solver.py

# Copyright (c) 2014 Evalf
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"""
The solver module defines solvers for problems of the kind ``res = 0`` or
``∂inertia/∂t + res = 0``, where ``res`` is a
:class:`nutils.sample.Integral`.  To demonstrate this consider the following
setup:

>>> from nutils import mesh, function, solver
>>> ns = function.Namespace()
>>> domain, ns.x = mesh.rectilinear([4,4])
>>> ns.basis = domain.basis('spline', degree=2)
>>> cons = domain.boundary['left,top'].project(0, onto=ns.basis, geometry=ns.x, ischeme='gauss4')
project > constrained 11/36 dofs, error 0.00e+00/area
>>> ns.u = 'basis_n ?lhs_n'

Function ``u`` represents an element from the discrete space but cannot not
evaluated yet as we did not yet establish values for ``?lhs``. It can,
however, be used to construct a residual functional ``res``. Aiming to solve
the Poisson problem ``u_,kk = f`` we define the residual functional ``res = v,k
u,k + v f`` and solve for ``res == 0`` using ``solve_linear``:

>>> res = domain.integral('(basis_n,i u_,i + basis_n) d:x' @ ns, degree=2)
>>> lhs = solver.solve_linear('lhs', residual=res, constrain=cons)
solve > solving 25x25 system using direct solver
solve > solver returned with residual ...

The coefficients ``lhs`` represent the solution to the Poisson problem.

In addition to ``solve_linear`` the solver module defines ``newton`` and
``pseudotime`` for solving nonlinear problems, as well as ``impliciteuler`` for
time dependent problems.
"""

from . import function, cache, log, numeric, sample, types, util, matrix
import numpy, itertools, functools, numbers, collections, math


argdict = types.frozendict[types.strictstr,types.frozenarray]


class SolverError(Exception): pass


@types.apply_annotations
@cache.function
def solve_linear(target:types.strictstr, residual:sample.strictintegral, constrain:types.frozenarray=None, *, arguments:argdict={}, solveargs:types.frozendict={}):
  '''solve linear problem

  Parameters
  ----------
  target : :class:`str`
      Name of the target: a :class:`nutils.function.Argument` in ``residual``.
  residual : :class:`nutils.sample.Integral`
      Residual integral, depends on ``target``
  constrain : :class:`numpy.ndarray` with dtype :class:`float`
      Defines the fixed entries of the coefficient vector
  arguments : :class:`collections.abc.Mapping`
      Defines the values for :class:`nutils.function.Argument` objects in
      `residual`.  The ``target`` should not be present in ``arguments``.
      Optional.

  Returns
  -------
  :class:`numpy.ndarray`
      Array of ``target`` values for which ``residual == 0``'''

  jacobian = residual.derivative(target)
  if jacobian.contains(target):
    raise SolverError('problem is not linear')
  assert target not in arguments, '`target` should not be defined in `arguments`'
  argshape = residual._argshape(target)
  res, jac = sample.eval_integrals(residual, jacobian, **{target: numpy.zeros(argshape)}, **arguments)
  return jac.solve(-res, constrain=constrain, **solveargs)


def solve(gen_lhs_resnorm, tol=0., maxiter=float('inf')):
  '''execute nonlinear solver, return lhs

  Iterates over nonlinear solver until tolerance is reached. Example::

      lhs = solve(newton(target, residual), tol=1e-5)

  Parameters
  ----------
  gen_lhs_resnorm : :class:`collections.abc.Generator`
      Generates (lhs, resnorm) tuples
  tol : :class:`float`
      Target residual norm
  maxiter : :class:`int`
      Maximum number of iterations

  Returns
  -------
  :class:`numpy.ndarray`
      Coefficient vector that corresponds to a smaller than ``tol`` residual.
  '''

  lhs, info = solve_withinfo(gen_lhs_resnorm, tol=tol, maxiter=maxiter)
  return lhs


