pymor.algorithms.projection

Module Contents

class pymor.algorithms.projection.ProjectRules(range_basis, source_basis)

Bases: pymor.algorithms.rules.RuleTable

RuleTable for the project algorithm.

Methods

action_AdjointOperator
action_AffineOperator
action_BlockOperatorBase
action_ConcatenationOperator
action_ConstantOperator
action_EmpiricalInterpolatedOperator
action_LincombOperator
action_QuadraticFunctional
action_SelectionOperator
action_ZeroOperator
action_apply_basis

action_AdjointOperator(op)

action_AffineOperator(op)

action_BlockOperatorBase(op)

action_ConcatenationOperator(op)

action_ConstantOperator(op)

action_EmpiricalInterpolatedOperator(op)

action_LincombOperator(op)

action_QuadraticFunctional(op)

action_SelectionOperator(op)

action_ZeroOperator(op)

action_apply_basis(op)

class pymor.algorithms.projection.ProjectToSubbasisRules(dim_range, dim_source)

Bases: pymor.algorithms.rules.RuleTable

RuleTable for the project_to_subbasis algorithm.

Methods

action_BlockColumnOperator
action_BlockRowOperator
action_ConstantOperator
action_IdentityOperator
action_NumpyMatrixOperator
action_ProjectedEmpiricalInterpolatedOperator
action_ProjectedOperator
action_QuadraticFunctional
action_VectorArrayOperator
action_ZeroOperator
action_recurse

action_BlockColumnOperator(op)

action_BlockRowOperator(op)

action_ConstantOperator(op)

action_IdentityOperator(op)

action_NumpyMatrixOperator(op)

action_ProjectedEmpiricalInterpolatedOperator(op)

action_ProjectedOperator(op)

action_QuadraticFunctional(op)

action_VectorArrayOperator(op)

action_ZeroOperator(op)

action_recurse(op)

pymor.algorithms.projection.project(op, range_basis, source_basis, product=None)

Petrov-Galerkin projection of a given Operator.

Given an inner product ( ⋅, ⋅), source vectors b_1, ..., b_N and range vectors c_1, ..., c_M, the projection op_proj of op is defined by

[ op_proj(e_j) ]_i = ( c_i, op(b_j) )

for all i,j, where e_j denotes the j-th canonical basis vector of R^N.

In particular, if the c_i are orthonormal w.r.t. the given product, then op_proj is the coordinate representation w.r.t. the b_i/c_i bases of the restriction of op to span(b_i) concatenated with the orthogonal projection onto span(c_i).

From another point of view, if op is viewed as a bilinear form (see apply2) and ( ⋅, ⋅ ) is the Euclidean inner product, then op_proj represents the matrix of the bilinear form restricted to span(b_i) / span(c_i) (w.r.t. the b_i/c_i bases).

How the projection is realized will depend on the given Operator. While a projected NumpyMatrixOperator will again be a NumpyMatrixOperator, only a generic ProjectedOperator can be returned in general. The exact algorithm is specified in ProjectRules.

Parameters:

• range_basis – The vectors c_1, ..., c_M as a VectorArray. If None, no projection in the range space is performed.

• source_basis – The vectors b_1, ..., b_N as a VectorArray or None. If None, no restriction of the source space is performed.

• product – An Operator representing the inner product. If None, the Euclidean inner product is chosen.

Returns:

The projected Operator op_proj.

pymor.algorithms.projection.project_to_subbasis(op, dim_range=None, dim_source=None)

Project already projected Operator to a subbasis.

The purpose of this method is to further project an operator that has been obtained through project to subbases of the original projection bases, i.e.

project_to_subbasis(project(op, r_basis, s_basis, prod), dim_range, dim_source)

should be the same as

project(op, r_basis[:dim_range], s_basis[:dim_source], prod)

For a NumpyMatrixOperator this amounts to extracting the upper-left (dim_range, dim_source) corner of its matrix.

The subbasis projection algorithm is specified in ProjectToSubbasisRules.

Parameters:

• dim_range – Dimension of the range subbasis.

• dim_source – Dimension of the source subbasis.

Returns:

The projected |Operator|.