_dot_r_new / rhat_dot_r
       rhat+=beta * d
       d=rhat

       rhat_dot_r = rhat_dot_r_new
       if rhat_dot_r<0: raise NegativeNorm,"negative norm."
       if verbose: print "PCG: iteration step %s: residual norm = %e"%(iter, math.sqrt(rhat_dot_r))
   if verbose: print "PCG: tolerance reached after %s steps."%iter
   return x,r

class Defect(object):
    """
    defines a non-linear defect F(x) of a variable x
    """
    def __init__(self):
        """
        initialize defect
        """
        self.setDerivativeIncrementLength()

    def bilinearform(self, x0, x1):
        """
        returns the inner product of x0 and x1
        @param x0: a value for x
        @param x1: a value for x
        @return: the inner product of x0 and x1
        @rtype: C{float}
        """
        return 0
      
    def norm(self,x):
        """
        the norm of argument C{x}
 
        @param x: a value for x
        @return: norm of argument x
        @rtype: C{float}
        @note: by default C{sqrt(self.bilinearform(x,x)} is retrurned.
        """
        s=self.bilinearform(x,x)
        if s<0: raise NegativeNorm,"negative norm."
        return math.sqrt(s)


    def eval(self,x):
        """
        returns the value F of a given x

        @param x: value for which the defect C{F} is evalulated.
        @return: value of the defect at C{x}
        """
        return 0

    def __call__(self,x):
        return self.eval(x)

    def setDerivativeIncrementLength(self,inc=math.sqrt(util.EPSILON)):
        """
        sets the relative length of the increment used to approximate the derivative of the defect
        the increment is inc*norm(x)/norm(v)*v in the direction of v with x as a staring point.

        @param inc: relative increment length
        @type inc: positive C{float}
        """
        if inc<=0: raise ValueError,"positive increment required."
        self.__inc=inc

    def getDerivativeIncrementLength(self):
        """
        returns the relative increment length used to approximate the derivative of the defect
        @return: value of the defect at C{x}
        @rtype: positive C{float}
        """
        return self.__inc

    def derivative(self, F0, x0, v, v_is_normalised=True):
        """
        returns the directional derivative at x0 in the direction of v

        @param F0: value of this defect at x0
        @param x0: value at which derivative is calculated.
        @param v: direction
        @param v_is_normalised: is true to indicate that C{v} is nomalized (self.norm(v)=0)
        @return: derivative of this defect at x0 in the direction of C{v}
        @note: by default numerical evaluation (self.eval(x0+eps*v)-F0)/eps is used but this method
        maybe oepsnew verwritten to use exact evalution.
        """
        normx=self.norm(x0)
        if normx>0:
             epsnew = self.getDerivativeIncrementLength() * normx
        else:
             epsnew = self.getDerivativeIncrementLength()
        if not v_is_normalised:
            normv=self.norm(v)
            if normv<=0:
               return F0*0
            else:
               epsnew /= normv
        F1=self.eval(x0 + epsnew * v)
        return (F1-F0)/epsnew

######################################    
def NewtonGMRES(defect, x, iter_max=100, sub_iter_max=20, atol=0,rtol=1.e-4, sub_tol_max=0.5, gamma=0.9, verbose=False):
   """
   solves a non-linear problem M{F(x)=0} for unknown M{x} using the stopping criterion:

   M{norm(F(x) <= atol + rtol * norm(F(x0)}
  
   where M{x0} is the initial guess.

   @param defect: object defining the the function M{F}, C{defect.norm} defines the M{norm} used in the stopping criterion.
   @type defect: L{Defect}
   @param x: initial guess for the solution, C{x} is altered.
   @type x: any object type allowing basic operations such as  L{numarray.NumArray}, L{Data}
   @param iter_max: maximum number of iteration steps 
   @type iter_max: positive C{int}
   @param sub_iter_max: 
   @type sub_iter_max:
   @param atol: absolute tolerance for the solution
   @type atol: positive C{float}
   @param rtol: relative tolerance for the solution
   @type rtol: positive C{float}
   @param gamma: tolerance safety factor for inner iteration
   @type gamma: positive C{float}, less than 1
   @param sub_tol_max: upper bound for inner tolerance.
   @type sub_tol_max: positive C{float}, less than 1
   @return: an approximation of the solution with the desired accuracy
   @rtype: same type as the initial guess.
   """
   lmaxit=iter_max
   if atol<0: raise ValueError,"atol needs to be non-negative."
   if rtol<0: raise ValueError,"rtol needs to be non-negative."
   if rtol+atol<=0: raise ValueError,"rtol or atol needs to be non-negative."
   if gamma<=0 or gamma>=1: raise ValueError,"tolerance safety factor for inner iteration (gamma =%s) needs to be positive and less than 1."%gamma
   if sub_tol_max<=0 or sub_tol_max>=1: raise ValueError,"upper bound for inner tolerance for inner iteration (sub_tol_max =%s) needs to be positive and less than 1."%sub_tol_max

   F=defect(x)
   fnrm=defect.norm(F)
   stop_tol=atol + rtol*fnrm
   sub_tol=sub_tol_max
   if verbose: print "NewtonGMRES: initial residual = %e."%fnrm
   if verbose: print "             tolerance = %e."%sub_tol
   iter=1
   #
   # main iteration loop
   #
   while not fnrm<=stop_tol:
            if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max 
            #
        #   adjust sub_tol_ 
        #
            if iter > 1:
           rat=fnrm/fnrmo
               sub_tol_old=sub_tol
           sub_tol=gamma*rat**2
           if gamma*sub_tol_old**2 > .1: sub_tol=max(sub_tol,gamma*sub_tol_old**2)
           sub_tol=max(min(sub_tol,sub_tol_max), .5*stop_tol/fnrm)
        #
        # calculate newton increment xc
            #     if iter_max in __FDGMRES is reached MaxIterReached is thrown
            #     if iter_restart -1 is returned as sub_iter
            #     if  atol is reached sub_iter returns the numer of steps performed to get there
            # 
            #   
            if verbose: print "             subiteration (GMRES) is called with relative tolerance %e."%sub_tol
            try:
               xc, sub_iter=__FDGMRES(F, defect, x, sub_tol*fnrm, iter_max=iter_max-iter, iter_restart=sub_iter_max)
            except MaxIterReached:
               raise MaxIterReached,"maximum number of %s steps reached."%iter_max
            if sub_iter<0:
               iter+=sub_iter_max
            else:
               iter+=sub_iter
            # ====
        x+=xc
            F=defect(x)
        iter+=1
            fnrmo, fnrm=fnrm, defect.norm(F)
            if verbose: print "             step %s: residual %e."%(iter,fnrm)
   if verbose: print "NewtonGMRES: completed after %s steps."%iter
   return x

