esys.escript.pdetools Package — esys.escript 5.6 documentation

# esys.escript.pdetools Package

## Classes

- `ArithmeticTuple`
- `CorrectionFailed`
- `Defect`
- `HomogeneousSaddlePointProblem`
- `IndefinitePreconditioner`
- `IterationBreakDown`
- `Locator`
- `MaxIterReached`
- `NegativeNorm`
- `NoPDE`
- `Projector`
- `SolverSchemeException`
- `TimeIntegrationManager`

class `esys.escript.pdetools.` `ArithmeticTuple`(*args)

  Bases: `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 numpy import array
      a=eData(...)
      b=array([1.,4.])
      x=ArithmeticTuple(a,b)
      y=5.*x

  `__init__`(*args)

    Initializes object with elements `args`.

    Parameters: | args – tuple of objects that support inplace add (x+=y) and scaling (x=a*y)

class `esys.escript.pdetools.` `CorrectionFailed`

  Bases: `esys.escriptcore.pdetools.SolverSchemeException`

  Exception thrown if no convergence has been achieved in the solution correction scheme.

  `__init__`()

    Initialize self. See help(type(self)) for accurate signature.

  `args`
  `with_traceback`()

    Exception.with_traceback(tb) – set self.__traceback__ to tb and return self.

class `esys.escript.pdetools.` `Defect`

  Bases: `object`

  Defines a non-linear defect F(x) of a variable x. This class includes two functions (bilinearform and eval) that must be overridden by subclassing before use.

  `__init__`()

    Initializes defect.

  `bilinearform`(x0, x1)

    Returns the inner product of x0 and x1

    NOTE: MUST BE OVERRIDDEN BY A SUBCLASS

    Parameters: | x0 – value for x0 x1 – value for x1
    Returns: | the inner product of x0 and x1
    Return type: | `float`

  `derivative`(F0, x0, v, v_is_normalised=True)

    Returns the directional derivative at `x0` in the direction of `v`.

    Parameters: | F0 – value of this defect at x0 x0 – value at which derivative is calculated v – direction v_is_normalised – True to indicate that `v` is nomalized (self.norm(v)=0)
    Returns: | derivative of this defect at x0 in the direction of `v`
    Note: | by default numerical evaluation (self.eval(x0+eps*v)-F0)/eps is used but this method maybe overwritten to use exact evaluation.

  `eval`(x)

    Returns the value F of a given `x`.

    NOTE: MUST BE OVERRIDDEN BY A SUBCLASS

    Parameters: | x – value for which the defect `F` is evaluated
    Returns: | value of the defect at `x`

  `getDerivativeIncrementLength`()

    Returns the relative increment length used to approximate the derivative of the defect. :return: value of the defect at `x` :rtype: positive `float`

  `norm`(x)

    Returns the norm of argument `x`.

    Parameters: | x – a value
    Returns: | norm of argument x
    Return type: | `float`
    Note: | by default `sqrt(self.bilinearform(x,x)` is returned.

  `setDerivativeIncrementLength`(inc=1.4901161193847656e-05)

    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 starting point.

    Parameters: | inc (positive `float`) – relative increment length

class `esys.escript.pdetools.` `HomogeneousSaddlePointProblem`(**kwargs)

  Bases: `object`

  This class provides a framework for solving linear homogeneous saddle point problems 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. A may depend weakly on v and p.

  `__init__`(**kwargs)

    initializes the saddle point problem

  `Bv`(v, tol)

    Returns Bv with accuracy `tol` (overwrite)

    Return type: | equal to the type of p
    Note: | boundary conditions on p should be zero!

  `getAbsoluteTolerance`()

    Returns the absolute tolerance.

    Returns: | absolute tolerance
    Return type: | `float`

  `getDV`(p, v, tol)

    return a correction to the value for a given v and a given p with accuracy `tol` (overwrite)

    Parameters: | p – pressure v – pressure
    Returns: | dv given as dv= A^{-1} (f-A v-B^*p)
    Note: | Only A may depend on v and p

  `getTolerance`()

    Returns the relative tolerance.

    Returns: | relative tolerance
    Return type: | `float`

  `inner_p`(p0, p1)

    Returns inner product of p0 and p1 (overwrite).

    Parameters: | p0 – a pressure p1 – a pressure
    Returns: | inner product of p0 and p1
    Return type: | `float`

  `inner_pBv`(p, Bv)

    Returns inner product of element p and Bv (overwrite).

