---
title: Discrete Empirical Interpolation Method (DEIM)
url: https://www.emergentmind.com/topics/discrete-empirical-interpolation-method-deim
type: topic
---

# Discrete Empirical Interpolation Method (DEIM)

The Discrete Empirical Interpolation Method (DEIM) is a computational technique for the efficient approximation of nonlinear functions in high-dimensional settings, with rigorous error bounds and robust algorithmic formulations. DEIM enables the reduction of the computational complexity encountered in reduced-order models and matrix factorizations by enforcing interpolation constraints at a small set of adaptively chosen coordinates, leveraging proper orthogonal decomposition (POD) or other reduced bases. Recent developments, such as the Q-DEIM selection operator based on column-pivoted QR factorization, provide improved conditioning, tighter a priori error bounds, and practical randomized sampling variants for large-scale problems [1505.00370].

## 1. Classical Formulation and Purpose

Consider a nonlinear vector-valued function \( f:\mathcal{T}\rightarrow \mathbb{R}^n \) and an \( n\times m \) matrix \( U \) with orthonormal columns (\( m\ll n \)), spanning the subspace in which \( f \) is well-approximated. The DEIM approximation of \( f(\tau) \) is
\[
\widehat{f}(\tau) = U\,(P^T U)^{-1} P^T f(\tau)
\]
where \( P \in \mathbb{R}^{n\times m} \) is a selection matrix whose columns are standard basis vectors (\( e_{p_1},\dots,e_{p_m} \)) corresponding to interpolation indices \( p_1,\dots,p_m \) [1505.00370]. The method enforces \( P^T \widehat{f}(\tau) = P^T f(\tau) \), restricting \( \widehat{f}(\tau) \) to the subspace \(\mathrm{Range}(U)\).

This principle is particularly powerful in "hyper-reduction" for model order reduction of nonlinear dynamical systems, where DEIM allows for the approximation of nonlinear terms from evaluations at a small set of spatial locations without evaluating the full-dimensional function [1211.2521].

## 2. Interpolation Index Selection: Classical and Q-DEIM Approaches

The core performance of DEIM depends on the conditioning of \( P^T U \). The original (greedy) DEIM algorithm selects indices sequentially as follows:

1. \( p_1 = \arg\max_i |u_1(i)| \) (largest-magnitude entry of the first mode).
2. For \( j = 2, \dots, m \):
   - Solve \( (U(p_1:\,p_{j-1}, 1:j-1))^T z = U(p_1:\,p_{j-1}, j) \)
   - \( r_j = u_j - U(:,1:j-1)z \)
   - \( p_j = \arg\max_i |r_j(i)| \)

The resulting selection matrix \( P \) ensures \( P^T U \) is nonsingular.

Q-DEIM, an advancement over the classical approach, formulates the selection using a column-pivoted QR decomposition on \( U^T \):
\[
U^T \Pi = Q R
\]
where \( \Pi \) is a permutation matrix. The first \( m \) columns \( \Pi(1:m) \) specify the interpolation indices (\( p_1,\dots,p_m \)) [1505.00370]. Q-DEIM enjoys practical computational efficiency as the selection is accomplished in a single QR call using standard high-performance libraries, and it leads to near-optimal conditioning in practice, outperforming the original greedy approach for many applications.

## 3. Rigorous A Priori Error Bounds

The central DEIM error bound for any \( f \in \mathbb{C}^n \) is
\[
\| f - U (P^T U)^{-1} P^T f \|_2 \leq \| (P^T U)^{-1} \|_2 \cdot \| f - UU^T f \|_2
\]
where the first term is the conditioning amplification ("DEIM constant") and the second is the best approximation error onto \( \mathrm{Range}(U) \) [1505.00370].

For the original DEIM selection, the amplification factor can grow rapidly:
\[
\| (P^T U)^{-1} \|_2 \leq \sqrt{n} (1+\sqrt{2n})^{m-1}
\]

Q-DEIM and related pivoted QR-based methods achieve a much sharper bound:
\[
\| (P^T U)^{-1} \|_2 \leq \sqrt{n-m+1} \cdot g(m), \quad g(m) = \sqrt{\frac{4^m + 6m - 1}{3}}
\]
with further improvement possible using max-volume submatrix selection:
\[
\| (P^T U)^{-1} \|_2 \leq \sqrt{1+m(n-m)}
\]
In moderate dimensions, the actual growth of \( g(m) \) is mild, and QR-based choices are nearly optimal.

Notably, the selection via Q-DEIM is invariant to arbitrary unitary transformations of the basis (\( U \rightarrow U \Omega, \; \Omega \) unitary), making the procedure a true function of the subspace \( \mathrm{Range}(U) \), not the specific basis [1505.00370].

## 4. Randomized and Scalable DEIM

For problems with extremely large \( n \), restricted or randomized Q-DEIM (Q-DEIM\(^r\)) can be employed, wherein only a randomly sampled set of rows of \( U \) (with \( k \gtrsim m \)) are used for pivoted QR. Sampling utilizes selection probabilities
\[
p_i = \frac{\|U(i,:)\|_2^2}{\sum_{j=1}^n \|U(j,:)\|_2^2}
\]
and with \( k = O(m\log m) \) rows, the DEIM constant \( \|(P^T U)^{-1}\| \) remains uniformly bounded with high probability:
\[
\|(P^T U)^{-1}\|_2 \leq \frac{1}{1-\varepsilon}
\]
with probability at least \( 1-\delta \) for suitable \( k \) [1505.00370]. This enables near-optimal interpolation while only accessing a subset of the data, critical for large-scale settings.

## 5. Implementation and Computational Considerations

Q-DEIM is highly amenable to robust, efficient, and parallel software implementation using established linear algebra libraries (e.g., LAPACK's xGEQP3, ScaLAPACK's PxGEQPF). In MATLAB, the index selection is a single call:
```matlab
[~,~,P] = qr(U','vector');
idx = P(1:m);
```
Once the indices are known, the online phase (interpolation and projection to \( \mathrm{Range}(U) \)) involves solving one triangular system (\( O(m^3) \)) and applying row extractions and matrix-vector products (\( O(nm) \)).

## 6. Generalizations and Extensions

- **Interpolation Invariance**: DEIM's selection and its corresponding projection operator depend only on the subspace \( \mathcal{U} = \mathrm{Range}(U) \), not on the particular orthonormal basis.
- **Related Methods**: The Q-DEIM framework encompasses and improves upon deterministically-guided index selection methods such as strong Rank-Revealing QR (sRRQR), which further tighten error guarantees.
- **Robustness for Model Order Reduction**: DEIM, and particularly Q-DEIM, are widely adopted for efficient evaluation of nonlinear terms in reduced-order modeling frameworks (e.g., POD-ROM), enabling dramatic computational savings without substantial loss of accuracy.

## 7. Significance and Impact

The introduction of Q-DEIM represents a significant advance in the stability and efficiency of empirical interpolation for nonlinear model reduction. Its favorable error analysis, subspace invariance, and direct applicability to large-scale and parallel computation make it a foundational tool across scientific computing, data-driven modeling, and core low-rank approximation tasks [1505.00370]. Its randomized variants offer further scalability without sacrificing approximation accuracy, broadening its reach in the era of large-scale data and high-dimensional parameter spaces.

---

**Reference:**  
Drmač, Z., & Gugercin, S. "A New Selection Operator for the Discrete Empirical Interpolation Method -- improved a priori error bound and extensions" [1505.00370]

Source: https://www.emergentmind.com/topics/discrete-empirical-interpolation-method-deim