def __givapp(c,s,vin):
    """
    apply a sequence of Givens rotations (c,s) to the recuirsively to the vector vin
    @warning: C{vin} is altered.
    """
    vrot=vin 
    if isinstance(c,float):
        vrot=[c*vrot[0]-s*vrot[1],s*vrot[0]+c*vrot[1]]
    else:
        for i in range(len(c)):
            w1=c[i]*vrot[i]-s[i]*vrot[i+1]
        w2=s[i]*vrot[i]+c[i]*vrot[i+1]
            vrot[i:i+2]=w1,w2
    return vrot

def __FDGMRES(F0, defect, x0, atol, iter_max=100, iter_restart=20):
   h=numarray.zeros((iter_restart,iter_restart),numarray.Float64)
   c=numarray.zeros(iter_restart,numarray.Float64)
   s=numarray.zeros(iter_restart,numarray.Float64)
   g=numarray.zeros(iter_restart,numarray.Float64)
   v=[]

   rho=defect.norm(F0)
   if rho<=0.: return x0*0
   
   v.append(-F0/rho)
   g[0]=rho
   iter=0
   while rho > atol and iter<iter_restart-1:

    if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max

        p=defect.derivative(F0,x0,v[iter], v_is_normalised=True)
    v.append(p)

    v_norm1=defect.norm(v[iter+1])

        # Modified Gram-Schmidt 
    for j in range(iter+1): 
         h[j,iter]=defect.bilinearform(v[j],v[iter+1])   
         v[iter+1]-=h[j,iter]*v[j]
       
    h[iter+1,iter]=defect.norm(v[iter+1])
    v_norm2=h[iter+1,iter]

        # Reorthogonalize if needed
    if v_norm1 + 0.001*v_norm2 == v_norm1:   #Brown/Hindmarsh condition (default)
        for j in range(iter+1):  
           hr=defect.bilinearform(v[j],v[iter+1])
               h[j,iter]=h[j,iter]+hr 
               v[iter+1] -= hr*v[j]

        v_norm2=defect.norm(v[iter+1])
        h[iter+1,iter]=v_norm2
        #   watch out for happy breakdown 
        if not v_norm2 == 0:
                v[iter+1]=v[iter+1]/h[iter+1,iter]

        #   Form and store the information for the new Givens rotation
    if iter > 0 :
        hhat=numarray.zeros(iter+1,numarray.Float64)
        for i in range(iter+1) : hhat[i]=h[i,iter]
        hhat=__givapp(c[0:iter],s[0:iter],hhat);
            for i in range(iter+1) : h[i,iter]=hhat[i]

    mu=math.sqrt(h[iter,iter]*h[iter,iter]+h[iter+1,iter]*h[iter+1,iter])

    if mu!=0 :
        c[iter]=h[iter,iter]/mu
        s[iter]=-h[iter+1,iter]/mu
        h[iter,iter]=c[iter]*h[iter,iter]-s[iter]*h[iter+1,iter]
        h[iter+1,iter]=0.0
        g[iter:iter+2]=__givapp(c[iter],s[iter],g[iter:iter+2])

        # Update the residual norm
        rho=abs(g[iter+1])
    iter+=1

   # At this point either iter > iter_max or rho < tol.
   # It's time to compute x and leave.        
   if iter > 0 : 
     y=numarray.zeros(iter,numarray.Float64)    
     y[iter-1] = g[iter-1] / h[iter-1,iter-1]
     if iter > 1 :  
        i=iter-2   
        while i>=0 :
          y[i] = ( g[i] - numarray.dot(h[i,i+1:iter], y[i+1:iter])) / h[i,i]
          i=i-1
     xhat=v[iter-1]*y[iter-1]
     for i in range(iter-1):
    xhat += v[i]*y[i]
   else : 
      xhat=v[0] * 0

   if iter<iter_restart-1: 
      stopped=iter
   else: 
      stopped=-1

   return xhat,stopped

##############################################
def GMRES(b, Aprod, bilinearform, atol=0, rtol=1.e-8, x=None, iter_max=100, iter_restart=20, verbose=False):
################################################
   m=iter_restart
   iter=0
   if x == None:
      xc=b*0 
   else:
      xc=x*1
   if rtol>0:
      r_dot_r = bilinearform(b, b)
      if r_dot_r<0: raise NegativeNorm,"negative norm."
      atol2=atol+rtol*math.sqrt(r_dot_r)
      if verbose: print "GMRES: norm of right hand side = %e (absolute tolerance = %e)"%(math.sqrt(r_dot_r), atol2)
   else:
      atol2=atol
      if verbose: print "GMRES: absolute tolerance = %e"%atol2
   
   while True:
      if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached"%iter_max
      xc,stopped=__GMRESm(b, Aprod, bilinearform, atol2, x=xc, iter_max=iter_max-iter, iter_restart=m, verbose=verbose)
      iter+=iter_restart    
      if stopped: break
      if verbose: print "GMRES: restart."
   if verbose: print "GMRES: tolerance has reached."
   return xc