    Parameters: | p – a pressure increment Bv – a residual
    Returns: | inner product of element p and Bv
    Return type: | `float`
    Note: | used if PCG is applied.

  `norm_Bv`(Bv)

    Returns the norm of Bv (overwrite).

    Return type: | equal to the type of p
    Note: | boundary conditions on p should be zero!

  `norm_p`(p)

    calculates the norm of `p`

    Parameters: | p – a pressure
    Returns: | the norm of `p` using the inner product for pressure
    Return type: | `float`

  `norm_v`(v)

    Returns the norm of v (overwrite).

    Parameters: | v – a velovity
    Returns: | norm of v
    Return type: | non-negative `float`

  `resetControlParameters`(K_p=1.0, K_v=1.0, rtol_max=0.01, rtol_min=1e-07, chi_max=0.5, reduction_factor=0.3, theta=0.1)

    sets a control parameter

    Parameters: | K_p (`float`) – initial value for constant to adjust pressure tolerance K_v (`float`) – initial value for constant to adjust velocity tolerance rtol_max (`float`) – maximuim relative tolerance used to calculate presssure and velocity increment. chi_max (`float`) – maximum tolerable converegence rate. reduction_factor (`float`) – reduction factor for adjustment factors.

  `setAbsoluteTolerance`(tolerance=0.0)

    Sets the absolute tolerance.

    Parameters: | tolerance (non-negative `float`) – tolerance to be used

  `setControlParameter`(K_p=None, K_v=None, rtol_max=None, rtol_min=None, chi_max=None, reduction_factor=None, theta=None)

    sets a control parameter

    Parameters: | K_p (`float`) – initial value for constant to adjust pressure tolerance K_v (`float`) – initial value for constant to adjust velocity tolerance rtol_max (`float`) – maximuim relative tolerance used to calculate presssure and velocity increment. chi_max (`float`) – maximum tolerable converegence rate.

  `setTolerance`(tolerance=0.0001)

    Sets the relative tolerance for (v,p).

    Parameters: | tolerance (non-negative `float`) – tolerance to be used

  `solve`(v, p, max_iter=20, verbose=False, usePCG=True, iter_restart=20, max_correction_steps=10)

    Solves the saddle point problem using initial guesses v and p.

    Parameters: | v (`Data`) – initial guess for velocity p (`Data`) – initial guess for pressure usePCG (`bool`) – indicates the usage of the PCG rather than GMRES scheme. max_iter (`int`) – maximum number of iteration steps per correction attempt verbose (`bool`) – if True, shows information on the progress of the saddlepoint problem solver. iter_restart (`int`) – restart the iteration after `iter_restart` steps (only used if useUzaw=False)
    Return type: | `tuple` of `Data` objects
    Note: | typically this method is overwritten by a subclass. It provides a wrapper for the `_solve` method.

  `solve_AinvBt`(dp, tol)

    Solves A dv=B^*dp with accuracy `tol`

    Parameters: | dp – a pressure increment
    Returns: | the solution of A dv=B^*dp
    Note: | boundary conditions on dv should be zero! A is the operator used in `getDV` and must not be altered.

  `solve_prec`(Bv, tol)

    Provides a preconditioner for (BA^{-1}B^ * ) applied to Bv with accuracy `tol`

    Return type: | equal to the type of p
    Note: | boundary conditions on p should be zero!

class `esys.escript.pdetools.` `IndefinitePreconditioner`

  Bases: `esys.escriptcore.pdetools.SolverSchemeException`

  Exception thrown if the preconditioner is not positive definite.

  `__init__`()

    Initialize self. See help(type(self)) for accurate signature.

  `args`
  `with_traceback`()

    Exception.with_traceback(tb) – set self.__traceback__ to tb and return self.

class `esys.escript.pdetools.` `IterationBreakDown`

  Bases: `esys.escriptcore.pdetools.SolverSchemeException`

  Exception thrown if the iteration scheme encountered an incurable breakdown.

  `__init__`()

    Initialize self. See help(type(self)) for accurate signature.

  `args`
  `with_traceback`()

    Exception.with_traceback(tb) – set self.__traceback__ to tb and return self.

class `esys.escript.pdetools.` `Locator`(where, x=array([0., 0., 0.]))

  Bases: `object`

  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 which is closest to the given point x.

