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

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.

💡 Research Summary

This paper introduces a certified reduced‑basis (RB) method for parameter‑dependent parabolic partial differential equations (PDEs) built upon a space‑time least‑squares formulation. The authors extend the recent least‑squares space‑time approach of Hinze, Kahle, and Stahl (2023) to the parametric setting, thereby obtaining a variational problem whose bilinear form is symmetric, uniformly continuous, and uniformly coercive for all parameters µ in a compact set P.

The underlying functional setting uses a Gelfand triple (V, H, V*). For each µ, a symmetric inner product a(µ;·,·) on V defines the Riesz operator A(µ):V→V* and its inverse R(µ). The parabolic equation y_t + A(µ)y = f(µ), y(0)=y₀(µ) is recast as a least‑squares minimization of the residual in L²(0,T;V*) together with the initial‑condition mismatch. The first‑order optimality condition yields a space‑time variational formulation
 b(µ; y, w) = ℓ(µ; w) ∀ w ∈ W(0,T),
where W(0,T) = {v ∈ L²(0,T;V) | v_t ∈ L²(0,T;V*)} and the bilinear form b(µ;·,·) consists of three terms: the L²‑inner product of time derivatives (via the Riesz lift), the spatial inner product, and a terminal‑time contribution. By construction b(µ;·,·) is symmetric, continuous, and coercive with constants independent of µ, guaranteeing well‑posedness via Lax–Milgram.

For numerical treatment the authors reformulate the problem as an equivalent saddle‑point system (P_µ) involving an auxiliary variable p that approximates the Riesz lift of the time derivative. This saddle‑point structure enables a compact block‑matrix representation and facilitates the construction of a discrete Riesz lift R_d(µ) on a finite‑dimensional test space Q_d.

The high‑fidelity discretization employs tensor‑product space‑time finite elements: piecewise linear continuous elements in time (space K_M) together with piecewise constant elements (space J_P) for the test space, and a standard spatial basis {φ_n} for V. The resulting discrete spaces are
 W_d = K_M ⊗ V_N, Q_d = J_P ⊗ V_N.
Galerkin projection of (P_µ) onto (W_d, Q_d) yields a linear system S_d(µ)