def __GMRESm(b, Aprod, bilinearform, atol, x, iter_max=100, iter_restart=20, verbose=False):
   iter=0
   
   h=numarray.zeros((iter_restart+1,iter_restart),numarray.Float64)
   c=numarray.zeros(iter_restart,numarray.Float64)
   s=numarray.zeros(iter_restart,numarray.Float64)
   g=numarray.zeros(iter_restart+1,numarray.Float64)
   v=[]

   r=b-Aprod(x)
   r_dot_r = bilinearform(r, r)
   if r_dot_r<0: raise NegativeNorm,"negative norm."
   rho=math.sqrt(r_dot_r)
   
   v.append(r/rho)
   g[0]=rho

   if verbose: print "GMRES: initial residual %e (absolute tolerance = %e)"%(rho,atol)
   while not (rho<=atol or iter==iter_restart):

    if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max

    p=Aprod(v[iter])
    v.append(p)

    v_norm1=math.sqrt(bilinearform(v[iter+1], v[iter+1]))  

# Modified Gram-Schmidt 
    for j in range(iter+1): 
      h[j,iter]=bilinearform(v[j],v[iter+1])   
      v[iter+1]-=h[j,iter]*v[j]
       
    h[iter+1,iter]=math.sqrt(bilinearform(v[iter+1],v[iter+1])) 
    v_norm2=h[iter+1,iter]

# Reorthogonalize if needed
    if v_norm1 + 0.001*v_norm2 == v_norm1:   #Brown/Hindmarsh condition (default)
     for j in range(iter+1):  
        hr=bilinearform(v[j],v[iter+1])
            h[j,iter]=h[j,iter]+hr 
            v[iter+1] -= hr*v[j]

     v_norm2=math.sqrt(bilinearform(v[iter+1], v[iter+1]))  
     h[iter+1,iter]=v_norm2

#   watch out for happy breakdown 
        if not v_norm2 == 0:
         v[iter+1]=v[iter+1]/h[iter+1,iter]

#   Form and store the information for the new Givens rotation
    if iter > 0: h[:iter+1,iter]=__givapp(c[:iter],s[:iter],h[:iter+1,iter])
    mu=math.sqrt(h[iter,iter]*h[iter,iter]+h[iter+1,iter]*h[iter+1,iter])

    if mu!=0 :
        c[iter]=h[iter,iter]/mu
        s[iter]=-h[iter+1,iter]/mu
        h[iter,iter]=c[iter]*h[iter,iter]-s[iter]*h[iter+1,iter]
        h[iter+1,iter]=0.0
        g[iter:iter+2]=__givapp(c[iter],s[iter],g[iter:iter+2])
# Update the residual norm
               
        rho=abs(g[iter+1])
        if verbose: print "GMRES: iteration step %s: residual %e"%(iter,rho)
    iter+=1

# At this point either iter > iter_max or rho < tol.
# It's time to compute x and leave.        

   if verbose: print "GMRES: iteration stopped after %s step."%iter
   if iter > 0 : 
     y=numarray.zeros(iter,numarray.Float64)    
     y[iter-1] = g[iter-1] / h[iter-1,iter-1]
     if iter > 1 :  
        i=iter-2   
        while i>=0 :
          y[i] = ( g[i] - numarray.dot(h[i,i+1:iter], y[i+1:iter])) / h[i,i]
          i=i-1
     xhat=v[iter-1]*y[iter-1]
     for i in range(iter-1):
    xhat += v[i]*y[i]
   else: 
     xhat=v[0] * 0

   x += xhat
   if iter<iter_restart-1: 
      stopped=True 
   else: 
      stopped=False

   return x,stopped

#################################################
def MINRES(b, Aprod, Msolve, bilinearform, atol=0, rtol=1.e-8, x=None, iter_max=100):
#################################################
    #
    #  minres solves the system of linear equations Ax = b
    #  where A is a symmetric matrix (possibly indefinite or singular)
    #  and b is a given vector.
    #  
    #  "A" may be a dense or sparse matrix (preferably sparse!)
    #  or the name of a function such that
    #               y = A(x)
    #  returns the product y = Ax for any given vector x.
    #
    #  "M" defines a positive-definite preconditioner M = C C'.
    #  "M" may be a dense or sparse matrix (preferably sparse!)
    #  or the name of a function such that
    #  solves the system My = x for any given vector x.
    #
    #
    
    #------------------------------------------------------------------
    # Set up y and v for the first Lanczos vector v1.
    # y  =  beta1 P' v1,  where  P = C**(-1).
    # v is really P' v1.
    #------------------------------------------------------------------
    if x==None:
      x=0*b
    else:
      b += (-1)*Aprod(x) 

    r1    = b
    y = Msolve(b)
    beta1 = bilinearform(y,b)
 
    if beta1< 0: raise NegativeNorm,"negative norm."

    #  If b = 0 exactly, stop with x = 0.
    if beta1==0: return x*0.

    if beta1> 0:
      beta1  = math.sqrt(beta1)       

    #------------------------------------------------------------------
    # Initialize quantities.
    # ------------------------------------------------------------------
    iter   = 0
    Anorm = 0
    ynorm = 0
    oldb   = 0
    beta   = beta1
    dbar   = 0
    epsln  = 0
    phibar = beta1
    rhs1   = beta1
    rhs2   = 0
    rnorm  = phibar
    tnorm2 = 0
    ynorm2 = 0
    cs     = -1
    sn     = 0
    w      = b*0.
    w2     = b*0.
    r2     = r1
    eps    = 0.0001

    #---------------------------------------------------------------------
    # Main iteration loop.
    # --------------------------------------------------------------------
    while not rnorm<=atol+rtol*Anorm*ynorm:    #  checks ||r|| < (||A|| ||x||) * TOL

    if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
        iter    = iter  +  1