  `__init__`(where, x=array([0., 0., 0.]))

    Initializes a Locator to access values in Data objects on the Doamin or FunctionSpace for the sample point which is closest to the given point x.

    Parameters: | where (`escript.FunctionSpace`) – function space x (`numpy.ndarray` or `list` of `numpy.ndarray`) – location(s) of the Locator

  `getFunctionSpace`()

    Returns the function space of the Locator.

  `getId`(item=None)

    Returns the identifier of the location.

  `getValue`(data)

    Returns the value of `data` at the Locator if `data` is a `Data` object otherwise the object is returned.

  `getX`()

    Returns the exact coordinates of the Locator.

  `setValue`(data, v)

    Sets the value of the `data` at the Locator.

class `esys.escript.pdetools.` `MaxIterReached`

  Bases: `esys.escriptcore.pdetools.SolverSchemeException`

  Exception thrown if the maximum number of iteration steps is reached.

  `__init__`()

    Initialize self. See help(type(self)) for accurate signature.

  `args`
  `with_traceback`()

    Exception.with_traceback(tb) – set self.__traceback__ to tb and return self.

class `esys.escript.pdetools.` `NegativeNorm`

  Bases: `esys.escriptcore.pdetools.SolverSchemeException`

  Exception thrown if a norm calculation returns a negative norm.

  `__init__`()

    Initialize self. See help(type(self)) for accurate signature.

  `args`
  `with_traceback`()

    Exception.with_traceback(tb) – set self.__traceback__ to tb and return self.

class `esys.escript.pdetools.` `NoPDE`(domain, D=None, Y=None, q=None, r=None)

  Bases: `object`

  Solves the following problem for u:

  kronecker[i,j]*D[j]*u[j]=Y[i]

  with constraint

  u[j]=r[j] where q[j]>0

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

  In the case of scalars this takes the form

  D*u=Y

  with constraint

  u=r where q>0

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

  The constraint overwrites any other condition.

  Note: | This class is similar to the `linearPDEs.LinearPDE` class with A=B=C=X=0 but has the intention that all input parameters are given in `Solution` or `ReducedSolution`.

  `__init__`(domain, D=None, Y=None, q=None, r=None)

    Initializes the problem.

    Parameters: | domain (`Domain`) – domain of the PDE D (`float`, `int`, `numpy.ndarray`, `Data`) – coefficient of the solution Y (`float`, `int`, `numpy.ndarray`, `Data`) – right hand side q (`float`, `int`, `numpy.ndarray`, `Data`) – location of constraints r (`float`, `int`, `numpy.ndarray`, `Data`) – value of solution at locations of constraints

  `getSolution`()

    Returns the solution.

    Returns: | the solution of the problem
    Return type: | `Data` object in the `FunctionSpace` `Solution` or `ReducedSolution`

  `setReducedOff`()

    Sets the `FunctionSpace` of the solution to `Solution`.

  `setReducedOn`()

    Sets the `FunctionSpace` of the solution to `ReducedSolution`.

  `setValue`(D=None, Y=None, q=None, r=None)

    Assigns values to the parameters.

    Parameters: | D (`float`, `int`, `numpy.ndarray`, `Data`) – coefficient of the solution Y (`float`, `int`, `numpy.ndarray`, `Data`) – right hand side q (`float`, `int`, `numpy.ndarray`, `Data`) – location of constraints r (`float`, `int`, `numpy.ndarray`, `Data`) – value of solution at locations of constraints

class `esys.escript.pdetools.` `Projector`(domain, reduce=True, fast=True)

  Bases: `object`

  The Projector is a factory which projects a discontinuous function onto a continuous function on a given domain.

  `__init__`(domain, reduce=True, fast=True)

    Creates a continuous function space projector for a domain.

    Parameters: | domain – Domain of the projection. reduce – Flag to reduce projection order fast – Flag to use a fast method based on matrix lumping

  `getSolverOptions`()

    Returns the solver options of the PDE solver.

    Return type: | `linearPDEs.SolverOptions`

  `getValue`(input_data)

    Projects `input_data` onto a continuous function.

    Parameters: | input_data – the data to be projected

class `esys.escript.pdetools.` `SolverSchemeException`

  Bases: `Exception`

  This is a generic exception thrown by solvers.

  `__init__`()

    Initialize self. See help(type(self)) for accurate signature.

