A New Selection Operator for the Discrete Empirical Interpolation Method -- improved a priori error bound and extensions (1505.00370v2)

Abstract: This paper introduces a new framework for constructing the Discrete Empirical Interpolation Method DEIM projection operator. The interpolation node selection procedure is formulated using the QR factorization with column pivoting, and it enjoys a sharper error bound for the DEIM projection error. Furthermore, for a subspace $\mathcal{U}$ given as the range of an orthonormal $U$, the DEIM projection does not change if $U$ is replaced by $U \Omega$ with arbitrary unitary matrix $\Omega$. In a large-scale setting, the new approach allows modifications that use only randomly sampled rows of $U$, but with the potential of producing good approximations with corresponding probabilistic error bounds. Another salient feature of the new framework is that robust and efficient software implementation is easily developed, based on readily available high performance linear algebra packages.

• The paper proposes a new DEIM selection operator based on QR factorization with column pivoting, yielding improved theoretical error bounds compared to the standard method.
• A key advantage is that the proposed selection operator maintains invariance under orthonormal transformations, ensuring robustness across different basis choices.
• The method offers computational advantages for large-scale systems using sampled rows and is efficiently implemented with standard libraries.

A New Selection Operator for the Discrete Empirical Interpolation Method

The paper under review presents an innovative approach to constructing the Discrete Empirical Interpolation Method (DEIM) projection operator, employing QR factorization with column pivoting to achieve improved a priori error bounds and flexibility in nonlinear model reduction. This approach addresses the challenge of efficiently approximating nonlinear dynamical systems, where the scale and complexity of state-space dimensions necessitate reduced-order models that maintain fidelity while being computationally tractable.

Key Contributions

1. QR-Based Selection Framework: The paper introduces a selection operator for the DEIM projection matrix using QR factorization with column pivoting. This enhances the theoretical error bounds compared to the traditional DEIM selection, indicating a tighter and potentially more reliable error characterization in practice.
2. Basis Invariance: A critical feature of the proposed selection operator is that it maintains invariance under orthonormal transformations. This implies that the DEIM projection's properties remain unaffected by changes in the choice of orthonormal bases, which is significant for ensuring the robustness and general applicability of the method.
3. Reduced Computational Complexity in Large-Scale Systems: In large-dimensional settings, the proposed approach can be effectively adapted by employing only randomly sampled rows of the data matrix. This yields a computational advantage while still yielding sufficiently accurate approximations and associated probabilistic error bounds.
4. Algorithmic and Software Implementation: The technique is amenable to efficient software implementation using high-performance linear algebra libraries such as LAPACK and ScaLAPACK. The emphasis on leveraging BLAS level-3 operations suggests the potential for significant performance gains in large-scale applications.

Numerical Results and Analysis

The paper provides several numerical experiments that demonstrate the efficacy of the QR-based DEIM selection strategy:

• FitzHugh-Nagumo System: The reduced-order models generated using the new selection operator outperform traditional DEIM in terms of reconstruction accuracy, with documented errors suggesting more precise trajectory approximations within the specified reduction dimensions.
• Nonlinear RC Model: Similar improvements in error performance are recorded. The new strategy achieves notably lower reconstruction errors, indicating better nonlinear snapshot approximation quality compared to conventional DEIM methods.

Implications and Future Directions

The proposed DEIM framework has direct implications for the field of nonlinear model reduction, where accurate yet computationally efficient approximations are crucial. The refined error bounds and efficient implementation strategies put forward new pathways for effective real-world applications, particularly in scenarios where large-scale dynamical systems are prevalent.

Potential future developments could explore combining this method with other data-driven model reduction techniques to enhance robustness and extend applicability. Additionally, the incorporation of randomized sampling techniques could be further analyzed to refine computational savings and error characteristics. These extensions could broaden the horizon for DEIM applications in stochastic and high-dimensional uncertainty quantification problems.

In conclusion, the QR-based DEIM framework proposed in this paper represents a significant step forward in nonlinear model reduction methodologies. It offers both theoretical advancements and practical solutions for handling large-scale dynamical systems, serving as a valuable tool for researchers and practitioners aiming for high-fidelity reduced order models in complex systems analysis.