        #-----------------------------------------------------------------
        # Obtain quantities for the next Lanczos vector vk+1, k = 1, 2,...
        # The general iteration is similar to the case k = 1 with v0 = 0:
        #
        #   p1      = Operator * v1  -  beta1 * v0,
        #   alpha1  = v1'p1,
        #   q2      = p2  -  alpha1 * v1,
        #   beta2^2 = q2'q2,
        #   v2      = (1/beta2) q2.
        #
        # Again, y = betak P vk,  where  P = C**(-1).
        #-----------------------------------------------------------------
        s = 1/beta                 # Normalize previous vector (in y).
        v = s*y                    # v = vk if P = I
     
        y      = Aprod(v)
    
        if iter >= 2:
          y = y - (beta/oldb)*r1

        alfa   = bilinearform(v,y)              # alphak
        y      += (- alfa/beta)*r2 
        r1     = r2
        r2     = y
        y = Msolve(r2)
        oldb   = beta                           # oldb = betak
        beta   = bilinearform(y,r2)             # beta = betak+1^2
        if beta < 0: raise NegativeNorm,"negative norm."

        beta   = math.sqrt( beta )
        tnorm2 = tnorm2 + alfa*alfa + oldb*oldb + beta*beta
        
        if iter==1:                 # Initialize a few things.
          gmax   = abs( alfa )      # alpha1
          gmin   = gmax             # alpha1

        # Apply previous rotation Qk-1 to get
        #   [deltak epslnk+1] = [cs  sn][dbark    0   ]
        #   [gbar k dbar k+1]   [sn -cs][alfak betak+1].
    
        oldeps = epsln
        delta  = cs * dbar  +  sn * alfa  # delta1 = 0         deltak
        gbar   = sn * dbar  -  cs * alfa  # gbar 1 = alfa1     gbar k
        epsln  =               sn * beta  # epsln2 = 0         epslnk+1
        dbar   =            -  cs * beta  # dbar 2 = beta2     dbar k+1

        # Compute the next plane rotation Qk

        gamma  = math.sqrt(gbar*gbar+beta*beta)  # gammak
        gamma  = max(gamma,eps) 
        cs     = gbar / gamma             # ck
        sn     = beta / gamma             # sk
        phi    = cs * phibar              # phik
        phibar = sn * phibar              # phibark+1

        # Update  x.

        denom = 1/gamma 
        w1    = w2 
        w2    = w 
        w     = (v - oldeps*w1 - delta*w2) * denom
        x     +=  phi*w

        # Go round again.

        gmax   = max(gmax,gamma)
        gmin   = min(gmin,gamma)
        z      = rhs1 / gamma
        ynorm2 = z*z  +  ynorm2
        rhs1   = rhs2 -  delta*z
        rhs2   =      -  epsln*z

        # Estimate various norms and test for convergence.

        Anorm  = math.sqrt( tnorm2 ) 
        ynorm  = math.sqrt( ynorm2 ) 

        rnorm  = phibar

    return x

def TFQMR(b, Aprod, Msolve, bilinearform, atol=0, rtol=1.e-8, x=None, iter_max=100):

# TFQMR solver for linear systems
#
#
# initialization
#
  errtol = math.sqrt(bilinearform(b,b))
  norm_b=errtol
  kmax  = iter_max
  error = []

  if math.sqrt(bilinearform(x,x)) != 0.0:
    r = b - Aprod(x)
  else:
    r = b

  r=Msolve(r)

  u1=0
  u2=0
  y1=0
  y2=0

  w = r
  y1 = r 
  iter = 0 
  d = 0
  
  v = Msolve(Aprod(y1))
  u1 = v
  
  theta = 0.0;
  eta = 0.0;
  tau = math.sqrt(bilinearform(r,r))
  error = [ error, tau ] 
  rho = tau * tau
#
#  TFQMR iteration
#
#  while ( iter < kmax-1 ):
   
  while not tau<=atol+rtol*norm_b:
    if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max

    sigma = bilinearform(r,v)

    if ( sigma == 0.0 ):
      raise 'TFQMR breakdown, sigma=0'
    

    alpha = rho / sigma

    for j in range(2):
#
#   Compute y2 and u2 only if you have to
#
      if ( j == 1 ):
        y2 = y1 - alpha * v
        u2 = Msolve(Aprod(y2))
 
      m = 2 * (iter+1) - 2 + (j+1)
      if j==0: 
         w = w - alpha * u1
         d = y1 + ( theta * theta * eta / alpha ) * d
      if j==1:
         w = w - alpha * u2
         d = y2 + ( theta * theta * eta / alpha ) * d

      theta = math.sqrt(bilinearform(w,w))/ tau
      c = 1.0 / math.sqrt ( 1.0 + theta * theta )
      tau = tau * theta * c
      eta = c * c * alpha
      x = x + eta * d
#
#  Try to terminate the iteration at each pass through the loop
#
     # if ( tau * math.sqrt ( m + 1 ) <= errtol ):
     #   error = [ error, tau ]
     #   total_iters = iter
     #   break
      

    if ( rho == 0.0 ):
      raise 'TFQMR breakdown, rho=0'
    

    rhon = bilinearform(r,w)
    beta = rhon / rho;
    rho = rhon;
    y1 = w + beta * y2;
    u1 = Msolve(Aprod(y1))
    v = u1 + beta * ( u2 + beta * v )
    error = [ error, tau ]
    total_iters = iter
    
    iter = iter + 1

  return x


#############################################

class ArithmeticTuple(object):
   """
   tuple supporting inplace update x+=y and scaling x=a*y where x,y is an ArithmeticTuple and a is a float.

   example of usage:

   from esys.escript import Data
   from numarray import array
   a=Data(...)
   b=array([1.,4.])
   x=ArithmeticTuple(a,b)
   y=5.*x

   """
   def __init__(self,*args):
       """
       initialize object with elements args.

