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 25 dof system to machine precision 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={}, **linargs)

solve linear problem

Parameters
Returns

Array of target values for which residual == 0

Return type

numpy.ndarray

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=1e-5)
Parameters
  • tol (float) – Target residual norm

  • maxiter (int) – Maximum number of iterations

Returns

Coefficient vector that corresponds to a smaller than tol residual.

Return type

numpy.ndarray

solve_withinfo(self, tol=0.0, maxiter=inf)

execute nonlinear solver, return lhs and info

Like 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.

class nutils.solver.newton(target, residual, jacobian=None, lhs0=None, relax0=1.0, constrain=None, linesearch=None, failrelax=1e-06, arguments={}, solveargs={}, **linargs)

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 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 (str) – Name of the target: a nutils.function.Argument in residual.

  • residual (nutils.sample.Integral) –

  • lhs0 (numpy.ndarray) – Coefficient vector, starting point of the iterative procedure.

  • relax0 (float) – Initial relaxation value.

  • constrain (numpy.ndarray with dtype bool or 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).

  • linesearch (nutils.solver.LineSearch) – Callable that defines relaxation logic.

  • failrelax (float) – Fail with exception if relaxation reaches this lower limit.

  • arguments (collections.abc.Mapping) – Defines the values for nutils.function.Argument objects in residual. The target should not be present in arguments. Optional.

Yields

numpy.ndarray – Coefficient vector that approximates residual==0 with increasing accuracy

class nutils.solver.LineSearch(*args, **kwargs)

Bases: nutils.types.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: nutils.solver.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: nutils.solver.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.

class nutils.solver.minimize(target, energy, lhs0=None, constrain=None, rampup=0.5, rampdown=-1.0, failrelax=-10.0, arguments={}, solveargs={}, **linargs)

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 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 (str) – Name of the target: a nutils.function.Argument in residual.

  • residual (nutils.sample.Integral) –

  • lhs0 (numpy.ndarray) – Coefficient vector, starting point of the iterative procedure.

  • constrain (numpy.ndarray with dtype bool or 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).

  • rampup (float) – Value to increase the relaxation power by in case energy is decreasing.

  • rampdown (float) – Value to decrease the relaxation power by in case energy is increasing.

  • failrelax (float) – Fail with exception if relaxation reaches this lower limit.

  • arguments (collections.abc.Mapping) – Defines the values for nutils.function.Argument objects in residual. The target should not be present in arguments. Optional.

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={}, **linargs)

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
Yields

numpy.ndarray with dtype float – Tuple of coefficient vector and residual norm

class nutils.solver.thetamethod(target, residual, inertia, timestep, lhs0, theta, target0='_thetamethod_target0', constrain=None, newtontol=1e-10, 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: a nutils.function.Argument in residual.

  • residual (nutils.sample.Integral) –

  • inertia (nutils.sample.Integral) –

  • timestep (float) – Initial time step, will scale up as residual decreases

  • lhs0 (numpy.ndarray) – Coefficient vector, starting point of the iterative procedure.

  • theta (float) – Theta value (theta=1 for implicit Euler, theta=0.5 for Crank-Nicolson)

  • residual0 (nutils.sample.Integral) – Optional additional residual component evaluated in previous timestep

  • constrain (numpy.ndarray with dtype bool or 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 (float) – Residual tolerance of individual timesteps

  • arguments (collections.abc.Mapping) – Defines the values for nutils.function.Argument objects in residual. The target should not be present in arguments. Optional.

  • timetarget (str) – Name of the nutils.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, *, tol=0.0, arguments={}, droptol=None, constrain=None, lhs0=None, relax0=1.0, solveargs={}, linesearch=None, failrelax=1e-06, **linargs)

find the minimizer of a given functional

Parameters
  • target (str) – Name of the target: a nutils.function.Argument in residual.

  • functional (scalar nutils.sample.Integral) – The functional the should be minimized by varying target

  • tol (float) – Target residual norm.

  • arguments (collections.abc.Mapping) – Defines the values for nutils.function.Argument objects in residual. The target should not be present in arguments. 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 dtype float) – Defines the fixed entries of the coefficient vector

  • lhs0 (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