DEIM Framework in Nonlinear Model Reduction

• DEIM is a numerical method for nonlinear model reduction that approximates nonlinear functions using a low-dimensional interpolation subspace selected via greedy algorithms.
• It decouples the evaluation of costly nonlinear terms from the full high-dimensional system by projecting onto dominant POD modes and using a selection matrix.
• Empirical results demonstrate significant CPU time reductions (up to 73× speedups) with controlled error, making DEIM effective for large-scale geophysical and fluid models.

The Discrete Empirical Interpolation Method (DEIM) is a widely adopted framework for nonlinear model order reduction, originally developed to overcome the computational bottleneck in reduced-order modeling of high-dimensional nonlinear dynamical systems. DEIM enables efficient online evaluation of nonlinear terms by projecting them onto a carefully selected low-dimensional interpolation subspace, decoupling the computational cost from the full system dimension. The method has seen extensive application in computational fluid dynamics, geophysical modeling, data assimilation, and matrix/tensor approximations, and it has been subject to invigorated theoretical and algorithmic development since its introduction.

1. Fundamental Principles and Mathematical Formulation

The DEIM framework arises in the context of projection-based reduction of large-scale nonlinear systems, such as those arising from the discretization of the shallow water equations (SWE) or general parametric PDEs. In classical Proper Orthogonal Decomposition (POD)-Galerkin methods, the solution is approximated as $w(t) \approx U \widetilde{w}(t)$ with $U \in \mathbb{R}^{n \times k}$, $k \ll n$. After Galerkin projection, nonlinear terms such as $U^T f(U\widetilde{w})$ still require evaluating $f$ at the full system dimension $n$.

DEIM approximates a nonlinear function $f(\tau)\in\mathbb{R}^{n}$ by interpolating it in the subspace spanned by the dominant $m$-dimensional POD modes $U_f$ of nonlinear "snapshots": $f(\tau) \approx U_f (P^T U_f)^{-1} P^T f(\tau)$ where $P \in \mathbb{R}^{n \times m}$ is a selection matrix extracting the $m$ interpolation indices chosen by a greedy or pivoted QR strategy. This reduces the online computational cost to evaluating $f$ at $m \ll n$ entries, subsequently reconstructing the full nonlinear term via projection.

2. DEIM in Nonlinear POD-Galerkin Model Reduction

In complex physical models, e.g., the shallow water equations discretized with alternating direction implicit (ADI) or explicit Euler schemes, the primary challenge post-POD reduction is the nonlinear term evaluation. The DEIM framework is applied by constructing a separate POD basis for the nonlinear snapshots (e.g., for terms $F_{ij}$ in the SWE), and expressing the reduced nonlinear term as

$$U^T F_{11}(U\widetilde{u}, \widetilde{\varphi}) \approx (U^T U^{F_{11}}) (P_{F_{11}}^T U^{F_{11}})^{-1} P_{F_{11}}^T F_{11}(U\widetilde{u}, \widetilde{\varphi})$$

where $U^{F_{11}}$ and $P_{F_{11}}$ are the nonlinear POD basis and interpolation matrix for $F_{11}$. All dominant nonlinear terms are treated analogously.

The framework leads to substantial decreases in computational complexity:

• Without DEIM: Nonlinear evaluation is $O(nk)$, where $n$ is the number of spatial grid points.
• With DEIM: Nonlinear evaluation is $O(mk)$, independent of $n$ ($m$ is the number of DEIM points, typically $m=50$–$90$ for practical accuracy).

3. Implementation Algorithms and Workflow

The standard DEIM workflow involves:

1. Offline Stage:
   • Simulate the full-order model; collect state and nonlinear term snapshots.
   • Compute POD bases $U$ for the state and $U_f$ for each nonlinear function.
   • Select interpolation indices via a greedy algorithm or QR with pivoting, forming selection matrices $P, P_{f}$.
2. Online Stage:
   • For each time step, compute the nonlinear term only at $m$ interpolation points: evaluate $P^T f(\cdot)$.
   • Project and reconstruct the nonlinear term via DEIM: $f(\tau) \approx U_f (P^T U_f)^{-1} P^T f(\tau)$.
   • Assemble and solve the reduced-order system.

The key DEIM formula: $f(\tau) \approx U_f (P^T U_f)^{-1} P^T f(\tau)$ In projection-based ROMs for systems such as the SWE, this formula is embedded within the Galerkin or Petrov-Galerkin projection structure.

4. Performance Metrics and Empirical Results

Empirical evaluation of DEIM in large-scale geophysical models has demonstrated:

• For spatial discretizations with $n > 10,000$ and $m=90$, the CPU time for a POD/DEIM-ADI (implicit) SWE scheme is reduced by a factor of ~10 relative to non-interpolated POD ROMs.
• For explicit schemes, speedups reach a factor of ~15.
• On very fine spatial grids ($n > 66,000$), speedups up to 73× (implicit) and 69× (explicit) are observed.
• When $m$ is increased to $50$ or above, the approximation errors between POD/DEIM and full POD ROMs equilibrate (same order of magnitude in average relative and RMSE errors).

This establishes DEIM as the key enabler for practical nonlinear reduced-order modeling at high spatial resolution.

5. Trade-Offs: Selection of DEIM Points and Accuracy

The selection of $m$, the number of interpolation points, is critical:

• Low $m$ improves speed but may introduce larger errors if the nonlinear behavior is not adequately captured.
• Numerical results show that for $m \gtrsim 50$, accuracy plateaus; further increases in $m$ offer diminishing returns.

Physical interpretation and model size dictate the lower bound on $m$. For high-resolution models where the dimension $n$ exceeds $10^4$, the computational gain from decoupling the online cost from $n$ is especially pronounced.

6. Generalizations and Broader Impact

The DEIM framework has influenced a broad class of nonlinear model reduction and matrix/tensor approximation schemes:

• Extension to data assimilation, adjoint models, and optimization, where the cost of repeatedly evaluating nonlinearities is a major bottleneck (Ştefănescu et al., 2014).
• Application in CUR matrix factorizations and large-scale low-rank data analysis (Sorensen et al., 2014).
• Integration with generalized Petrov-Galerkin projections (including weighted inner products and dictionary-based interpolation) (Drmač et al., 2017).
• Robustification and improved error control through advanced selection schemes (QR with pivoting, randomized sampling, block DEIM) (Drmac et al., 2015, Gidisu et al., 2022).

DEIM has become foundational in enabling efficient real-time and many-query simulations (e.g., uncertainty quantification, inverse problems) where direct evaluation of the full-order nonlinearities is prohibitive.

7. Summary Table: DEIM Parameters and Key Effects

Parameter	Typical Range	Effect on Model
Grid size ($n$)	$10^3$–$10^5$	Sets FOM cost
POD basis size ($k$)	$10$–$100$	Reduced state
DEIM points ($m$)	$50$–$90$	Controls nonlinear cost/accuracy
CPU time speedup	10–73×	For $n\sim 10^4$–$6\times10^4$
Error w.r.t. POD ROM	$< 1\%$	For $m \gtrsim 50$

For models with large $n$, the computational efficiency of DEIM scales independently of $n$ once $m$ is chosen appropriately. The balance between accuracy and speed is controlled by $m$, firmly grounded in the singular value decay of nonlinear snapshots.

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By integrating the DEIM framework with POD-based reduction, the computational cost of evaluating nonlinearities in high-dimensional geophysical models is reduced to depend only on a modest number of interpolation points, yielding robust and efficient reduced-order models with negligible loss of fidelity. This integration is a core innovation enabling practical nonlinear ROMs for large-scale systems (Stefanescu et al., 2012).