       @param args: tuple of object that support implace add (x+=y) and scaling (x=a*y)
       """
       self.__items=list(args)

   def __len__(self):
       """
       number of items

       @return: number of items
       @rtype: C{int}
       """
       return len(self.__items)

   def __getitem__(self,index):
       """
       get an item

       @param index: item to be returned
       @type index: C{int}
       @return: item with index C{index}
       """
       return self.__items.__getitem__(index)

   def __mul__(self,other):
       """
       scaling from the right 

       @param other: scaling factor
       @type other: C{float}
       @return: itemwise self*other
       @rtype: L{ArithmeticTuple}
       """
       out=[]
       try:  
           l=len(other)
           if l!=len(self):
               raise ValueError,"length of of arguments don't match."
           for i in range(l): out.append(self[i]*other[i])
       except TypeError:
       for i in range(len(self)): out.append(self[i]*other)
       return ArithmeticTuple(*tuple(out))

   def __rmul__(self,other):
       """
       scaling from the left

       @param other: scaling factor
       @type other: C{float}
       @return: itemwise other*self
       @rtype: L{ArithmeticTuple}
       """
       out=[]
       try:  
           l=len(other)
           if l!=len(self):
               raise ValueError,"length of of arguments don't match."
           for i in range(l): out.append(other[i]*self[i])
       except TypeError:
       for i in range(len(self)): out.append(other*self[i])
       return ArithmeticTuple(*tuple(out))

   def __div__(self,other):
       """
       dividing from the right 

       @param other: scaling factor
       @type other: C{float}
       @return: itemwise self/other
       @rtype: L{ArithmeticTuple}
       """
       return self*(1/other)

   def __rdiv__(self,other):
       """
       dividing from the left

       @param other: scaling factor
       @type other: C{float}
       @return: itemwise other/self
       @rtype: L{ArithmeticTuple}
       """
       out=[]
       try:  
           l=len(other)
           if l!=len(self):
               raise ValueError,"length of of arguments don't match."
           for i in range(l): out.append(other[i]/self[i])
       except TypeError:
       for i in range(len(self)): out.append(other/self[i])
       return ArithmeticTuple(*tuple(out))
  
   def __iadd__(self,other):
       """
       in-place add of other to self

       @param other: increment
       @type other: C{ArithmeticTuple}
       """
       if len(self) != len(other):
           raise ValueError,"tuple length must match."
       for i in range(len(self)):
           self.__items[i]+=other[i]
       return self

   def __add__(self,other):
       """
       add other to self

       @param other: increment
       @type other: C{ArithmeticTuple}
       """
       out=[]
       try:  
           l=len(other)
           if l!=len(self):
               raise ValueError,"length of of arguments don't match."
           for i in range(l): out.append(self[i]+other[i])
       except TypeError:
       for i in range(len(self)): out.append(self[i]+other)
       return ArithmeticTuple(*tuple(out))

   def __sub__(self,other):
       """
       subtract other from self

       @param other: increment
       @type other: C{ArithmeticTuple}
       """
       out=[]
       try:  
           l=len(other)
           if l!=len(self):
               raise ValueError,"length of of arguments don't match."
           for i in range(l): out.append(self[i]-other[i])
       except TypeError:
       for i in range(len(self)): out.append(self[i]-other)
       return ArithmeticTuple(*tuple(out))
   
   def __isub__(self,other):
       """
       subtract other from self

       @param other: increment
       @type other: C{ArithmeticTuple}
       """
       if len(self) != len(other):
           raise ValueError,"tuple length must match."
       for i in range(len(self)):
           self.__items[i]-=other[i]
       return self

   def __neg__(self):
       """
       negate 

       """
       out=[]
       for i in range(len(self)):
           out.append(-self[i])
       return ArithmeticTuple(*tuple(out))


class HomogeneousSaddlePointProblem(object):
      """
      This provides a framwork for solving linear homogeneous saddle point problem of the form

             Av+B^*p=f
             Bv    =0

      for the unknowns v and p and given operators A and B and given right hand side f.
      B^* is the adjoint operator of B.
      """
      def __init__(self,**kwargs):
        self.setTolerance()
        self.setAbsoluteTolerance()
        self.setSubToleranceReductionFactor()

      #=============================================================
      def initialize(self):
        """
        initialize the problem (overwrite)
        """
        pass

      def B(self,v):
        """
        returns Bv (overwrite)
        @rtype: equal to the type of p

        @note: boundary conditions on p should be zero!
        """
        raise NotImplementedError,"no operator B implemented."

      def inner_pBv(self,p,Bv):
         """
         returns inner product of element p and Bv  (overwrite)
         
         @type p: equal to the type of p
         @type Bv: equal to the type of result of operator B
         @rtype: C{float}

         @rtype: equal to the type of p
         """
         raise NotImplementedError,"no inner product for p implemented."

      def inner_p(self,p0,p1):
         """
         returns inner product of element p0 and p1  (overwrite)
         
         @type p0: equal to the type of p
         @type p1: equal to the type of p
         @rtype: equal to the type of p
         """
         raise NotImplementedError,"no inner product for p implemented."

      def inner_v(self,v0,v1):
         """
         returns inner product of two element v0 and v1  (overwrite)
         
         @type v0: equal to the type of v
         @type v1: equal to the type of v
         @rtype: C{float}

         @rtype: equal to the type of v
         """
         raise NotImplementedError,"no inner product for v implemented."
         pass

      def solve_A(self,u,p):
         """
         solves Av=f-Au-B^*p with accuracy self.getSubProblemTolerance() (overwrite) 

         @rtype: equal to the type of v
         @note: boundary conditions on v should be zero!
         """
         raise NotImplementedError,"no operator A implemented."

      def solve_prec(self,p):
         """
         provides a preconditioner for BA^{-1}B^* with accuracy self.getSubProblemTolerance() (overwrite)

