MOR-T L : A Novel Model Order Reduction Method for Parametrized Problems with Application to Seismic Wave Propagation

TL;DR

MOR-T_L introduces a Taylor-polynomial–based reduced-order framework that leverages Fréchet derivatives to construct robust ROM bases across parameter variations in parametrized PDEs. By solving a sequence of RHS problems for derivative orders and assembling an augmented snapshot then applying Gram-Schmidt-based QR reduction, the method achieves accurate solutions at new parameter values with substantially reduced online costs. Numerical experiments on a 2D acoustic/seismic wave problem show that MOR-T_L attains high accuracy (relative $L^2$ errors below 0.01%) and significant speedups over spectral-element methods, especially during line-search steps in full waveform inversion. The approach is demonstrated to be memory-efficient, parallelizable, and effective under both homogeneous and heterogeneous media, making it a promising tool for large-scale seismic data assimilation and inversion workflows.

Abstract

This paper presents an efficient strategy for constructing Reduced-Order Model (ROM) bases using Taylor polynomial expansions and Fr{é}chet derivatives with respect to model parameters. The proposed approach enables the construction of ROM bases with minimal additional computational cost. By exploiting Fr{é}chet derivatives -solution to the same problem with distinct right-hand sides -the method introduces a streamlined multiple-right-hand-side (RHS) strategy for ROM bases construction. This approach not only reduces overall computational expenses but also improves accuracy during model parameter updates. Numerical experiments on a two-dimensional wave problem demonstrate significant efficiency gains and enhanced performance, highlighting the potential of the proposed method to advance computational cost-effectiveness, particularly in seismic inversion applications.

Figures (14)

• Figure 1: Sample of basis functions: SVD (left) v.s. QR (right).
• Figure 2: Pressure Field Comparison at $\boldsymbol{x_r = (39.05, 15.42)}$ for a Homogeneous Medium. Pressure values over $[0,4×10^3]$ are compared between SEM (reference), MOR-T$_0$, and POD for a homogeneous medium with boundary condition $B_\theta= I$.
• Figure 3: Relative Errors Between MOR-T$_0$, POD, and SEM. Relative errors over the time interval $[0, 4\times10^3]$ are shown for MOR-T$_0$, and POD compared to SEM. The error is computed as the standard relative L$^2$-norm error across the computational domain $\Omega$ at each time step, for a homogeneous medium $\Omega$ with boundary condition $B_\theta = I$.
• Figure 4: Pressure Field Comparison at $\mathbf{x}_r = (39.05, 15.42)$ for a Homogeneous Medium. Pressure values over $[0,4×10^3]$ are compared between SEM (reference), MOR-T$_0$, and POD for a homogeneous medium with boundary condition $B_\theta= \sqrt{\theta}\frac{\partial }{\partial t} + \frac{\partial }{\partial \bf{n}}$.
• Figure 5: Relative Errors Between MOR-T$_0$, POD, and SEM. Relative errors over the time interval $[0, 4\times10^3]$ are shown for MOR-T$_0$ and POD compared to SEM. The error is computed as the standard relative L$^2$-norm error across the computational domain $\Omega$ at each time step, for a homogeneous medium $\Omega$ with boundary condition $B_\theta = \sqrt{\theta}\frac{\partial }{\partial t} + \frac{\partial }{\partial \bf{n}}$.

Theorems & Definitions (6)

• Remark 3.1
• Remark 3.2
• Remark 4.1
• Remark 4.2
• Remark 4.3
• Remark 4.4