@types.apply_annotations
@cache.function
def solve_withinfo(gen_lhs_resnorm, tol:types.strictfloat=0., maxiter:types.strictfloat=float('inf')):
  '''execute nonlinear solver, return lhs and info

  Like :func:`solve`, but return a 2-tuple of the solution and the
  corresponding info object which holds information about the final residual
  norm and other generator-dependent information.
  '''

  iterator = iter(gen_lhs_resnorm)
  lhs, info = next(iterator)
  if tol:
    iiter = 0
    resnorm0 = info.resnorm
    while info.resnorm > tol:
      if iiter > maxiter:
        raise SolverError('failed to reach target tolerance')
      with log.context('iter {} ({:.0f}%)'.format(iiter, 100 * numpy.log(resnorm0/info.resnorm) / numpy.log(resnorm0/tol))):
        lhs, info = next(iterator)
      iiter += 1
  elif info.resnorm:
    lhs, info = next(iterator)
    if info.resnorm:
      raise SolverError('nonlinear problem requires a nonzero tolerance')
  return lhs, info

class RecursionWithSolve(cache.Recursion):
  '''add a .solve method to (lhs,resnorm) iterators

  Introduces the convenient form::

      newton(target, residual).solve(tol)

  Shorthand for::

      solve(newton(target, residual), tol)
  '''

  __slots__ = ()

  solve_withinfo = solve_withinfo
  solve = solve


class newton(RecursionWithSolve, length=1):
  '''iteratively solve nonlinear problem by gradient descent

  Generates targets such that residual approaches 0 using Newton procedure with
  line search based on the residual norm. Suitable to be used inside ``solve``.

  An optimal relaxation value is computed based on the following cubic
  assumption::

      |res(lhs + r * dlhs)|^2 = A + B * r + C * r^2 + D * r^3

  where ``A``, ``B``, ``C`` and ``D`` are determined based on the current
  residual and tangent, the new residual, and the new tangent. If this value is
  found to be close to 1 then the newton update is accepted.

  Parameters
  ----------
  target : :class:`str`
      Name of the target: a :class:`nutils.function.Argument` in ``residual``.
  residual : :class:`nutils.sample.Integral`
  lhs0 : :class:`numpy.ndarray`
      Coefficient vector, starting point of the iterative procedure.
  constrain : :class:`numpy.ndarray` with dtype :class:`bool` or :class:`float`
      Equal length to ``lhs0``, masks the free vector entries as ``False``
      (boolean) or NaN (float). In the remaining positions the values of
      ``lhs0`` are returned unchanged (boolean) or overruled by the values in
      `constrain` (float).
  searchrange : :class:`tuple` of two floats
      The lower bound (>=0) and upper bound (<=1) for line search relaxation
      updates. If the estimated optimum relaxation (determined by polynomial
      interpolation) is above upper bound of the current relaxation value then
      the newton update is accepted. Below it, the functional is re-evaluated
      at the new relaxation value or at the lower bound, whichever is largest.
  rebound : :class:`float`
      Factor by which the relaxation value grows after every update until it
      reaches unity.
  droptol : :class:`float`
      Threshold for leaving entries in the return value at NaN if they do not
      contribute to the value of the functional.
  failrelax : :class:`float`
      Fail with exception if relaxation reaches this lower limit.
  arguments : :class:`collections.abc.Mapping`
      Defines the values for :class:`nutils.function.Argument` objects in
      `residual`.  The ``target`` should not be present in ``arguments``.
      Optional.

  Yields
  ------
  :class:`numpy.ndarray`
      Coefficient vector that approximates residual==0 with increasing accuracy
  '''