         @rtype: equal to the type of p
         @note: boundary conditions on p should be zero!
         """
         raise NotImplementedError,"no preconditioner for Schur complement implemented."
      #=============================================================
      def __Aprod_PCG(self,p):
          # return (v,Bv) with v=A^-1B*p 
          #solve Av =B^*p as Av =f-Az-B^*(-p)
          v=self.solve_A(self.__z,-p)
          return ArithmeticTuple(v, self.B(v))

      def __inner_PCG(self,p,r):
         return self.inner_pBv(p,r[1])

      def __Msolve_PCG(self,r):
          return self.solve_prec(r[1])
      #=============================================================
      def __Aprod_GMRES(self,x):
          # return w,q  from v, p
          # solve Aw =Av+B^*p as Aw =f-A(z-v)-B^*(-p)
          #  and  Sq=B(v-w)
          v=x[0]
          p=x[1]
          w=self.solve_A(self.__z-v,-p)
          Bw=self.B(w-v)
      q=self.solve_prec(Bw)
          return ArithmeticTuple(w,q)

      def __inner_GMRES(self,a1,a2):
         return self.inner_p(a1[1],a2[1])+self.inner_v(a1[0],a2[0])

      #=============================================================
      def norm(self,v,p):
        f=self.inner_p(p,p)+self.inner_v(v,v)
        if f<0:
            raise ValueError,"negative norm."
        return math.sqrt(f)

      def solve(self,v,p,max_iter=20, verbose=False, show_details=False, useUzawa=True, iter_restart=20,max_correction_steps=3):
         """
         solves the saddle point problem using initial guesses v and p.

         @param v: initial guess for velocity
         @param p: initial guess for pressure
         @type v: L{Data}
         @type p: L{Data}
         @param useUzawa: indicate the usage of the Uzawa scheme. Otherwise the problem is solved in coupled form.
         @param max_iter: maximum number of iteration steps per correction attempt.
         @param verbose: show information on the progress of the saddlepoint problem solver.
         @param show_details: show details of the sub problems solves
         @param iter_restart: restart the iteration after C{iter_restart} steps (only used if useUzaw=False)
         @param max_correction_steps: maximum number of iteration steps in the attempt get |Bv| to zero.
         @return: new approximations for velocity and pressure
         @type useUzawa: C{bool}
         @type max_iter: C{int}
         @type verbose: C{bool}
         @type show_details: C{bool}
         @type iter_restart: C{int}
         @type max_correction_steps: C{int}
         @rtype: C{tuple} of L{Data} objects
         """
         self.verbose=verbose
         self.show_details=show_details and self.verbose
         #=================================================================================
         # Az=f is solved as A(z-v)=f-Av-B^*0 (typically with z-v=0 on boundary)
         self.setSubProblemTolerance(self.getTolerance(), apply_reduction=True)
         self.__z=v+self.solve_A(v,p*0)
         Bz=self.B(self.__z) 
         p0=self.solve_prec(Bz)
         # tolerances:
         rtol=self.getTolerance()
         atol=self.getAbsoluteTolerance()
         f0=self.norm(self.__z,p0)
         if not f0>0:
            return self.__z, p*0
         ATOL=rtol*f0+atol
         if not ATOL>0:
            raise ValueError,"overall absolute tolerance needs to be positive."
         if self.verbose: print "saddle point solver: initial residual %e, tolerance = %e."%(f0,ATOL)
         # initialization
         self.iter=0
         correction_step=0
         converged=False
         # initial guess:
         q=p*1
         if not useUzawa:
           u=v
         while not converged :
            if useUzawa:
               self.setSubProblemTolerance(ATOL/f0, apply_reduction=True)
               # assume p is known: then v=z-A^-1B^*p
               #      
               # which leads to BA^-1B^*p = Bz
               #
               # note that the residual r=Bz-BA^-1B^*p = B(z-A^-1B^*p) = Bu
               #
               # note that q=BA^-1B^*p = B (A^-1B^*p)=Bw  with Aw=B^*p = f - Az - B^*(-p)
               #
               if self.verbose: print "enter PCG method (iter_max=%s, atol=%e, subtolerance=%e)"%(max_iter,ATOL, self.getSubProblemTolerance())
               q,r=PCG(ArithmeticTuple(self.__z,Bz),self.__Aprod_PCG,self.__Msolve_PCG,self.__inner_PCG,atol=ATOL, rtol=0.,iter_max=max_iter, x=q, verbose=self.verbose)
           u=r[0]  
           Bu=r[1]  
            else:
               self.setSubProblemTolerance(ATOL/f0, apply_reduction=True)
               #
               #  with v=dv+z
               #
               #   A dv + B p = A (v-z) + Bp = 0
               #   B dv       =  - Bz 
               #
               # apply the preconditioner [[A^{-1} 0][(S^{-1} B A^{-1})  -S^{-1}]] to the problem 
               # the right hand side is then [0, S^{-1} Bz ]
               #
               #           
               if self.verbose: print "enter GMRES (iter_max=%s, atol=%e, subtolerance=%e)"%(max_iter,ATOL,self.getSubProblemTolerance())
               x=GMRES(ArithmeticTuple(0*self.__z,p0),self.__Aprod_GMRES,self.__inner_GMRES,atol=ATOL, rtol=0.,iter_max=max_iter, x=ArithmeticTuple(u-self.__z,q),iter_restart=iter_restart, verbose=self.verbose)
               self.setSubProblemTolerance(ATOL/f0, apply_reduction=False)
           u=self.__z+x[0]
               q=x[1]
           Bu=self.B(u)
            # now we check if |Bu| ~ 0 or more precise |Bu|  <= rtol * |v| 
            nu=self.inner_v(u,u)
            nz=self.inner_v(self.__z,self.__z)
            p2=self.solve_prec(Bu)
            nBu=self.inner_p(p2,p2)
            if not nu>=0 and not nBu>=0: raise NegativeNorm,"negative norm."
            nu=math.sqrt(nu)
            nBu=math.sqrt(nBu)
            if self.verbose: print "saddle point solver: norm v= %e (Bv = %e)"%(nu,nBu)
            QTOL=atol+nu*rtol
            if nBu <= QTOL:
                converged=True
            else:
                ATOL=QTOL/nBu*ATOL*0.3
                if self.verbose: print "correction step %s: tolerance reduced to %e."%(correction_step,ATOL)
                converged=False
            correction_step+=1
            if correction_step>max_correction_steps:
               raise CorrectionFailed,"Given up after %d correction steps."%correction_step
         if self.verbose: print "saddle point solver: tolerance reached."
     return u,q

