# 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
Returns

Array of target values for which residual == 0

Return type

numpy.ndarray

nutils.solver.solve(gen_lhs_resnorm, tol=0.0, maxiter=inf)

execute nonlinear solver, return lhs

Iterates over nonlinear solver until tolerance is reached. Example:

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

Parameters
Returns

Coefficient vector that corresponds to a smaller than tol residual.

Return type

numpy.ndarray

nutils.solver.solve_withinfo(gen_lhs_resnorm, 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.RecursionWithSolve(*args, **kwargs)

add a .solve method to (lhs,resnorm) iterators

Introduces the convenient form:

newton(target, residual).solve(tol)


Shorthand for:

solve(newton(target, residual), tol)

solve_withinfo(gen_lhs_resnorm, 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.

solve(gen_lhs_resnorm, tol=0.0, maxiter=inf)

execute nonlinear solver, return lhs

Iterates over nonlinear solver until tolerance is reached. Example:

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

Parameters
Returns

Coefficient vector that corresponds to a smaller than tol residual.

Return type

numpy.ndarray

class nutils.solver.newton(target, residual, jacobian=None, lhs0=None, constrain=None, searchrange=(0.01, 0.6666666666666666), droptol=None, rebound=2.0, failrelax=1e-06, arguments={}, solveargs={})

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
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=1e-06, arguments={}, solveargs={}, maxinc=1e-14)

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).

• 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 re-evaluated 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 for nutils.function.Argument objects in residual. The target should not be present in arguments. 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={})

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)

solve time dependent problem using the theta method

Parameters
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
Yields

numpy.ndarray – Coefficient vector corresponding to the functional optimum