  @types.apply_annotations
  def __init__(self, target:types.strictstr, residual:sample.strictintegral, jacobian:sample.strictintegral=None, lhs0:types.frozenarray[types.strictfloat]=None, constrain:types.frozenarray=None, searchrange:types.tuple[float]=(.01,2/3), droptol:types.strictfloat=None, rebound:types.strictfloat=2., failrelax:types.strictfloat=1e-6, arguments:argdict={}, solveargs:types.frozendict={}):
    super().__init__()
    if target in arguments:
      raise ValueError('`target` should not be defined in `arguments`')
    self.target = target
    self.residual = residual
    self.jacobian = _derivative(residual, target, jacobian)
    self.lhs0, self.constrain = _parse_lhs_cons(lhs0, constrain, residual.shape)
    self.minscale, self.maxscale = searchrange
    self.rebound = rebound
    self.droptol = droptol
    self.failrelax = failrelax
    self.arguments = arguments
    self.solveargs = solveargs
    self.islinear = not self.jacobian.contains(self.target)

  def _eval(self, lhs):
    return sample.eval_integrals(self.residual, self.jacobian, **{self.target: lhs}, **self.arguments)

  def resume(self, history):
    if history:
      lhs, info = history[-1]
      if self.droptol is not None:
        lhs = numpy.choose(numpy.isnan(lhs), [lhs, self.lhs0])
      res, jac = self._eval(lhs)
      resnorm = numpy.linalg.norm(res[~self.constrain])
      assert resnorm == info.resnorm
      relax = info.relax
    else:
      lhs = self.lhs0
      res, jac = self._eval(lhs)
      resnorm = numpy.linalg.norm(res[~self.constrain])
      relax = 1
      nosupp = self.droptol is not None and ~(jac.rowsupp(self.droptol)|self.constrain)
      yield _nan_at(lhs, nosupp), types.attributes(resnorm=resnorm, relax=relax)

    while resnorm:
      nosupp = self.droptol is not None and ~(jac.rowsupp(self.droptol)|self.constrain)
      dlhs = -jac.solve(res, constrain=self.constrain|nosupp, **self.solveargs)
      if self.islinear:
        yield _nan_at(lhs+dlhs, nosupp), types.attributes(resnorm=0, relax=1)
        return
      for irelax in itertools.count():
        newlhs = lhs+relax*dlhs
        res, jac = self._eval(newlhs)
        newresnorm = numpy.linalg.norm(res[~self.constrain])
        if not numpy.isfinite(newresnorm):
          log.info('residual norm {} / {}'.format(newresnorm, round(relax, 5)))
          relax *= self.minscale
          continue
        # To determine optimal relaxation we minimize a polynomial estimation for the residual norm:
        # P(scale) = A + B scale + C scale^2 + D scale^3 ~= |res(lhs+scale*relax*dlhs)|^2
        scale = _minimize2(p0=resnorm**2, q0=-2*resnorm**2*relax, p1=newresnorm**2, q1=2*relax*(jac@dlhs)[~self.constrain].dot(res[~self.constrain]))
        log.info('residual norm {:+.2f}% / {} with estimated minimum at {:.0f}%'.format(100*(newresnorm/resnorm-1), round(relax, 5), scale*100))
        if newresnorm < resnorm and scale > self.maxscale:
          relax = min(relax * min(scale, self.rebound), 1)
          break
        relax *= max(scale, self.minscale)
        if not numpy.isfinite(relax) or relax <= self.failrelax:
          raise SolverError('stuck in local minimum')
      yield _nan_at(newlhs, nosupp), types.attributes(resnorm=newresnorm, relax=relax)
      lhs, resnorm = newlhs, newresnorm


class minimize(RecursionWithSolve, length=1, version=3):
  '''iteratively minimize nonlinear functional by gradient descent

  Generates targets such that residual approaches 0 using Newton procedure with
  line search based on the energy. Suitable to be used inside ``solve``.

  An optimal relaxation value is computed based on the following assumption::

      energy(lhs + r * dlhs) = A + B * r + C * r^2 + D * r^3 + E * r^4 + F * r^5

  where ``A``, ``B``, ``C``, ``D``, ``E`` and ``F`` are determined based on the
  current and new energy, residual and tangent. If this value is found to be
  close to 1 then the newton update is accepted.