      #==========================================================================================================================
      def setTolerance(self,tolerance=1.e-4):
         """
         sets the relative tolerance for (v,p)

         @param tolerance: tolerance to be used 
         @type tolerance: non-negative C{float}
         """
         if tolerance<0:
             raise ValueError,"tolerance must be positive."
         self.__rtol=tolerance
      def getTolerance(self):
         """
         returns the relative tolerance

         @return: relative tolerance 
         @rtype: C{float}
         """
         return self.__rtol
      def setAbsoluteTolerance(self,tolerance=0.):
         """
         sets the absolute tolerance 

         @param tolerance: tolerance to be used 
         @type tolerance: non-negative C{float}
         """
         if tolerance<0:
             raise ValueError,"tolerance must be non-negative."
         self.__atol=tolerance
      def getAbsoluteTolerance(self):
         """
         returns the absolute tolerance

         @return: absolute tolerance 
         @rtype: C{float}
         """
         return self.__atol
      def setSubToleranceReductionFactor(self,reduction=None):
         """
         sets the reduction factor for the tolerance used to solve the subproblem in each iteration step.
         if set to C{None} the square of the problem tolerance is used.
    
         @param reduction: reduction factor.
         @type reduction: positive C{float} less or equal one or C{None}
         """
         if not reduction == None:
              if reduction<=0 or reduction>1:
                  raise ValueError,"reduction factor (=%e) must be positive and less or equal one."%reduction
         self.__reduction=reduction
      def getSubToleranceReductionFactor(self):
         """
         returns the subproblem reduction factor

         @return: subproblem reduction factor
         @rtype: C{float} or C{None}
         """
         return self.__reduction

      def setSubProblemTolerance(self,rtol,apply_reduction=False):
         """
         sets the relative tolerance to solve the subproblem(s). 

         @param rtol: relative tolerence
         @type rtol: positive C{float}
         @param apply_reduction: if set the the reduction factor is applied.
         @type apply_reduction: C{bool}
         """
         if rtol<=0:
             raise ValueError,"tolerance must be positive."
         if apply_reduction:
             if self.__reduction == None:
                 self.__sub_tol=max(rtol**2,util.EPSILON*10)
             else:
                 self.__sub_tol=max(self.__reduction*rtol,util.EPSILON*10)
         else:
             self.__sub_tol=rtol
      def getSubProblemTolerance(self):
         """
         returns the subproblem reduction factor

         @return: subproblem reduction factor
         @rtype: C{float} 
         """
         return self.__sub_tol

def MaskFromBoundaryTag(domain,*tags):
   """
   create a mask on the Solution(domain) function space which one for samples 
   that touch the boundary tagged by tags.

   usage: m=MaskFromBoundaryTag(domain,"left", "right")

   @param domain: a given domain
   @type domain: L{escript.Domain}
   @param tags: boundray tags
   @type tags: C{str}
   @return: a mask which marks samples that are touching the boundary tagged by any of the given tags.
   @rtype: L{escript.Data} of rank 0
   """
   pde=linearPDEs.LinearPDE(domain,numEquations=1, numSolutions=1)
   d=escript.Scalar(0.,escript.FunctionOnBoundary(domain))
   for t in tags: d.setTaggedValue(t,1.)
   pde.setValue(y=d)
   return util.whereNonZero(pde.getRightHandSide())
#============================================================================================================================================
class SaddlePointProblem(object):
   """
   This implements a solver for a saddlepoint problem

   M{f(u,p)=0}
   M{g(u)=0}

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

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

   where Q_f is an approximation of the Jacobiean 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 jacobiean 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.
   """
   def __init__(self,verbose=False,*args):
       """
       initializes the problem

       @param verbose: switches on the printing out some information
       @type verbose: C{bool}
       @note: this method may be overwritten by a particular saddle point problem
       """
       print "SaddlePointProblem should not be used anymore!"
       if not isinstance(verbose,bool):
            raise TypeError("verbose needs to be of type bool.")
       self.__verbose=verbose
       self.relaxation=1.
       DeprecationWarning("SaddlePointProblem should not be used anymore.")

   def trace(self,text):
       """
       prints text if verbose has been set

       @param text: a text message
       @type text: C{str}
       """
       if self.__verbose: print "%s: %s"%(str(self),text)

   def solve_f(self,u,p,tol=1.e-8):
       """
       solves 

       A_f du = f(u,p) 

       with tolerance C{tol} and return du. A_f is Jacobiean of f with respect to u.

