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A greedy non-intrusive reduced order model for shallow water equations

Sourav Dutta, Matthew W. Farthing, Emma Perracchione, Gaurav Savant, Mario Putti

TL;DR

This work tackles the computational burden of high-fidelity shallow water simulations by combining POD with RBF interpolation to form a non-intrusive reduced order model (NIROM). The reduced coefficients’ time derivatives are interpolated with kernel methods, and three greedy strategies (p-greedy, f-greedy, and the novel psr-greedy) optimize the RBF center distribution. The approach achieves accuracy comparable to NPOD while delivering substantial online speedups in coastal and riverine 2D SWE problems, with psr-greedy consistently yielding the best balance of accuracy and efficiency. The results demonstrate practical potential for fast-replay, many-query, and real-time hydrodynamic analyses, with open-source code planned for public release.

Abstract

In this work, we develop Non-Intrusive Reduced Order Models (NIROMs) that combine Proper Orthogonal Decomposition (POD) with a Radial Basis Function (RBF) interpolation method to construct efficient reduced order models for time-dependent problems arising in large scale environmental flow applications. The performance of the POD-RBF NIROM is compared with a traditional nonlinear POD (NPOD) model by evaluating the accuracy and robustness for test problems representative of riverine flows. Different greedy algorithms are studied in order to determine a near-optimal distribution of interpolation points for the RBF approximation. A new power-scaled residual greedy (psr-greedy) algorithm is proposed to address some of the primary drawbacks of the existing greedy approaches. The relative performances of these greedy algorithms are studied with numerical experiments using realistic two-dimensional (2D) shallow water flow applications involving coastal and riverine dynamics.

A greedy non-intrusive reduced order model for shallow water equations

TL;DR

This work tackles the computational burden of high-fidelity shallow water simulations by combining POD with RBF interpolation to form a non-intrusive reduced order model (NIROM). The reduced coefficients’ time derivatives are interpolated with kernel methods, and three greedy strategies (p-greedy, f-greedy, and the novel psr-greedy) optimize the RBF center distribution. The approach achieves accuracy comparable to NPOD while delivering substantial online speedups in coastal and riverine 2D SWE problems, with psr-greedy consistently yielding the best balance of accuracy and efficiency. The results demonstrate practical potential for fast-replay, many-query, and real-time hydrodynamic analyses, with open-source code planned for public release.

Abstract

In this work, we develop Non-Intrusive Reduced Order Models (NIROMs) that combine Proper Orthogonal Decomposition (POD) with a Radial Basis Function (RBF) interpolation method to construct efficient reduced order models for time-dependent problems arising in large scale environmental flow applications. The performance of the POD-RBF NIROM is compared with a traditional nonlinear POD (NPOD) model by evaluating the accuracy and robustness for test problems representative of riverine flows. Different greedy algorithms are studied in order to determine a near-optimal distribution of interpolation points for the RBF approximation. A new power-scaled residual greedy (psr-greedy) algorithm is proposed to address some of the primary drawbacks of the existing greedy approaches. The relative performances of these greedy algorithms are studied with numerical experiments using realistic two-dimensional (2D) shallow water flow applications involving coastal and riverine dynamics.

Paper Structure

This paper contains 25 sections, 2 theorems, 48 equations, 25 figures, 6 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. High-fidelity formulation for the shallow water equations
  3. Continuous formulation
  4. Stabilized finite element scheme for the semi-discrete weak formulation
  5. Time discretization
  6. Global basis reduced order model
  7. Proper Orthogonal Decomposition
  8. RBF-POD reduced order model formulation
  9. Remarks on reproducing kernel Hilbert spaces
  10. RBF approximation of the reduced coefficients
  11. Optimal distribution of RBF interpolation points
  12. p-greedy algorithm
  13. f-greedy algorithm
  14. psr-greedy algorithm
  15. Combined RBF-POD reduced order model
  16. ...and 10 more sections

Key Result

Lemma Appendix A.1

($V$-orthonormal basis property). The functions $N_1,\ldots,N_r$ are, by construction, an orthonormal basis for the span of translates $\Phi(\mathbf{x}_1,\cdot), \ldots, \Phi(\mathbf{x}_r,\cdot)$ such that

Figures (25)

  • Figure 1: San Diego Bay
  • Figure 2: Solutions for the x-velocity (left column) and the corresponding errors (right column) with respect to the true solution obtained using the NIROM (top row) and the NPOD (bottom row) model at $t=2.55\times10^4$.
  • Figure 3: Error analysis for the San Diego Bay example
  • Figure 4: Kissimmee River
  • Figure 5: Semi-logarithmic plot of the singular values for the Kissimmee River example
  • ...and 20 more figures

Theorems & Definitions (3)

  • Definition Appendix A.1
  • Lemma Appendix A.1
  • Lemma Appendix A.2