---
title: DEIM Framework in Nonlinear Model Reduction
url: https://www.emergentmind.com/topics/deim-framework
type: topic
---

# DEIM Framework in Nonlinear Model Reduction

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 [1402.5992].
- Application in CUR matrix factorizations and large-scale low-rank data analysis [1407.5516].
- Integration with generalized Petrov-Galerkin projections (including weighted inner products and dictionary-based interpolation) [1704.06606].
- Robustification and improved error control through advanced selection schemes (QR with pivoting, randomized sampling, block DEIM) [1505.00370, 2208.02213].

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 [1211.2521].

Source: https://www.emergentmind.com/topics/deim-framework