  Parameters
  ----------
  target : :class:`str`
      Name of the target: a :class:`nutils.function.Argument` in ``residual``.
  residual : :class:`nutils.sample.Integral`
  lhs0 : :class:`numpy.ndarray`
      Coefficient vector, starting point of the iterative procedure.
  constrain : :class:`numpy.ndarray` with dtype :class:`bool` or :class:`float`
      Equal length to ``lhs0``, masks the free vector entries as ``False``
      (boolean) or NaN (float). In the remaining positions the values of
      ``lhs0`` are returned unchanged (boolean) or overruled by the values in
      `constrain` (float).
  searchrange : :class:`tuple` of two floats
      The lower bound (>=0) and upper bound (<=1) for line search relaxation
      updates. If the estimated optimum relaxation (determined by polynomial
      interpolation) is above upper bound of the current relaxation value then
      the newton update is accepted. Below it, the functional is re-evaluated
      at the new relaxation value or at the lower bound, whichever is largest.
  rebound : :class:`float`
      Factor by which the relaxation value grows after every update until it
      reaches unity.
  droptol : :class:`float`
      Threshold for leaving entries in the return value at NaN if they do not
      contribute to the value of the functional.
  failrelax : :class:`float`
      Fail with exception if relaxation reaches this lower limit.
  arguments : :class:`collections.abc.Mapping`
      Defines the values for :class:`nutils.function.Argument` objects in
      `residual`.  The ``target`` should not be present in ``arguments``.
      Optional.
  maxinc : :class:`float`
      Permitted energy increase; a small nonzero value is required to avoid
      getting stuck on numerical noise close to the energy minimum.

  Yields
  ------
  :class:`numpy.ndarray`
      Coefficient vector that approximates residual==0 with increasing accuracy
  '''

  @types.apply_annotations
  def __init__(self, target:types.strictstr, energy:sample.strictintegral, lhs0:types.frozenarray[types.strictfloat]=None, constrain:types.frozenarray=None, searchrange:types.tuple[float]=(.01,.5), rebound:types.strictfloat=2., droptol:types.strictfloat=None, failrelax:types.strictfloat=1e-6, arguments:argdict={}, solveargs:types.frozendict={}, maxinc=1e-14):
    super().__init__()
    if target in arguments:
      raise ValueError('`target` should not be defined in `arguments`')
    if energy.shape != ():
      raise ValueError('`energy` should be scalar')
    self.target = target
    self.energy = energy
    self.residual = energy.derivative(target)
    self.jacobian = _derivative(self.residual, target)
    self.lhs0, self.constrain = _parse_lhs_cons(lhs0, constrain, self.residual.shape)
    self.minscale, self.maxscale = searchrange
    self.rebound = rebound
    self.droptol = droptol
    self.failrelax = failrelax
    self.arguments = arguments
    self.solveargs = solveargs
    self.islinear = not self.jacobian.contains(target)
    self.maxinc = maxinc

  def _eval(self, lhs):
    return sample.eval_integrals(self.energy, self.residual, self.jacobian, **{self.target: lhs}, **self.arguments)

  def resume(self, history):
    if history:
      lhs, info = history[-1]
      if self.droptol is not None:
        lhs = numpy.choose(numpy.isnan(lhs), [lhs, self.lhs0])
      nrg, res, jac = self._eval(lhs)
      assert nrg == info.energy
      resnorm = numpy.linalg.norm(res[~self.constrain])
      assert resnorm == info.resnorm
      relax = info.relax
    else:
      lhs = self.lhs0
      nrg, res, jac = self._eval(lhs)
      resnorm = numpy.linalg.norm(res[~self.constrain])
      relax = 1
      nosupp = self.droptol is not None and ~(jac.rowsupp(self.droptol)|self.constrain)
      yield _nan_at(lhs, nosupp), types.attributes(resnorm=resnorm, energy=nrg, relax=relax, shift=0)

