rom.deim.bilinear_form_hyperrom_deim

Purpose: Implements DEIM-based hyperreduction for finite element bilinear forms—enabling reduced-order matrix assembly that is 100–1000× faster than classical projection approaches. By assembling only a small, carefully selected subset of elements and using DEIM interpolation to reconstruct the full reduced operator, this module enables real-time and large-scale nonlinear parametric ROM simulations with dramatic savings in both computation time and memory.

Summary: Provides the BilinearFormHYPERROM_deim class, which combines element selection (driven by DEIM sampling points) with sparse assembly and reduced-basis projection. Assembly cost scales with the number of selected elements rather than the total mesh size—making this approach particularly effective when the operator is approximately low-rank with respect to parameters, or when most of the mesh contributes negligibly to the parametric variation.

Author: Suparno Bhattacharyya

Classes

Name Description
BilinearFormHYPERROM_deim DEIM-based hyperreduced bilinear form assembly for fast ROM operator construction.

BilinearFormHYPERROM_deim

rom.deim.bilinear_form_hyperrom_deim.BilinearFormHYPERROM_deim(
    form,
    elem_weight,
    ubasis,
    lob,
    rob,
    sampled_rows,
    deim_mat,
    vbasis=None,
    free_dofs=None,
    mean=None,
    nthreads=0,
    dtype=np.float64,
)

Summary: Constructs a hyperreduced bilinear form operator by restricting element assembly to only the elements connected to DEIM-selected degrees of freedom. The reduced operator is reconstructed from these sparse samples using the DEIM interpolation matrix and projected onto the reduced basis—producing the final ( r r ) system matrix at a fraction of the cost of full assembly.

The assembly process follows four key stages:

  1. Element Selection: Only elements linked to DEIM-identified critical DOFs are assembled, reducing the active set from potentially thousands of elements to just a few dozen.
  2. Sparse Assembly: Local element matrices for the active subset are assembled into a sparse global matrix using COO-to-CSR conversion, minimizing memory usage.
  3. DEIM Reconstruction: The DEIM interpolation matrix is applied to the sampled rows of the sparse global matrix to reconstruct the full reduced operator.
  4. Basis Projection: The reconstructed operator is projected onto the right reduced basis to yield the final ROM system matrix.

This approach is most advantageous when operators are approximately low-rank in the parameter space, or when real-time ROM evaluation is required for large-scale nonlinear problems.

Parameters

Name Type Description Default
form callable Full-order bilinear form function; must accept basis objects and return element-wise matrices. required
elem_weight array_like, shape (n_elements,) Element selection vector from DEIM; entries are 1 for selected elements and 0 for skipped ones. required
ubasis Basis Full FE basis for trial functions, containing mesh, quadrature, and connectivity information. required
lob ndarray, shape (n_free, r) Left (test) reduced basis; included for interface compatibility. required
rob ndarray, shape (n_free, r) Right (trial) reduced basis used for projecting the reconstructed operator. required
sampled_rows array_like of int, shape (n_samp,) Global DOF indices selected by DEIM as the most informative interpolation points. required
deim_mat ndarray, shape (r, n_samp) DEIM interpolation matrix used to reconstruct the full reduced operator from sampled values. required
vbasis Basis or None FE basis for test functions; defaults to ubasis if not provided. None
free_dofs ndarray of int or None Indices of non-Dirichlet (free) DOFs for boundary condition handling. None
mean ndarray or None Mean snapshot field subtracted prior to POD/SVD basis construction. None
nthreads int Number of threads for parallel element assembly; 0 uses single-core execution. 0
dtype numpy.dtype Numeric precision for assembly operations. np.float64

Attributes

Name Type Description
weight ndarray, shape (n_elements,) Copy of the element selection weight vector identifying active elements.
nonzero_elements ndarray of int Indices of elements that are active (selected for assembly by DEIM).
ubasis_rom Basis FE basis restricted to only the active elements of the hyperreduced mesh.
sampled_rows ndarray of int, shape (n_samp,) Global DOF indices at which DEIM interpolation is performed.
n_samp int Number of DEIM sampling points (length of sampled_rows).
deim_mat ndarray, shape (r, n_samp) DEIM interpolation matrix for operator reconstruction.
edofs ndarray, shape (n_active_elements, n_local_dofs) Local-to-global DOF connectivity table for the active element subset.
n_elems int Number of active elements in the hyperreduced mesh.
n_loc int Number of local DOFs per element.
n_dofs int Total number of global DOFs in the restricted mesh.
rows, cols ndarray Row and column index arrays for local-to-global sparse matrix assembly.
row_flat, col_flat ndarray Flattened index arrays for fast COO sparse matrix construction.

