solver¶
The solver module defines solvers for problems of the kind res = 0
or
∂inertia/∂t + res = 0
, where res
is a
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.

nutils.solver.
solve_linear
(target, residual, constrain=None, *, arguments={}, solveargs={})¶ solve linear problem
 Parameters
target (
str
) – Name of the target: anutils.function.Argument
inresidual
.residual (
nutils.sample.Integral
) – Residual integral, depends ontarget
constrain (
numpy.ndarray
with dtypefloat
) – Defines the fixed entries of the coefficient vectorarguments (
collections.abc.Mapping
) – Defines the values fornutils.function.Argument
objects in residual. Thetarget
should not be present inarguments
. Optional.
 Returns
Array of
target
values for whichresidual == 0
 Return type

class
nutils.solver.
RecursionWithSolve
(*args, **kwargs)¶ Bases:
nutils.cache.Recursion
add a .solve method to (lhs,resnorm) iterators

solve
(self, tol=0.0, maxiter=inf)¶ execute nonlinear solver, return lhs
Iterates over nonlinear solver until tolerance is reached. Example:
lhs = newton(target, residual).solve(tol=1e5)
 Parameters
 Returns
Coefficient vector that corresponds to a smaller than
tol
residual. Return type


class
nutils.solver.
newton
(target, residual, jacobian=None, lhs0=None, constrain=None, searchrange=(0.01, 0.6666666666666666), droptol=None, rebound=2.0, failrelax=1e06, arguments={}, solveargs={})¶ Bases:
nutils.solver.RecursionWithSolve
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
andD
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 (
str
) – Name of the target: anutils.function.Argument
inresidual
.residual (
nutils.sample.Integral
) –lhs0 (
numpy.ndarray
) – Coefficient vector, starting point of the iterative procedure.constrain (
numpy.ndarray
with dtypebool
orfloat
) – Equal length tolhs0
, masks the free vector entries asFalse
(boolean) or NaN (float). In the remaining positions the values oflhs0
are returned unchanged (boolean) or overruled by the values in constrain (float).searchrange (
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 reevaluated at the new relaxation value or at the lower bound, whichever is largest.rebound (
float
) – Factor by which the relaxation value grows after every update until it reaches unity.droptol (
float
) – Threshold for leaving entries in the return value at NaN if they do not contribute to the value of the functional.failrelax (
float
) – Fail with exception if relaxation reaches this lower limit.arguments (
collections.abc.Mapping
) – Defines the values fornutils.function.Argument
objects in residual. Thetarget
should not be present inarguments
. Optional.
 Yields
numpy.ndarray
– Coefficient vector that approximates residual==0 with increasing accuracy

class
nutils.solver.
minimize
(target, energy, lhs0=None, constrain=None, searchrange=(0.01, 0.5), rebound=2.0, droptol=None, failrelax=1e06, arguments={}, solveargs={}, maxinc=1e14)¶ Bases:
nutils.solver.RecursionWithSolve
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
andF
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 (
str
) – Name of the target: anutils.function.Argument
inresidual
.residual (
nutils.sample.Integral
) –lhs0 (
numpy.ndarray
) – Coefficient vector, starting point of the iterative procedure.constrain (
numpy.ndarray
with dtypebool
orfloat
) – Equal length tolhs0
, masks the free vector entries asFalse
(boolean) or NaN (float). In the remaining positions the values oflhs0
are returned unchanged (boolean) or overruled by the values in constrain (float).searchrange (
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 reevaluated at the new relaxation value or at the lower bound, whichever is largest.rebound (
float
) – Factor by which the relaxation value grows after every update until it reaches unity.droptol (
float
) – Threshold for leaving entries in the return value at NaN if they do not contribute to the value of the functional.failrelax (
float
) – Fail with exception if relaxation reaches this lower limit.arguments (
collections.abc.Mapping
) – Defines the values fornutils.function.Argument
objects in residual. Thetarget
should not be present inarguments
. Optional.maxinc (
float
) – Permitted energy increase; a small nonzero value is required to avoid getting stuck on numerical noise close to the energy minimum.
 Yields
numpy.ndarray
– Coefficient vector that approximates residual==0 with increasing accuracy

class
nutils.solver.
pseudotime
(target, residual, inertia, timestep, lhs0=None, constrain=None, arguments={}, solveargs={})¶ Bases:
nutils.solver.RecursionWithSolve
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 (
str
) – Name of the target: anutils.function.Argument
inresidual
.residual (
nutils.sample.Integral
) –inertia (
nutils.sample.Integral
) –timestep (
float
) – Initial time step, will scale up as residual decreaseslhs0 (
numpy.ndarray
) – Coefficient vector, starting point of the iterative procedure.constrain (
numpy.ndarray
with dtypebool
orfloat
) – Equal length tolhs0
, masks the free vector entries asFalse
(boolean) or NaN (float). In the remaining positions the values oflhs0
are returned unchanged (boolean) or overruled by the values in constrain (float).arguments (
collections.abc.Mapping
) – Defines the values fornutils.function.Argument
objects in residual. Thetarget
should not be present inarguments
. Optional.
 Yields
numpy.ndarray
with dtypefloat
– Tuple of coefficient vector and residual norm

class
nutils.solver.
thetamethod
(target, residual, inertia, timestep, lhs0, theta, target0='_thetamethod_target0', constrain=None, newtontol=1e10, arguments={}, newtonargs={}, timetarget=None, time0=0.0)¶ Bases:
nutils.solver.RecursionWithSolve
solve time dependent problem using the theta method
 Parameters
target (
str
) – Name of the target: anutils.function.Argument
inresidual
.residual (
nutils.sample.Integral
) –inertia (
nutils.sample.Integral
) –timestep (
float
) – Initial time step, will scale up as residual decreaseslhs0 (
numpy.ndarray
) – Coefficient vector, starting point of the iterative procedure.theta (
float
) – Theta value (theta=1 for implicit Euler, theta=0.5 for CrankNicolson)residual0 (
nutils.sample.Integral
) – Optional additional residual component evaluated in previous timestepconstrain (
numpy.ndarray
with dtypebool
orfloat
) – Equal length tolhs0
, masks the free vector entries asFalse
(boolean) or NaN (float). In the remaining positions the values oflhs0
are returned unchanged (boolean) or overruled by the values in constrain (float).newtontol (
float
) – Residual tolerance of individual timestepsarguments (
collections.abc.Mapping
) – Defines the values fornutils.function.Argument
objects in residual. Thetarget
should not be present inarguments
. Optional.timetarget (
str
) – Name of thenutils.function.Argument
that represents time. Optional.time0 (
float
) – The intial time. Default:0.0
.
 Yields
numpy.ndarray
– Coefficient vector for all timesteps after the initial condition.

nutils.solver.
optimize
(target, functional, *, newtontol=0.0, arguments={}, **kwargs)¶ find the minimizer of a given functional
 Parameters
target (
str
) – Name of the target: anutils.function.Argument
inresidual
.functional (scalar
nutils.sample.Integral
) – The functional the should be minimized by varying targetnewtontol (
float
) – Residual tolerance of Newton procedure (if applicable)arguments (
collections.abc.Mapping
) – Defines the values fornutils.function.Argument
objects in residual. Thetarget
should not be present inarguments
. Optional.**kwargs – Additional arguments for
minimize
 Yields
numpy.ndarray
– Coefficient vector corresponding to the functional optimum