    while resnorm:
      nosupp = self.droptol is not None and ~(jac.rowsupp(self.droptol)|self.constrain)
      dlhs = -jac.solve(res, constrain=self.constrain|nosupp, **self.solveargs)
      if self.islinear:
        yield _nan_at(lhs+dlhs, nosupp), types.attributes(resnorm=0, energy=nrg+.5*res.dot(dlhs), relax=1, shift=0)
        return
      shift = 0
      while res.dot(dlhs) > 0:
        # Energy is locally increasing, an adjustment is required to maintain
        # gradient descent. Because the minimum Rayleigh quotient of a matrix
        # equals its smallest eigenvalue, we have:
        #   mineig(invjac) < res.invjac.res/res.res = -dlhs.res/res.res < 0
        # We aim to modify jac by adding a sufficiently large diagonal term
        #   modjac = jac + shift eye
        # Knowing that modjac becomes SPD if shift > -mineig(jac), we know that
        # res.invmodjac.res > 0 for shift sufficiently large. Since mineig(jac)
        # is unknown, and a complete push to SPD is typically not required for
        # positivity, an iterative procedure is used to shift the spectrum in
        # steps, using the available upper bound for mineig(invjac) as a
        # reciprocal lower bound for at least one negative eigenvalue of jac.
        shift += res[~self.constrain].dot(res[~self.constrain]) / res.dot(dlhs)
        log.warning('negative eigenvalue detected; shifting spectrum by {:.2e}'.format(shift))
        dlhs = -(jac + shift * matrix.eye(len(dlhs))).solve(res, constrain=self.constrain|nosupp, **self.solveargs)
      if not shift:
        relax = min(relax, 1)
      for irelax in itertools.count():
        newlhs = lhs+relax*dlhs
        newnrg, newres, newjac = self._eval(newlhs)
        resnorm = numpy.linalg.norm(newres[~self.constrain])
        if not numpy.isfinite(newnrg):
          log.info('energy {} / {}'.format(newnrg, round(relax, 5)))
          relax *= self.minscale
          continue
        # To determine optimal relaxation we minimize a polynomial estimator for the energy:
        # nrg(lhs+scale*relax*dlhs) ~= A + B scale + C scale^2 + D scale^3 + E scale^4 + F scale^5
        scale = _minimize3(p0=nrg, q0=relax*res.dot(dlhs)-self.maxinc, r0=-relax**2*(res+shift*dlhs).dot(dlhs),
          p1=newnrg-self.maxinc, q1=relax*newres.dot(dlhs)-self.maxinc, r1=relax**2*(newjac@dlhs).dot(dlhs))
        log.info('energy {:+.1e} / {} with estimated minimum at {:.0f}%'.format(newnrg - nrg, round(relax, 5), scale*100))
        if newnrg < nrg+self.maxinc and scale > self.maxscale:
          relax = min(relax * min(scale, self.rebound), 1)
          break
        relax *= max(scale, self.minscale)
        if relax <= self.failrelax:
          raise SolverError('stuck in local minimum')
      else:
        log.warning('failed to', 'decrease' if newnrg > nrg else 'optimize', 'energy')
      yield _nan_at(newlhs, nosupp), types.attributes(resnorm=resnorm, energy=newnrg, relax=relax, shift=shift)
      nrg, res, jac = newnrg, newres, newjac
      lhs = numpy.choose(nosupp, [newlhs, self.lhs0])


class pseudotime(RecursionWithSolve, length=1):
  '''iteratively solve nonlinear problem by pseudo time stepping

  Generates targets such that residual approaches 0 using hybrid of Newton and
  time stepping. Requires an inertia term and initial timestep. Suitable to be
  used inside ``solve``.

  Parameters
  ----------
  target : :class:`str`
      Name of the target: a :class:`nutils.function.Argument` in ``residual``.
  residual : :class:`nutils.sample.Integral`
  inertia : :class:`nutils.sample.Integral`
  timestep : :class:`float`
      Initial time step, will scale up as residual decreases
  lhs0 : :class:`numpy.ndarray`
      Coefficient vector, starting point of the iterative procedure.
  constrain : :class:`numpy.ndarray` with dtype :class:`bool` or :class:`float`
      Equal length to ``lhs0``, masks the free vector entries as ``False``
      (boolean) or NaN (float). In the remaining positions the values of
      ``lhs0`` are returned unchanged (boolean) or overruled by the values in
      `constrain` (float).
  arguments : :class:`collections.abc.Mapping`
      Defines the values for :class:`nutils.function.Argument` objects in
      `residual`.  The ``target`` should not be present in ``arguments``.
      Optional.