Methods

Name Description
assemble_deim Construct the DEIM hyperreduced bilinear form matrix ready for ROM solve.
deim_elem_assembly Assemble a sparse global matrix restricted to the DEIM-selected elements.
extract_element_matrices_rom Compute local stiffness matrices for the active (DEIM-selected) elements only.

assemble_deim

rom.deim.bilinear_form_hyperrom_deim.BilinearFormHYPERROM_deim.assemble_deim(**kwargs)

Summary: Orchestrates the full DEIM hyperreduction pipeline to produce the final reduced-order operator matrix. Executes the following four steps in sequence:

  1. Calls deim_elem_assembly to build a sparse global matrix from only the active elements.
  2. Extracts the rows corresponding to DEIM sampling points (sampled_rows) from the sparse matrix.
  3. Applies the DEIM interpolation matrix to reconstruct the full reduced operator from the sampled values.
  4. Projects the result onto the right reduced basis rob to obtain the ROM-ready matrix.

The assembled operator follows:

[ A_{} = D A_{}[,, :] V ]

where ( D ) is the DEIM matrix, ( A_{} ) is the sparse global matrix from selected elements, and ( V ) is the right reduced basis rob.

Parameters
Name Type Description Default
**kwargs dict Additional keyword arguments forwarded to element assembly. {}
Returns
Name Type Description
A_reduced ndarray, shape (r, r) Hyperreduced bilinear form matrix assembled via DEIM, ready for ROM linear solve.

deim_elem_assembly

rom.deim.bilinear_form_hyperrom_deim.BilinearFormHYPERROM_deim.deim_elem_assembly(**kwargs)

Summary: Assembles a sparse global stiffness matrix by accumulating contributions from only the DEIM-selected elements. Local element matrices are obtained via extract_element_matrices_rom and inserted into a global COO sparse matrix, which is then converted to CSR format for efficient downstream row extraction. Zero entries are optionally skipped to further reduce memory usage. The resulting matrix is equivalent to what would be obtained from a full assembly if only the active elements were present in the mesh.

Parameters
Name Type Description Default
**kwargs dict Additional keyword arguments forwarded to element matrix extraction. {}
Returns
Name Type Description
K scipy.sparse.csr_matrix, shape (n_dofs, n_dofs) Sparse global stiffness matrix assembled over the DEIM-selected element subset.

extract_element_matrices_rom

rom.deim.bilinear_form_hyperrom_deim.BilinearFormHYPERROM_deim.extract_element_matrices_rom(
    ubasis,
    vbasis=None,
    elem_indices=None,
    **kwargs,
)

Summary: Evaluates the bilinear form element by element and returns the local stiffness matrices for the active (DEIM-selected) elements only. Supports both serial and parallel execution. For each selected element ( e ), the local matrix entry is:

[ K_e[i,j] = _{_e} _i (_j) , d ]

where ( _i ) and ( _j ) are local basis functions and ( ) denotes the bilinear operator. Only the elements identified by elem_indices (defaulting to all active elements) are integrated, significantly reducing quadrature cost relative to full-mesh evaluation.

Parameters
Name Type Description Default
ubasis Basis Trial FE basis with correct mesh, quadrature rule, and element connectivity. required
vbasis Basis or None Test FE basis; defaults to ubasis if not provided. None
elem_indices array of int or None Indices of elements to integrate; defaults to all active (DEIM-selected) elements. None
**kwargs dict Additional keyword arguments forwarded to the form evaluation. {}
Returns
Name Type Description
element_matrices ndarray, shape (n_elements, n_local_dofs, n_local_dofs) Stack of local stiffness matrices for each selected element.
Raises
Type Condition
ValueError Raised if trial and test bases are incompatible due to a quadrature rule mismatch.

Notes

BilinearFormHYPERROM_deim is the recommended class for DEIM-accelerated ROM operator assembly when the bilinear form varies with parameters. It reduces assembly cost from ((N_{})) to ((N_{})), where ( N_{} N_{} ) in typical applications. It is most effective for parametric nonlinear problems where the operator cannot be precomputed offline in its entirety.