  `args`
  `with_traceback`()

    Exception.with_traceback(tb) – set self.__traceback__ to tb and return self.

class `esys.escript.pdetools.` `TimeIntegrationManager`(*inital_values, **kwargs)

  Bases: `object`

  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.

  `__init__`(*inital_values, **kwargs)

    Sets up the value manager where `inital_values` are the initial values and p is the order used for extrapolation.

  `checkin`(dt, *values)

    Adds new values to the manager. The p+1 last values are lost.

  `extrapolate`(dt)

    Extrapolates to `dt` forward in time.

  `getTime`()
  `getValue`()

## Functions

`esys.escript.pdetools.` `BoundaryValuesFromVolumeTag`(domain, **values)

  Creates a mask on the Solution(domain) function space where the value is one for samples that touch regions tagged by tags.

  Usage: m=BoundaryValuesFromVolumeTag(domain, ham=1, f=6)

  Parameters: | domain (`escript.Domain`) – domain to be used
  Returns: | a mask which marks samples that are touching the boundary tagged by any of the given tags
  Return type: | `escript.Data` of rank 0

`esys.escript.pdetools.` `GMRES`(r, Aprod, x, bilinearform, atol=0, rtol=1e-08, iter_max=100, iter_restart=20, verbose=False, P_R=None)

  Solver for

  Ax=b

  with a general operator A (more details required!). It uses the generalized minimum residual method (GMRES).

  The iteration is terminated if

  |r| <= atol+rtol*|r0|

  where r0 is the initial residual and |.| is the energy norm. In fact

  |r| = sqrt( bilinearform(r,r))

  Parameters: | r (any object supporting inplace add (x+=y) and scaling (x=scalar*y)) – initial residual r=b-Ax. `r` is altered. x (same like `r`) – an initial guess for the solution Aprod (function `Aprod(x)` where `x` is of the same object like argument `x`. The returned object needs to be of the same type like argument `r`.) – returns the value Ax bilinearform (function `bilinearform(x,r)` where `x` is of the same type like argument `x` and `r`. The returned value is a `float`.) – inner product `<x,r>` atol (non-negative `float`) – absolute tolerance rtol (non-negative `float`) – relative tolerance iter_max (`int`) – maximum number of iteration steps iter_restart (`int`) – in order to save memory the orthogonalization process is terminated after `iter_restart` steps and the iteration is restarted.
  Returns: | the solution approximation and the corresponding residual
  Return type: | `tuple`
  Warning: | `r` and `x` are altered.

`esys.escript.pdetools.` `MINRES`(r, Aprod, x, Msolve, bilinearform, atol=0, rtol=1e-08, iter_max=100)

  Solver for

  Ax=b

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

  The iteration is terminated if

  |r| <= atol+rtol*|r0|

  where r0 is the initial residual and |.| is the energy norm. In fact

  |r| = sqrt( bilinearform(Msolve(r),r))

  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”.

  Parameters: | r (any object supporting inplace add (x+=y) and scaling (x=scalar*y)) – initial residual r=b-Ax. `r` is altered. x (any object supporting inplace add (x+=y) and scaling (x=scalar*y)) – an initial guess for the solution Aprod (function `Aprod(x)` where `x` is of the same object like argument `x`. The returned object needs to be of the same type like argument `r`.) – returns the value Ax Msolve (function `Msolve(r)` where `r` is of the same type like argument `r`. The returned object needs to be of the same type like argument `x`.) – solves Mx=r bilinearform (function `bilinearform(x,r)` where `x` is of the same type like argument `x` and `r` is. The returned value is a `float`.) – inner product `<x,r>` atol (non-negative `float`) – absolute tolerance rtol (non-negative `float`) – relative tolerance iter_max (`int`) – maximum number of iteration steps
  Returns: | the solution approximation and the corresponding residual
  Return type: | `tuple`
  Warning: | `r` and `x` are altered.

`esys.escript.pdetools.` `MaskFromBoundaryTag`(domain, *tags)

  Creates a mask on the Solution(domain) function space where the value is one for samples that touch the boundary tagged by tags.

  Usage: m=MaskFromBoundaryTag(domain, “left”, “right”)

  Parameters: | domain (`escript.Domain`) – domain to be used tags (`str`) – boundary tags
  Returns: | a mask which marks samples that are touching the boundary tagged by any of the given tags
  Return type: | `escript.Data` of rank 0

`esys.escript.pdetools.` `MaskFromTag`(domain, *tags)

  Creates a mask on the Solution(domain) function space where the value is one for samples that touch regions tagged by tags.