  Yields
  ------
  :class:`numpy.ndarray` with dtype :class:`float`
      Tuple of coefficient vector and residual norm
  '''

  @types.apply_annotations
  def __init__(self, target:types.strictstr, residual:sample.strictintegral, inertia:sample.strictintegral, timestep:types.strictfloat, lhs0:types.frozenarray[types.strictfloat]=None, constrain:types.frozenarray=None, arguments:argdict={}, solveargs:types.frozendict={}):
    super().__init__()
    if target in arguments:
      raise ValueError('`target` should not be defined in `arguments`')
    if inertia.shape != residual.shape:
      raise ValueError('expected `inertia` with shape {} but got {}'.format(residual.shape, inertia.shape))
    self.target = target
    self.residual = residual
    self.jacobian = _derivative(residual, target)
    self.inertia = inertia
    self.jacobiant = _derivative(inertia, target)
    self.lhs0, self.constrain = _parse_lhs_cons(lhs0, constrain, residual.shape)
    self.timestep = timestep
    self.arguments = arguments
    self.solveargs = solveargs

  def _eval(self, lhs, timestep):
    return sample.eval_integrals(self.residual, self.jacobian+self.jacobiant/timestep, **{self.target: lhs}, **self.arguments)

  def resume(self, history):
    if history:
      lhs, info = history[-1]
      resnorm0 = info.resnorm0
      timestep = info.timestep
      res, jac = self._eval(lhs, timestep)
      resnorm = numpy.linalg.norm(res[~self.constrain])
      assert resnorm == info.resnorm
    else:
      lhs = self.lhs0
      timestep = self.timestep
      res, jac = self._eval(lhs, timestep)
      resnorm = resnorm0 = numpy.linalg.norm(res[~self.constrain])
      yield numpy.array(lhs), types.attributes(resnorm=resnorm, timestep=timestep, resnorm0=resnorm0)

    lhs = numpy.array(lhs)
    while True:
      lhs -= jac.solve(res, constrain=self.constrain, **self.solveargs)
      timestep = self.timestep * (resnorm0/resnorm)
      log.info('timestep: {:.0e}'.format(timestep))
      res, jac = self._eval(lhs, timestep)
      resnorm = numpy.linalg.norm(res[~self.constrain])
      yield lhs.copy(), types.attributes(resnorm=resnorm, timestep=timestep, resnorm0=resnorm0)


class thetamethod(RecursionWithSolve, length=1, version=1):
  '''solve time dependent problem using the theta method

  Parameters
  ----------
  target : :class:`str`
      Name of the target: a :class:`nutils.function.Argument` in ``residual``.
  residual : :class:`nutils.sample.Integral`
  inertia : :class:`nutils.sample.Integral`
  timestep : :class:`float`
      Initial time step, will scale up as residual decreases
  lhs0 : :class:`numpy.ndarray`
      Coefficient vector, starting point of the iterative procedure.
  theta : :class:`float`
      Theta value (theta=1 for implicit Euler, theta=0.5 for Crank-Nicolson)
  residual0 : :class:`nutils.sample.Integral`
      Optional additional residual component evaluated in previous timestep
  constrain : :class:`numpy.ndarray` with dtype :class:`bool` or :class:`float`
      Equal length to ``lhs0``, masks the free vector entries as ``False``
      (boolean) or NaN (float). In the remaining positions the values of
      ``lhs0`` are returned unchanged (boolean) or overruled by the values in
      `constrain` (float).
  newtontol : :class:`float`
      Residual tolerance of individual timesteps
  arguments : :class:`collections.abc.Mapping`
      Defines the values for :class:`nutils.function.Argument` objects in
      `residual`.  The ``target`` should not be present in ``arguments``.
      Optional.
  timetarget : :class:`str`
      Name of the :class:`nutils.function.Argument` that represents time.
      Optional.
  time0 : :class:`float`
      The intial time.  Default: ``0.0``.

