Discrete Empirical Interpolation Method (DEIM)

• DEIM is a computational method that approximates nonlinear functions in high-dimensional spaces by selecting key interpolation indices from a reduced basis.
• It leverages techniques like proper orthogonal decomposition and Q-DEIM to ensure near-optimal error bounds and efficient performance in complex model reduction tasks.
• Randomized and scalable variants of DEIM enable robust and efficient implementation for large-scale nonlinear dynamical systems and high-dimensional data problems.

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 (Drmac et al., 2015).

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 $p_1,\dots,p_m$ (Drmac et al., 2015). 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 (Stefanescu et al., 2012).

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 ($p_1,\dots,p_m$) (Drmac et al., 2015). 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 $\mathrm{Range}(U)$ (Drmac et al., 2015).

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 (Drmac et al., 2015).

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 $k$ (Drmac et al., 2015). 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:

[~,~,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 (Drmac et al., 2015). 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.

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Reference:

Drmač, Z., & Gugercin, S. "A New Selection Operator for the Discrete Empirical Interpolation Method -- improved a priori error bound and extensions" (Drmac et al., 2015)