solver¶
The solver module defines solvers for problems of the kind res = 0
or
∂inertia/∂t + res = 0
, where res
is a
nutils.evaluable.AsEvaluableArray
. 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 25 dof system to machine precision using arnoldi 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.single_or_multiple(f)¶
add support for legacy string target + array return value
- class nutils.solver.iterable(target, *args, **kwargs)¶
Bases:
object
iterable equivalent of single_or_multiple
- __weakref__¶
list of weak references to the object (if defined)
- class nutils.solver.withsolve(target, *args, **kwargs)¶
Bases:
iterable
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=1e-5)
- Parameters:
- Returns:
Coefficient vector that corresponds to a smaller than
tol
residual.- Return type:
- class nutils.solver.LineSearch(*args, **kwargs)¶
Bases:
Immutable
Line search abstraction for gradient based optimization.
A line search object is a callable that takes four arguments: the current residual and directional derivative, and the candidate residual and directional derivative, with derivatives normalized to unit length; and returns the optimal scaling and a boolean flag that marks whether the candidate should be accepted.
- class nutils.solver.NormBased(minscale=0.01, acceptscale=0.6666666666666666, maxscale=2.0)¶
Bases:
LineSearch
Line search abstraction for Newton-like iterations, computing relaxation values that correspond to greatest reduction of the residual norm.
- Parameters:
minscale (
float
) – Minimum relaxation scaling per update. Must be strictly greater than zero.acceptscale (
float
) – Relaxation scaling that is considered close enough to optimality to to accept the current Newton update. Must lie between minscale and one.maxscale (
float
) – Maximum relaxation scaling per update. Must be greater than one, and therefore always coincides with acceptance, determining how fast relaxation values rebound to one if not bounded by optimality.
- class nutils.solver.MedianBased(minscale=0.01, acceptscale=0.6666666666666666, maxscale=2.0, quantile=0.5)¶
Bases:
LineSearch
Line search abstraction for Newton-like iterations, computing relaxation values such that half (or any other configurable quantile) of the residual vector has its optimal reduction beyond it. Unline the
NormBased
approach this is invariant to constant scaling of the residual items.- Parameters:
minscale (
float
) – Minimum relaxation scaling per update. Must be strictly greater than zero.acceptscale (
float
) – Relaxation scaling that is considered close enough to optimality to to accept the current Newton update. Must lie between minscale and one.maxscale (
float
) – Maximum relaxation scaling per update. Must be greater than one, and therefore always coincides with acceptance, determining how fast relaxation values rebound to one if not bounded by optimality.quantile (
float
) – Fraction of the residual vector that is aimed to have its optimal reduction at a smaller relaxation value. The default value of one half corresponds to the median. A value close to zero means tighter control, resulting in strong relaxation.
- nutils.solver.solve_linear(target, residual, *, constrain=None, lhs0=None, arguments={}, **kwargs)¶
solve linear problem
- Parameters:
target (
str
) – Name of the target: anutils.function.Argument
inresidual
.residual (
nutils.evaluable.AsEvaluableArray
) – 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.newton(target, residual, jacobian=None, lhs0=None, relax0=1.0, constrain=None, linesearch=None, failrelax=1e-06, arguments={}, **kwargs)¶
Bases:
withsolve
add a .solve method to (lhs,resnorm) iterators
- class nutils.solver.minimize(target, energy, lhs0=None, constrain=None, rampup=0.5, rampdown=-1.0, failrelax=-10.0, arguments={}, **kwargs)¶
Bases:
withsolve
add a .solve method to (lhs,resnorm) iterators
- class nutils.solver.pseudotime(target, residual, inertia, timestep, lhs0=None, constrain=None, arguments={}, **kwargs)¶
Bases:
withsolve
add a .solve method to (lhs,resnorm) iterators
- __wrapped__¶
alias of
pseudotime
- class nutils.solver.thetamethod(target, residual, inertia, timestep, theta, lhs0=None, target0=None, constrain=None, newtontol=1e-10, arguments={}, newtonargs={}, timetarget='_thetamethod_time', time0=0.0, historysuffix='0')¶
Bases:
iterable
iterable equivalent of single_or_multiple
- __wrapped__¶
alias of
thetamethod
- nutils.solver.optimize(target, functional, *, tol=0.0, arguments={}, droptol=None, constrain=None, lhs0=None, relax0=1.0, linesearch=None, failrelax=1e-06, **kwargs)¶
find the minimizer of a given functional
- Parameters:
target (
str
) – Name of the target: anutils.function.Argument
inresidual
.functional (scalar
nutils.evaluable.AsEvaluableArray
) – The functional the should be minimized by varying targettol (
float
) – Target residual norm.arguments (
collections.abc.Mapping
) – Defines the values fornutils.function.Argument
objects in residual. Thetarget
should not be present inarguments
. Optional.droptol (
float
) – Threshold for leaving entries in the return value at NaN if they do not contribute to the value of the functional.constrain (
numpy.ndarray
with dtypefloat
) – Defines the fixed entries of the coefficient vectorlhs0 (
numpy.ndarray
) – Coefficient vector, starting point of the iterative procedure.relax0 (
float
) – Initial relaxation value.linesearch (
nutils.solver.LineSearch
) – Callable that defines relaxation logic.failrelax (
float
) – Fail with exception if relaxation reaches this lower limit.
- Yields:
numpy.ndarray
– Coefficient vector corresponding to the functional optimum