rom.ecm.bilinear_form_hyperrom_ecm

Purpose: Implements ECM-based hyperreduction for finite element bilinear forms, enabling reduced-order stiffness matrix assembly at a fraction of the cost of full-domain integration. Unlike element-wise hyperreduction strategies such as ECSW, ECM operates at the Gauss-point level within each element, providing finer-grained control over the quadrature approximation and enabling higher efficiency and accuracy.

Summary: Provides the BilinearFormHYPERROM_ecm class, which assembles the hyperreduced bilinear form by restricting integration to ECM-selected Gauss points with their associated weights. The assembly pipeline restricts the active mesh to elements with nonzero ECM weights, groups elements by free local DOF count for memory-efficient processing, extracts per-Gauss-point contributions, projects onto the reduced basis, and accumulates into the final reduced-order operator matrix.

Assumption

ECM weights stored in gauss_weight are expected to multiply the raw per-Gauss-point contributions produced by scikit-fem:

[ () dx ]

If weights were obtained from an external source and represent absolute cubature values rather than multiplicative scaling factors, they must be converted first using the helper utility in skrom.rom.ecm.helpers before being passed to this class.

Classes

Name Description
BilinearFormHYPERROM_ecm Hyperreduced bilinear form assembled from ECM-selected Gauss points.

BilinearFormHYPERROM_ecm

rom.ecm.bilinear_form_hyperrom_ecm.BilinearFormHYPERROM_ecm(
    form,
    gauss_weight,
    ubasis,
    lob,
    rob,
    vbasis=None,
    free_dofs=None,
    mean=None,
    nthreads=0,
    dtype=np.float64,
)

Summary: Assembles the hyperreduced bilinear form for ROM use by integrating only over those Gauss points and elements selected by the ECM procedure. The assembly proceeds through five stages:

  1. Restrict: Identify elements with nonzero ECM weights and construct the active hyperreduced mesh.
  2. Cluster: Group active elements by their number of free local DOFs to optimize memory layout and vectorized processing.
  3. Extract: Retrieve local element matrix contributions at each relevant Gauss point without pre-summing across quadrature points, preserving the ability to apply ECM weights individually.
  4. Project: Project each weighted Gauss-point contribution onto the left and right reduced bases.
  5. Accumulate: Sum all projected contributions into the final reduced-order operator matrix ( K_r ^{r r} ).

This Gauss-point-level approach provides more precise hyperreduction than element-wise methods, particularly when the parametric variation is concentrated in specific regions or quadrature points of the mesh.

Methods

Name Description
assemble_weighted_ecm Assemble the reduced bilinear form matrix using ECM Gauss-point weights.
extract_element_matrices_qp_rom Extract per-Gauss-point local element matrices for ECM-weighted assembly.

assemble_weighted_ecm

rom.ecm.bilinear_form_hyperrom_ecm.BilinearFormHYPERROM_ecm.assemble_weighted_ecm(**kwargs)

Summary: Constructs the ROM’s reduced stiffness matrix by applying ECM weights at the individual Gauss-point level. Each active Gauss-point contribution is scaled by its corresponding ECM weight prior to basis projection. Because contributions are not pre-summed across quadrature points, ECM weights can be applied directly to the raw per-point integrands—preserving the correctness of the hyperreduced quadrature rule. The weighted contributions are then projected onto the reduced bases and accumulated into the final reduced operator.

The mathematical operation performed for the assembled reduced matrix is:

[ K_r = {e {}} {q} w{e,q}^{} V_L^K_{e,q} V_R ]

where ( w_{e,q}^{} ) is the ECM weight for Gauss point ( q ) in element ( e ), ( K_{e,q} ) is the raw per-point contribution, and ( V_L, V_R ) are the left and right reduced bases.

Parameters
Name Type Description Default
**kwargs dict Additional keyword arguments forwarded to element-level assembly. {}
Returns
Name Type Description
K_reduced ndarray, shape (r, r) Assembled hyperreduced bilinear form matrix ready for use in ROM linear solves.

extract_element_matrices_qp_rom

rom.ecm.bilinear_form_hyperrom_ecm.BilinearFormHYPERROM_ecm.extract_element_matrices_qp_rom(
    ubasis,
    vbasis=None,
    elem_indices=None,
    **kwargs,
)

Summary: Evaluates the bilinear form over the active elements and returns the local stiffness contributions resolved per Gauss point, without summing across quadrature points. This is the key distinction from standard element matrix extraction: by retaining the per-point granularity, ECM weights can be applied individually at each quadrature point before the summation step, enabling precise control over the reduced quadrature rule.

Each returned entry element_matrices_q[e, :, :, q] represents the contribution of the ( q )-th Gauss point in element ( e ) to the local (test × trial) matrix, already multiplied by basis.dx but not yet summed over ( q ). ECM weights should be applied to this axis before accumulation.

Parameters
Name Type Description Default
ubasis Basis Trial FE basis containing mesh, quadrature rule, and element connectivity for the active hyperreduced mesh. required
vbasis Basis or None Test FE basis; defaults to ubasis if not provided. None
elem_indices array_like of int or None Specific element indices to process; defaults to all active elements in the hyperreduced mesh. None
**kwargs dict Additional keyword arguments forwarded to the bilinear form evaluation. {}
Returns
Name Type Description
element_matrices_q ndarray, shape (n_elem, n_loc_test, n_loc_trial, n_q) Per-Gauss-point local element matrices. The axis ordering is (element, test DOF, trial DOF, quadrature point). Multiply ECM weights along the n_q axis before summing to obtain ECM-weighted element matrices.

Notes

Use BilinearFormHYPERROM_ecm when Gauss-point-level precision in hyperreduction is required—particularly when the parametric variation in your bilinear form is concentrated in specific spatial regions or quadrature points. Always verify that your ECM weights are compatible with the raw per-Gauss-point convention expected by this class; weights representing fully integrated element contributions must be converted using skrom.rom.ecm.helpers before use.