       @param u: current approximation of u
       @type u: L{escript.Data}
       @param p: current approximation of p
       @type p: L{escript.Data}
       @param tol: tolerance expected for du
       @type tol: C{float}
       @return: increment du
       @rtype: L{escript.Data}
       @note: this method has to be overwritten by a particular saddle point problem
       """
       pass

   def solve_g(self,u,tol=1.e-8):
       """
       solves 

       Q_g dp = g(u) 

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

       @param u: current approximation of u
       @type u: L{escript.Data}
       @param tol: tolerance expected for dp
       @type tol: C{float}
       @return: increment dp
       @rtype: L{escript.Data}
       @note: this method has to be overwritten by a particular saddle point problem
       """
       pass

   def inner(self,p0,p1):
       """
       inner product of p0 and p1 approximating p. Typically this returns integrate(p0*p1)
       @return: inner product of p0 and p1
       @rtype: C{float}
       """
       pass

   subiter_max=3
   def solve(self,u0,p0,tolerance=1.e-6,tolerance_u=None,iter_max=100,accepted_reduction=0.995,relaxation=None):
        """
        runs the solver

        @param u0: initial guess for C{u}
        @type u0: L{esys.escript.Data}
        @param p0: initial guess for C{p}
        @type p0: L{esys.escript.Data}
        @param tolerance: tolerance for relative error in C{u} and C{p}
        @type tolerance: positive C{float}
        @param tolerance_u: tolerance for relative error in C{u} if different from C{tolerance}
        @type tolerance_u: positive C{float}
        @param iter_max: maximum number of iteration steps.
        @type iter_max: C{int}
        @param accepted_reduction: if the norm  g cannot be reduced by C{accepted_reduction} backtracking to adapt the 
                                   relaxation factor. If C{accepted_reduction=None} no backtracking is used.
        @type accepted_reduction: positive C{float} or C{None}
        @param relaxation: initial relaxation factor. If C{relaxation==None}, the last relaxation factor is used.
        @type relaxation: C{float} or C{None}
        """
        tol=1.e-2
        if tolerance_u==None: tolerance_u=tolerance
        if not relaxation==None: self.relaxation=relaxation
        if accepted_reduction ==None:
              angle_limit=0.
        elif accepted_reduction>=1.:
              angle_limit=0.
        else:
              angle_limit=util.sqrt(1-accepted_reduction**2)
        self.iter=0
        u=u0
        p=p0
        #
        #   initialize things:
        #
        converged=False
        #
        #  start loop:
        #
        #  initial search direction is g
        #
        while not converged :
            if self.iter>iter_max:
                raise ArithmeticError("no convergence after %s steps."%self.iter)
            f_new=self.solve_f(u,p,tol)
            norm_f_new = util.Lsup(f_new)
            u_new=u-f_new
            g_new=self.solve_g(u_new,tol)
            self.iter+=1
            norm_g_new = util.sqrt(self.inner(g_new,g_new))
            if norm_f_new==0. and norm_g_new==0.: return u, p
            if self.iter>1 and not accepted_reduction==None:
               #
               #   did we manage to reduce the norm of G? I
               #   if not we start a backtracking procedure
               #
               # print "new/old norm = ",norm_g_new, norm_g, norm_g_new/norm_g
               if norm_g_new > accepted_reduction * norm_g:
                  sub_iter=0
                  s=self.relaxation
                  d=g
                  g_last=g
                  self.trace("    start substepping: f = %s, g = %s, relaxation = %s."%(norm_f_new, norm_g_new, s))
                  while sub_iter < self.subiter_max and  norm_g_new > accepted_reduction * norm_g:
                     dg= g_new-g_last
                     norm_dg=abs(util.sqrt(self.inner(dg,dg))/self.relaxation)
                     rad=self.inner(g_new,dg)/self.relaxation
                     # print "   ",sub_iter,": rad, norm_dg:",abs(rad), norm_dg*norm_g_new * angle_limit
                     # print "   ",sub_iter,": rad, norm_dg:",rad, norm_dg, norm_g_new, norm_g
                     if abs(rad) < norm_dg*norm_g_new * angle_limit:
                         if sub_iter>0: self.trace("    no further improvements expected from backtracking.")
                         break
                     r=self.relaxation
                     self.relaxation= - rad/norm_dg**2
                     s+=self.relaxation
                     #####
                     # a=g_new+self.relaxation*dg/r
                     # print "predicted new norm = ",util.sqrt(self.inner(a,a)),util.sqrt(self.inner(g_new,g_new)), self.relaxation
                     #####
                     g_last=g_new
                     p+=self.relaxation*d
                     f_new=self.solve_f(u,p,tol)
                     u_new=u-f_new
                     g_new=self.solve_g(u_new,tol)
                     self.iter+=1
                     norm_f_new = util.Lsup(f_new)
                     norm_g_new = util.sqrt(self.inner(g_new,g_new))
                     # print "   ",sub_iter," new g norm",norm_g_new
                     self.trace("    %s th sub-step: f = %s, g = %s, relaxation = %s."%(sub_iter, norm_f_new, norm_g_new, s))
                     #
                     #   can we expect reduction of g?
                     #
                     # u_last=u_new
                     sub_iter+=1
                  self.relaxation=s
            #
            #  check for convergence:
            #
            norm_u_new = util.Lsup(u_new)
            p_new=p+self.relaxation*g_new
            norm_p_new = util.sqrt(self.inner(p_new,p_new))
            self.trace("%s th step: f = %s, f/u = %s, g = %s, g/p = %s, relaxation = %s."%(self.iter, norm_f_new ,norm_f_new/norm_u_new, norm_g_new, norm_g_new/norm_p_new, self.relaxation))

            if self.iter>1:
               dg2=g_new-g
               df2=f_new-f
               norm_dg2=util.sqrt(self.inner(dg2,dg2))
               norm_df2=util.Lsup(df2)
               # print norm_g_new, norm_g, norm_dg, norm_p, tolerance
               tol_eq_g=tolerance*norm_dg2/(norm_g*abs(self.relaxation))*norm_p_new
               tol_eq_f=tolerance_u*norm_df2/norm_f*norm_u_new
               if norm_g_new <= tol_eq_g and norm_f_new <= tol_eq_f:
                   converged=True
            f, norm_f, u, norm_u, g, norm_g, p, norm_p = f_new, norm_f_new, u_new, norm_u_new, g_new, norm_g_new, p_new, norm_p_new
        self.trace("convergence after %s steps."%self.iter)
        return u,p
Name	Value
svn:eol-style	native
svn:keywords	Author Date Id Revision