rom.ecsw.linear_form_hyperrom_ecsw

Purpose: Implements ECSW-based hyperreduction for finite element linear forms, enabling fast reduced-order load vector assembly by projecting element-wise load contributions onto a reduced basis, clustering elements by free-DOF count after Dirichlet condensation, and assembling the global reduced load vector via vectorized weighted projections.

Summary: Provides the LinearFormHYPERROM_ecsw class, a reduced-order linear form that handles Dirichlet boundary conditions through full-to-free DOF mapping, applies element-wise ECSW weights during projection, and produces the final ( r )-dimensional reduced load vector. All operations are restricted to free DOFs, with Dirichlet and mean-field contributions reinserted during reconstruction. The broader ecsw package also includes:

Reduced-order bilinear forms (BilinearFormHYPERROM_ecsw) and linear forms (LinearFormHYPERROM_ecsw)
Routines for extracting element stiffness matrices and load vectors in a reduced basis
Utilities for efficient Dirichlet condition handling and element clustering
Support for weights, parallelization, and full-order data reconstruction

Author: Suparno Bhattacharyya

Classes

Name	Description
LinearFormHYPERROM_ecsw	Reduced-order linear form for ECSW hyperreduction of load vectors.

LinearFormHYPERROM_ecsw

rom.ecsw.linear_form_hyperrom_ecsw.LinearFormHYPERROM_ecsw(
    form,
    elem_weight,
    ubasis,
    lob,
    free_dofs=None,
    mean=None,
    nthreads=0,
    dtype=np.float64,
)

Summary: Assembles the hyperreduced load vector for ROM use by projecting element-level load contributions onto the reduced test basis and accumulating them with their corresponding ECSW weights. Dirichlet boundary conditions are handled via a mapping from global DOFs to free (non-Dirichlet) DOFs—all assembly operations are performed exclusively on free DOFs. Elements are clustered by their free-DOF count to enable vectorized batch projections, following the same clustering strategy as BilinearFormHYPERROM_ecsw. The mean field, if provided, is subtracted during basis construction and reinserted during full-order reconstruction.

The assembly operation performed is:

[ r = {e {}} w_e V_L^|{} e|{} ]

where ( w_e ) is the ECSW weight for element ( e ), ( V_L ) is the left reduced test basis lob, and ( _e ) is the local load vector restricted to free DOFs.

Parameters

Name	Type	Description	Default
form	callable	Full-order linear form function evaluating local load contributions for each element.	required
elem_weight	scalar or ndarray	Element-wise ECSW weights; either a scalar applied uniformly to all elements or an array of length equal to the number of elements.	required
ubasis	Basis	FE basis object containing the full DOF count and element connectivity for the original mesh.	required
lob	ndarray, shape (N_free, r) or (N, r)	Left (test) reduced basis matrix; shape `(N_free, r)` when `free_dofs` is provided, or `(N, r)` otherwise, where ( r ) is the reduced dimension.	required
free_dofs	ndarray of int or None	Indices of global DOFs that are free (non-Dirichlet); if provided, all assembly operations are restricted to these DOFs.	`None`
mean	ndarray or None	Mean snapshot vector of length `N_full`; subtracted during basis construction and reinserted during full-order reconstruction.	`None`
nthreads	int	Number of threads for parallel element-wise load evaluation; `0` uses serial execution.	`0`
dtype	numpy.dtype	NumPy data type for assembled vectors and intermediate arrays.	`np.float64`

Methods

Name	Description
assemble_weighted_ecsw	Assemble the ECSW-weighted reduced load vector.
extract_element_vector_rom	Extract local element load vectors over the hyperreduced element set.

assemble_weighted_ecsw

rom.ecsw.linear_form_hyperrom_ecsw.LinearFormHYPERROM_ecsw.assemble_weighted_ecsw(**kwargs)

Summary: Assembles the global reduced load vector by iterating over active elements, scaling each element load vector by its ECSW weight, projecting the result onto the left reduced test basis restricted to free DOFs, and summing all contributions into the final vector of length ( r ). The method forwards additional keyword arguments (such as previous solution states or material parameters) to the underlying element extraction routine.

Parameters

Name	Type	Description	Default
**kwargs	dict	Additional keyword arguments forwarded to `extract_element_vector_rom`, such as previous solution states or material parameters.	`{}`

Returns

Name	Type	Description
f_reduced	ndarray, shape (r,)	Assembled ECSW-weighted reduced load vector, ready for use in ROM linear solves.

extract_element_vector_rom

rom.ecsw.linear_form_hyperrom_ecsw.LinearFormHYPERROM_ecsw.extract_element_vector_rom(
    basis,
    elem_indices=None,
    **kwargs,
)

Summary: Evaluates the full-order linear form over the specified elements and returns the resulting local load vectors as an array of shape (n_elem, Nbfun), where Nbfun is the number of local basis functions per element. The basis is restricted to the specified element subset via with_elements before evaluation. Supports both serial and parallel execution depending on nthreads.

Parameters

Name	Type	Description	Default
basis	Basis	Basis object restricted via `with_elements` for the active element subset.	required
elem_indices	ndarray of int or None	Subset of element indices to include in the extraction; forwarded to `with_elements`. Defaults to all active elements.	`None`
**kwargs	dict	Additional keyword arguments forwarded to the low-level form evaluation.	`{}`

Returns

Name	Type	Description
element_vectors	ndarray, shape (n_elem, Nbfun)	Local load vectors for each element in the restricted element set; `Nbfun` is the number of local basis functions per element.

Raises

Type	Condition
ValueError	Raised if `basis` is `None` or improperly configured.

Functions

Name	Description
with_elements	Return a restricted basis on a specified subset of element indices.

with_elements

rom.ecsw.linear_form_hyperrom_ecsw.with_elements(self, elements=None)

Summary: Returns a new basis instance restricted to the specified subset of element indices. Used internally to construct element-restricted basis objects before load vector extraction, and is also available for direct use when restricting any compatible basis to an arbitrary element subset.

Parameters

Name	Type	Description	Default
elements	array_like of int or None	Indices of elements to retain in the restricted basis; if `None`, the full basis is returned unchanged.	`None`

Returns

Name	Type	Description
restricted_basis	Basis	New basis instance covering only the specified elements, with mesh connectivity adjusted accordingly.