tlm_adjoint.firedrake.solve

Finite element variational problem solution operations with Firedrake.

Module Contents

class tlm_adjoint.firedrake.solve.EquationSolver(eq, x, bcs=None, *, J=None, form_compiler_parameters=None, solver_parameters=None, adjoint_solver_parameters=None, tlm_solver_parameters=None, cache_jacobian=None, cache_adjoint_jacobian=None, cache_tlm_jacobian=None, cache_rhs_assembly=None, match_quadrature=None)

Represents the solution of a finite element variational problem.

Caching is based on the approach described in

  • J. R. Maddison and P. E. Farrell, ‘Rapid development and adjoining of transient finite element models’, Computer Methods in Applied Mechanics and Engineering, 276, 95–121, 2014, doi: 10.1016/j.cma.2014.03.010

The arguments eq, x, bcs, J, form_compiler_parameters, and solver_parameters are based on the interface for the DOLFIN dolfin.solve function (see e.g. FEniCS 2017.1.0).

Parameters:

  • eq – A ufl.equation.Equation defining the finite element variational problem.

  • x – A firedrake.function.Function defining the forward solution.

  • bcs – Dirichlet boundary conditions.

  • J – A ufl.Form defining a Jacobian matrix approximation to use in a non-linear forward solve.

  • form_compiler_parameters – Form compiler parameters.

  • solver_parameters – Linear or non-linear solver parameters.

  • adjoint_solver_parameters – Linear solver parameters to use in an adjoint solve.

  • tlm_solver_parameters – Linear solver parameters to use when solving tangent-linear problems.

  • cache_jacobian – Whether to cache the forward Jacobian matrix and linear solver data. Defaults to parameters[‘tlm_adjoint’][‘EquationSolver][‘cache_jacobian’]. If None then caching is autodetected.

  • cache_adjoint_jacobian – Whether to cache the adjoint Jacobian matrix and linear solver data. Defaults to cache_jacobian.

  • cache_tlm_jacobian – Whether to cache the Jacobian matrix and linear solver data when solving tangent-linear problems. Defaults to cache_jacobian.

  • cache_rhs_assembly – Whether to enable right-hand-side caching. If enabled then right-hand-side terms are divided into terms which are cached, terms which are converted into matrix multiplication by a cached matrix, and terms which are not cached. Defaults to parameters[‘tlm_adjoint’][‘EquationSolver’][‘cache_rhs_assembly’].

  • match_quadrature – Whether to set quadrature parameters consistently in the forward, adjoint, and tangent-linears. Defaults to parameters[‘tlm_adjoint’][‘EquationSolver’][‘match_quadrature’].

drop_references()

Drop references to variables which store values.

forward_solve(x, deps=None)

Compute the forward solution.

Can assume that the currently active EquationManager is paused.

Parameters:

  • X – A variable or a Sequence of variables storing the solution. May define an initial guess, and should be set by this method.

  • deps – A tuple of variables, defining values for dependencies. Only the elements corresponding to X may be modified. self.dependencies() should be used if not supplied.

subtract_adjoint_derivative_actions(adj_x, nl_deps, dep_Bs)

Subtract terms from other adjoint right-hand-sides.

Can be overridden for an optimized implementation, but otherwise uses Equation.adjoint_derivative_action().

Parameters:

  • adj_X – The adjoint solution. A variable or a Sequence of variables. Should not be modified.

  • nl_deps – A Sequence of variables defining values for non-linear dependencies. Should not be modified.

  • dep_Bs – A Mapping whose items are (dep_index, dep_B). Each dep_B is an AdjointRHS which should be updated by subtracting adjoint derivative information computed by differentiating with respect to self.dependencies()[dep_index].

adjoint_jacobian_solve(adj_x, nl_deps, b)

Compute an adjoint solution.

Parameters:

  • adj_X – Either None, or a variable or Sequence of variables defining the initial guess for an iterative solve. May be modified or returned.

  • nl_deps – A Sequence of variables defining values for non-linear dependencies. Should not be modified.

  • B – The right-hand-side. A variable or Sequence of variables storing the value of the right-hand-side. May be modified or returned.

Returns:

A variable or Sequence of variables storing the value of the adjoint solution. May return None to indicate a value of zero.

tangent_linear(tlm_map)

Derive an Equation corresponding to a tangent-linear operation.

Parameters:

tlm_map – A TangentLinearMap storing values for tangent-linear variables.

Returns:

An Equation, corresponding to the tangent-linear operation.

class tlm_adjoint.firedrake.solve.LocalEquationSolver(eq, x, *, form_compiler_parameters=None, cache_jacobian=None, cache_adjoint_jacobian=None, cache_tlm_jacobian=None, cache_rhs_assembly=None, match_quadrature=None)

Represents the solution of a linear finite element variational problem, for the case where the matrix is element-wise local block diagonal.

Remaining arguments are passed to the EquationSolver constructor.

tangent_linear(tlm_map)

Derive an Equation corresponding to a tangent-linear operation.

Parameters:

tlm_map – A TangentLinearMap storing values for tangent-linear variables.

Returns:

An Equation, corresponding to the tangent-linear operation.