  Usage: m=MaskFromTag(domain, “ham”)

  Parameters: | domain (`escript.Domain`) – domain to be used tags (`str`) – boundary tags
  Returns: | a mask which marks samples that are touching the boundary tagged by any of the given tags
  Return type: | `escript.Data` of rank 0

`esys.escript.pdetools.` `NewtonGMRES`(defect, x, iter_max=100, sub_iter_max=20, atol=0, rtol=0.0001, subtol_max=0.5, gamma=0.9, verbose=False)

  Solves a non-linear problem F(x)=0 for unknown x using the stopping criterion:

  norm(F(x) <= atol + rtol * norm(F(x0)

  where x0 is the initial guess.

  Parameters: | defect (`Defect`) – object defining the function F. `defect.norm` defines the norm used in the stopping criterion. x (any object type allowing basic operations such as `numpy.ndarray`, `Data`) – initial guess for the solution, `x` is altered. iter_max (positive `int`) – maximum number of iteration steps sub_iter_max (positive `int`) – maximum number of inner iteration steps atol (positive `float`) – absolute tolerance for the solution rtol (positive `float`) – relative tolerance for the solution gamma (positive `float`, less than 1) – tolerance safety factor for inner iteration subtol_max (positive `float`, less than 1) – upper bound for inner tolerance
  Returns: | an approximation of the solution with the desired accuracy
  Return type: | same type as the initial guess

`esys.escript.pdetools.` `PCG`(r, Aprod, x, Msolve, bilinearform, atol=0, rtol=1e-08, iter_max=100, initial_guess=True, verbose=False)

  Solver for

  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

  |r| <= atol+rtol*|r0|

  where r0 is the initial residual and |.| is the energy norm. In fact

  |r| = sqrt( bilinearform(Msolve(r),r))

  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”.

  Parameters: | r (any object supporting inplace add (x+=y) and scaling (x=scalar*y)) – initial residual r=b-Ax. `r` is altered. x (any object supporting inplace add (x+=y) and scaling (x=scalar*y)) – an initial guess for the solution Aprod (function `Aprod(x)` where `x` is of the same object like argument `x`. The returned object needs to be of the same type like argument `r`.) – returns the value Ax Msolve (function `Msolve(r)` where `r` is of the same type like argument `r`. The returned object needs to be of the same type like argument `x`.) – solves Mx=r bilinearform (function `bilinearform(x,r)` where `x` is of the same type like argument `x` and `r` is. The returned value is a `float`.) – inner product `<x,r>` atol (non-negative `float`) – absolute tolerance rtol (non-negative `float`) – relative tolerance iter_max (`int`) – maximum number of iteration steps
  Returns: | the solution approximation and the corresponding residual
  Return type: | `tuple`
  Warning: | `r` and `x` are altered.

`esys.escript.pdetools.` `TFQMR`(r, Aprod, x, bilinearform, atol=0, rtol=1e-08, iter_max=100)

  Solver for

  Ax=b

  with a general operator A (more details required!). It uses the Transpose-Free Quasi-Minimal Residual method (TFQMR).

  The iteration is terminated if

  |r| <= atol+rtol*|r0|

  where r0 is the initial residual and |.| is the energy norm. In fact

  |r| = sqrt( bilinearform(r,r))

  Parameters: | r (any object supporting inplace add (x+=y) and scaling (x=scalar*y)) – initial residual r=b-Ax. `r` is altered. x (same like `r`) – an initial guess for the solution Aprod (function `Aprod(x)` where `x` is of the same object like argument `x`. The returned object needs to be of the same type like argument `r`.) – returns the value Ax bilinearform (function `bilinearform(x,r)` where `x` is of the same type like argument `x` and `r`. The returned value is a `float`.) – inner product `<x,r>` atol (non-negative `float`) – absolute tolerance rtol (non-negative `float`) – relative tolerance iter_max (`int`) – maximum number of iteration steps
  Return type: | `tuple`
  Warning: | `r` and `x` are altered.

`esys.escript.pdetools.` `getInfLocator`(arg)

  Return a Locator for a point with the inf value over all arg.

`esys.escript.pdetools.` `getSupLocator`(arg)

  Return a Locator for a point with the sup value over all arg.

## Others

## Packages