A reduced basis method for parabolic PDEs based on a space-time least squares formulation

TL;DR

The paper addresses efficient, certified reduced-order modeling for parametrized parabolic PDEs by leveraging a space-time least-squares variational formulation that yields a symmetric, uniformly coercive bilinear form on $W(0,T)$. It adapts the POD-Greedy reduced basis method to this space-time setting, enabling an offline-online decomposition and space-time POD to construct low-dimensional spaces with provable error bounds. Absolute and relative a posteriori estimators are developed in the discrete $W(0,T)$ norm, with both exact-residual and offline-online computable variants demonstrated via two numerical examples, including a 3D minimal-regularity case. The approach achieves exponential convergence in the reduced-basis dimension and offers practical computational savings for parametric parabolic problems, making it suitable for rapid, reliable simulations in engineering and physics applications.

Abstract

In this work, we present a POD-greedy reduced basis method for parabolic partial differential equations (PDEs), based on the least squares space-time formulation proposed in [Hinze, Kahle, Stahl, A least-squares space-time approach for parabolic equations, 2023, arXiv:2305.03402] that assumes only minimal regularity. We extend this approach to the parameter-dependent case. The corresponding variational formulation then is based on a parameter-dependent, symmetric, uniformly coercive, and continuous bilinear form. We apply the reduced basis method to this formulation, following the well-developed techniques for parameterized coercive problems, as seen e.g. in reduced basis methods for parameterized elliptic PDEs. We present an offline-online decomposition and provide certification with absolute and relative error bounds. The performance of the method is demonstrated using selected numerical examples.

Figures (6)

• Figure 1: Sketch of $\Omega$ and its subdomains and boundaries in the thermal block example.
• Figure 2: Left: Absolute error of the reduced basis space $W_\texttt{L}$, as evaluated on a validation set, plotted against the dimension $\texttt{L}$ of the reduced basis space. As in a classical reduced basis approach, we observe exponential decay. Right: Effectivities of the error estimators $\eta_\star^\text{abs}$ (from Theorem \ref{['thm:RBerror_ex']}) and $\eta_\mathfrak{c}^\text{abs}$ (from Theorem \ref{['thm:RB_error_est_cc_sq']}), computed on the validation set, too. The estimator $\eta_\star^\text{abs}$, which evaluates the true residual, produces the most precise results due to effectivities close to 1, whereas the estimator $\eta_\mathfrak{c}^\text{abs}$ uses an estimation of the residual.
• Figure 3: Sketch of the domain $\Omega$ and their subdomains $\Omega_0$, $\Omega_1$, $\Omega_2$, $\Omega_3$ in the second example.
• Figure 4: Comparison of the error estimators $\eta_\mathfrak{c}^\text{abs}$ and $\eta_\star^\text{abs}$ (left) resp. $\eta_\mathfrak{c}^\text{rel}$ and $\eta_\star^\text{rel}$ (right) with the real errors $\epsilon^\text{abs}$ resp. $\epsilon^\text{rel}$ in dependence of the number $\texttt{L}$ of used basis functions for Example 2. The parameter is freshly selected by $\eta_\mathfrak{c}^\text{abs}$ resp. $\eta_\mathfrak{c}^\text{rel}$, and is different in every step. We plot the error for the parameter that maximizes the latter error estimators. The region where the assumption $\eta_\mathfrak{c}^\text{abs} \leq 1$ resp. $\eta_\star^\text{rel} \leq 1$ from \ref{['eq:RB_first_rel_est']} is not satisfied is plotted in gray.
• Figure 5: The first three basis functions $\xi_1$, $\xi_2$, $\xi_3$ generated by Algorithm \ref{['greedy']} for the second example. The domain is cut at $z = 0.1$ and the functions are plotted at $t = 0.4$. We choose a different color scaling for each function in order to highlight the differences and illustrate the most important information. White means small values, black means large values.

Theorems & Definitions (17)

• Remark 2.1
• Definition 2.2: Continuous problem
• Remark 3.1
• Definition 3.2: Energy inner product and norm
• Definition 4.1: Reduced problem
• Lemma 4.2
• proof
• Remark 4.3
• Remark 4.4: Formulation of space-time POD
• Theorem 5.1