  Yields
  ------
  :class:`numpy.ndarray`
      Coefficient vector for all timesteps after the initial condition.
  '''

  __cache__ = '_res_jac'

  @types.apply_annotations
  def __init__(self, target:types.strictstr, residual:sample.strictintegral, inertia:sample.strictintegral, timestep:types.strictfloat, lhs0:types.frozenarray, theta:types.strictfloat, target0:types.strictstr='_thetamethod_target0', constrain:types.frozenarray=None, newtontol:types.strictfloat=1e-10, arguments:argdict={}, newtonargs:types.frozendict={}, timetarget:types.strictstr=None, time0:types.strictfloat=0.):
    super().__init__()

    assert target != target0, '`target` should not be equal to `target0`'
    assert target not in arguments, '`target` should not be defined in `arguments`'
    assert target0 not in arguments, '`target0` should not be defined in `arguments`'
    self.target = target
    self.target0 = target0
    self.lhs0 = lhs0
    self.constrain = constrain
    self.newtonargs = newtonargs
    self.newtontol = newtontol
    self.arguments = arguments
    self.residual = residual
    self.inertia = inertia
    self.theta = theta
    self.timestep = timestep
    self.timetarget = timetarget or '_thetamethod_dummy'
    self.time0 = time0

  def _res_jac(self, timestep):
    res = (self.residual * self.theta + self.inertia / timestep).replace({self.timetarget: function.Argument(self.timetarget, ())+timestep}) \
        + (self.residual * (1-self.theta) - self.inertia / timestep).replace({self.target: function.Argument(self.target0, self.lhs0.shape)})
    return res, res.derivative(self.target)

  def _step(self, lhs, t, timestep):
    res, jac = self._res_jac(timestep)
    try:
      return newton(self.target, residual=res, jacobian=jac, lhs0=lhs, constrain=self.constrain,
        arguments=collections.ChainMap(self.arguments, {self.target0: lhs, self.timetarget: t}), **self.newtonargs).solve(tol=self.newtontol)
    except (SolverError, matrix.MatrixError) as e:
      log.error('error: {}; retrying with timestep {}'.format(e, timestep/2))
      return self._step(self._step(lhs, t, timestep/2), t+timestep/2, timestep/2)

  def resume_index(self, history, index):
    if history:
      lhs, = history
    else:
      lhs = self.lhs0
      yield lhs
    while True:
      lhs = self._step(lhs, self.time0+index*self.timestep, self.timestep)
      index += 1
      yield lhs

impliciteuler = functools.partial(thetamethod, theta=1)
cranknicolson = functools.partial(thetamethod, theta=0.5)


@log.withcontext
def optimize(target:types.strictstr, functional:sample.strictintegral, *, newtontol:types.strictfloat=0., arguments:argdict={}, **kwargs):
  '''find the minimizer of a given functional

  Parameters
  ----------
  target : :class:`str`
      Name of the target: a :class:`nutils.function.Argument` in ``residual``.
  functional : scalar :class:`nutils.sample.Integral`
      The functional the should be minimized by varying target
  newtontol : :class:`float`
      Residual tolerance of Newton procedure (if applicable)
  arguments : :class:`collections.abc.Mapping`
      Defines the values for :class:`nutils.function.Argument` objects in
      `residual`.  The ``target`` should not be present in ``arguments``.
      Optional.
  **kwargs
      Additional arguments for :class:`minimize`

  Yields
  ------
  :class:`numpy.ndarray`
      Coefficient vector corresponding to the functional optimum
  '''

  lhs = newton(target, functional.derivative(target), arguments=arguments, **kwargs).solve(newtontol)
  optimum = functional.eval(**{target: numpy.choose(numpy.isnan(lhs), [lhs, 0])}, **arguments)
  log.info('constrained {}/{} dofs, optimum value {:.2e}'.format(len(lhs)-numpy.isnan(lhs).sum(), len(lhs), optimum))
  return lhs


