`pymor.operators.ei`¶

Module Contents¶

class pymor.operators.ei.EmpiricalInterpolatedOperator(operator, interpolation_dofs, collateral_basis, triangular, solver=None, name=None)[source]¶

Bases: pymor.operators.interface.Operator

Interpolate an Operator using Empirical Operator Interpolation.

Let L be an Operator, 0 <= c_1, ..., c_M < L.range.dim indices of interpolation DOFs and let b_1, ..., b_M in R^(L.range.dim) be collateral basis vectors. If moreover ψ_j(U) denotes the j-th component of U, the empirical interpolation L_EI of L w.r.t. the given data is given by

|                M
|   L_EI(U, μ) = ∑ b_i⋅λ_i     such that
|               i=1
|
|   ψ_(c_i)(L_EI(U, μ)) = ψ_(c_i)(L(U, μ))   for i=0,...,M

Since the original operator only has to be evaluated at the given interpolation DOFs, EmpiricalInterpolatedOperator calls restricted to obtain a restricted version of the operator which is used to quickly obtain the required evaluations. If the restricted method, is not implemented, the full operator will be evaluated (which will lead to the same result, but without any speedup).

The interpolation DOFs and the collateral basis can be generated using the algorithms provided in the pymor.algorithms.ei module.

Parameters:

operator – The Operator to interpolate.
interpolation_dofs – List or 1D NumPy array of the interpolation DOFs c_1, ..., c_M.
collateral_basis – VectorArray containing the collateral basis b_1, ..., b_M.
triangular – If True, assume that ψ_(c_i)(b_j) = 0 for i < j, which means that the interpolation matrix is triangular.
solver – The Solver for the operator.
name – Name of the operator.

Methods

`apply`	Apply the operator to a `VectorArray`.
`jacobian`	Return the operator's Jacobian as a new `Operator`.

apply(U, mu=None)[source]¶

Apply the operator to a VectorArray.

Parameters:

U – VectorArray of vectors to which the operator is applied.
mu – The parameter values for which to evaluate the operator.

Returns:

|VectorArray| of the operator evaluations.

jacobian(U, mu=None)[source]¶

Return the operator’s Jacobian as a new Operator.

If the operator has a Solver, the Jacobian Operator will be equipped with the solver’s jacobian_solver.

Parameters:

U – Length 1 VectorArray containing the vector for which to compute the Jacobian.
mu – The parameter values for which to compute the Jacobian.

Returns:

Linear |Operator| representing the Jacobian.

class pymor.operators.ei.ProjectedEmpiricalInterpolatedOperator(restricted_operator, interpolation_matrix, source_basis_dofs, projected_collateral_basis, triangular, solver=None, name=None)[source]¶

Bases: pymor.operators.interface.Operator

A projected EmpiricalInterpolatedOperator.

Methods

`apply`	Apply the operator to a `VectorArray`.
`jacobian`	Return the operator's Jacobian as a new `Operator`.
`with_cb_dim`

apply(U, mu=None)[source]¶

Apply the operator to a VectorArray.

Parameters:

U – VectorArray of vectors to which the operator is applied.
mu – The parameter values for which to evaluate the operator.

Returns:

|VectorArray| of the operator evaluations.

jacobian(U, mu=None)[source]¶

Return the operator’s Jacobian as a new Operator.

If the operator has a Solver, the Jacobian Operator will be equipped with the solver’s jacobian_solver.

Parameters:

U – Length 1 VectorArray containing the vector for which to compute the Jacobian.
mu – The parameter values for which to compute the Jacobian.

Returns:

Linear |Operator| representing the Jacobian.

with_cb_dim(dim)[source]¶

pymor.operators.ei¶

Module Contents¶

`pymor.operators.ei`¶