"""
Defines an implicit Euler solver
.. codeauthor:: David Zwicker <david.zwicker@ds.mpg.de>
"""
from __future__ import annotations
from typing import Callable
import numba as nb
import numpy as np
from ..fields.base import FieldBase
from ..pdes.base import PDEBase
from ..tools.typing import BackendType
from .base import ConvergenceError, SolverBase
[docs]
class ImplicitSolver(SolverBase):
"""implicit (backward) Euler PDE solver"""
name = "implicit"
def __init__(
self,
pde: PDEBase,
maxiter: int = 100,
maxerror: float = 1e-4,
backend: BackendType = "auto",
):
"""
Args:
pde (:class:`~pde.pdes.base.PDEBase`):
The instance describing the pde that needs to be solved
maxiter (int):
The maximal number of iterations per step
maxerror (float):
The maximal error that is permitted in each step
backend (str):
Determines how the function is created. Accepted values are 'numpy` and
'numba'. Alternatively, 'auto' lets the code decide for the most optimal
backend.
"""
super().__init__(pde, backend=backend)
self.maxiter = maxiter
self.maxerror = maxerror
def _make_single_step_fixed_dt_deterministic(
self, state: FieldBase, dt: float
) -> Callable[[np.ndarray, float], None]:
"""return a function doing a deterministic step with an implicit Euler scheme
Args:
state (:class:`~pde.fields.base.FieldBase`):
An example for the state from which the grid and other information can
be extracted
dt (float):
Time step of the implicit step
"""
if self.pde.is_sde:
raise RuntimeError("Cannot use implicit stepper with stochastic equation")
self.info["function_evaluations"] = 0
self.info["scheme"] = "implicit-euler"
self.info["stochastic"] = False
self.info["dt_adaptive"] = False
rhs = self._make_pde_rhs(state, backend=self.backend)
maxiter = int(self.maxiter)
maxerror2 = self.maxerror**2
# handle deterministic version of the pde
def implicit_step(state_data: np.ndarray, t: float) -> None:
"""compiled inner loop for speed"""
nfev = 0 # count function evaluations
# save state at current time point t for stepping
state_t = state_data.copy()
# estimate state at next time point
state_data[:] = state_t + dt * rhs(state_data, t)
state_prev = np.empty_like(state_data)
# fixed point iteration for improving state after dt
for n in range(maxiter):
state_prev[:] = state_data # keep previous state to judge convergence
# another interation to improve estimate
state_data[:] = state_t + dt * rhs(state_data, t + dt)
# calculate mean squared error to judge convergence
err = 0.0
for j in range(state_data.size):
diff = state_data.flat[j] - state_prev.flat[j]
err += (diff.conjugate() * diff).real
err /= state_data.size
if err < maxerror2:
# fix point iteration converged
break
else:
with nb.objmode:
self._logger.warning(
"Implicit Euler step did not converge after %d iterations "
"at t=%g (error=%g)",
maxiter,
t,
err,
)
raise ConvergenceError("Implicit Euler step did not converge.")
nfev += n + 1
self._logger.info("Init implicit Euler stepper with dt=%g", dt)
return implicit_step
def _make_single_step_fixed_dt_stochastic(
self, state: FieldBase, dt: float
) -> Callable[[np.ndarray, float], None]:
"""return a function doing a step for a SDE with an implicit Euler scheme
Args:
state (:class:`~pde.fields.base.FieldBase`):
An example for the state from which the grid and other information can
be extracted
dt (float):
Time step of the implicit step
"""
self.info["function_evaluations"] = 0
self.info["scheme"] = "implicit-euler-maruyama"
self.info["stochastic"] = True
self.info["dt_adaptive"] = False
rhs = self.pde.make_pde_rhs(state, backend=self.backend) # type: ignore
rhs_sde = self._make_sde_rhs(state, backend=self.backend)
maxiter = int(self.maxiter)
maxerror2 = self.maxerror**2
# handle deterministic version of the pde
def implicit_step(state_data: np.ndarray, t: float) -> None:
"""compiled inner loop for speed"""
nfev = 0 # count function evaluations
# save state at current time point t for stepping
state_t = state_data.copy()
state_prev = np.empty_like(state_data)
# estimate state at next time point
evolution_rate, noise_realization = rhs_sde(state_data, t)
if noise_realization is not None:
# add the noise to the reference state at the current time point and
# adept the state at the next time point iteratively below
state_t += np.sqrt(dt) * noise_realization
state_data[:] = state_t + dt * evolution_rate # estimated new state
# fixed point iteration for improving state after dt
for n in range(maxiter):
state_prev[:] = state_data # keep previous state to judge convergence
# another interation to improve estimate
state_data[:] = state_t + dt * rhs(state_data, t + dt)
# calculate mean squared error to judge convergence
err = 0.0
for j in range(state_data.size):
diff = state_data.flat[j] - state_prev.flat[j]
err += (diff.conjugate() * diff).real
err /= state_data.size
if err < maxerror2:
# fix point iteration converged
break
else:
with nb.objmode:
self._logger.warning(
"Semi-implicit Euler-Maruyama step did not converge after %d "
"iterations at t=%g (error=%g)",
maxiter,
t,
err,
)
raise ConvergenceError(
"Semi-implicit Euler-Maruyama step did not converge."
)
nfev += n + 1
self._logger.info("Init semi-implicit Euler-Maruyama stepper with dt=%g", dt)
return implicit_step
def _make_single_step_fixed_dt(
self, state: FieldBase, dt: float
) -> Callable[[np.ndarray, float], None]:
"""return a function doing a single step with an implicit Euler scheme
Args:
state (:class:`~pde.fields.base.FieldBase`):
An example for the state from which the grid and other information can
be extracted
dt (float):
Time step of the implicit step
"""
if self.pde.is_sde:
return self._make_single_step_fixed_dt_stochastic(state, dt)
else:
return self._make_single_step_fixed_dt_deterministic(state, dt)