## HELPER FUNCTIONS

def _nan_at(vec, where):
  copy = vec.copy()
  if where is not False:
    copy[where] = numpy.nan
  return copy

def _parse_lhs_cons(lhs0, constrain, shape):
  if lhs0 is None:
    lhs0 = types.frozenarray.full(shape, fill_value=0.)
  elif lhs0.shape == shape:
    lhs0 = types.frozenarray(lhs0)
  else:
    raise ValueError('expected `lhs0` with shape {} but got {}'.format(shape, lhs0.shape))
  if constrain is None:
    constrain = types.frozenarray.full(shape, fill_value=False)
  elif constrain.shape != shape:
    raise ValueError('expected `constrain` with shape {} but got {}'.format(shape, constrain.shape))
  elif constrain.dtype == float:
    isnan = numpy.isnan(constrain)
    lhs0 = types.frozenarray(numpy.choose(isnan, [constrain, lhs0]), copy=False)
    constrain = types.frozenarray(~isnan, copy=False)
  elif constrain.dtype == bool:
    constrain = types.frozenarray(constrain)
  else:
    raise ValueError('`constrain` should have dtype bool or float but got {}'.format(constrain.dtype))
  return lhs0, constrain

def _derivative(residual, target, jacobian=None):
  if jacobian is None:
    jacobian = residual.derivative(target)
  if jacobian.shape != residual.shape * 2:
    raise ValueError('expected `jacobian` with shape {} but got {}'.format(residual.shape * 2, jacobian.shape))
  return jacobian

def _minimize2(p0, p1, q0, q1):
  'find smallest positive minimum of a 3rd degree polynomial defined by values (p) and derivatives (q) at x=0 and x=1'
  assert q0 < 0
  # Polynomial: P(x) = p0 + q0 x + c x^2 + d x^3
  c = math.fsum([-3*p0, 3*p1, -2*q0, -q1])
  d = math.fsum([2*p0, -2*p1, q0, q1])
  # To minimize P we need to determine the roots for P'(x) = q0 + 2 c x + 3 d x^2
  # For numerical stability we use Citardauq's formula: x = -q0 / (c +/- sqrt(D)),
  # with D the discriminant
  D = c**2 - 3 * q0 * d
  # If D <= 0 we have at most one duplicate root, which we ignore. For D > 0,
  # taking into account that q0 < 0, we distinguish three situations:
  # - d > 0 => sqrt(D) > abs(c): one negative, one positive root
  # - d = 0 => sqrt(D) = abs(c): one negative root
  # - d < 0 => sqrt(D) < abs(c): two roots of same sign as c
  return -q0 / (c + math.sqrt(D)) if D > 0 and (c > 0 or d > 0) else math.inf

def _minimize3(p0, p1, q0, q1, r0, r1):
  'find smallest positive minimum of a 5th degree polynomial defined by values (p), derivatives (q) and second derivatives (r) at x=0 and x=1'
  assert q0 < 0
  # Polynomial: P(x) = p0 + q0 x + r0/2 x^2 + d x^3 + e x^4 + f x^5
  d = math.fsum([-10*p0, -6*q0, -1.5*r0, 10*p1, -4*q1, .5*r1])
  e = math.fsum([15*p0, 8*q0, 1.5*r0, -15*p1, 7*q1, -r1])
  f = math.fsum([-6*p0, -3*q0, -.5*r0, 6*p1, -3*q1, .5*r1])
  # To minimize P we need to determine the roots for P'(x) = q0 + r0 x + 3 d x^2 + 4 e x^3 + 5 f x^4
  roots = [x.real for x in numpy.roots([5*f, 4*e, 3*d, r0, q0]) if not x.imag and x > 0]
  return min(roots) if roots else math.inf

# vim:sw=2:sts=2:et