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Space-time POD for linear parabolic evolution problems: Singular value based a-priori error estimation
Carmen Gräßle1, Jan Heiland2, Jannis Marquardt1
1: Institute for Partial Differential Equations, TU Braunschweig, Germany; 2: Department of Mathematics and Natural Sciences, Institute of Mathematics, TU Ilmenau, Germany
In this presentation, we recall the space-time POD method for linear evolution problems such as \[ \left\{ \begin{array}{rll} x_t + \mathcal{A} x &=\; f &\text{in } \Omega_T,\\ x &=\; 0 & \text{on } \Sigma_T,\\ x(0) &=\; x_0 & \text{in } \Omega, \end{array} \right. \] where \(\Omega\subseteq \mathbb R^d\) is open and bounded with a sufficiently smooth boundary \(\partial \Omega\), \(\Omega_T := (0,T] \times \Omega\) and \(\Sigma_T := (0,T] \times \partial \Omega\). The parabolic differential operator \(\mathcal A\) is defined by the (uniformly) continuous and coercive bilinear form \(a(t;v,w)=: \int_{\Omega_T} \mathcal{A}v \cdot w \;\mathrm{d}x \mathrm{d}t\).
The space-time POD method, as introduced in [2], is defined in the context of the tensor product spaces \(\mathcal{S}\cdot \mathcal{Y}\) for finite dimensional subspaces \(\mathcal{S}\subseteq H^1(0,T)\) and \(\mathcal{Y}\subseteq H^1_0(\Omega)\). Furthermore, the corresponding space-time POD subspace is denoted as \(\hat{\mathcal{S}}\cdot \hat{\mathcal{Y}}\) for reduced subspaces \(\hat{\mathcal{S}}\subseteq \mathcal{S}\) and \(\hat{\mathcal{Y}}\subseteq \mathcal{Y}\). While the application of space-time POD has already been introduced in [2], we will build on those results and present an a-priori error estimate for the error between a full order solution \(x \in \mathcal{S}\cdot \mathcal{Y}\) and a reduced order solution \(\hat{x} \in \hat{\mathcal{S}}\cdot \hat{\mathcal{Y}}\) as derived in [3]. The singular value based error estimate for space-time POD follows the idea of splitting the error as \[ \Vert x - \hat{x}\Vert_{L^2(\Omega_T)} \leq \Vert x - \mathcal{P}_{\hat{\mathcal{S}}\cdot \hat{\mathcal{Y}}} x \Vert_{L^2(\Omega_T)} + \Vert \mathcal{P}_{\hat{\mathcal{S}}\cdot \hat{\mathcal{Y}}} x - \hat{x}\Vert_{L^2(\Omega_T)} =: \Vert \varrho \Vert_{L^2(\Omega_T)} +\Vert \vartheta \Vert_{L^2(\Omega_T)}, \] where \(\mathcal{P}_{\hat{\mathcal{S}}\cdot \hat{\mathcal{Y}}} x\) denotes the projection onto \(\hat{\mathcal{S}}\cdot \hat{\mathcal{Y}}\). While such splitting is the same as for similar a-priori error estimates for standard POD, see e.g. [1], the estimation of the error terms \(\varrho\) and \(\vartheta\) differs in the context of space-time POD.
References:
[1] - S. Banholzer, D. Beermann, L. Mechelli, and S. Volkwein, Pod suboptimal control of evolution problems: Theory and applications, 2024. [2] - M. Baumann, P. Benner, and J. Heiland, Space-time galerkin pod with application in optimal control of semilinear partial differential equations, 2018. [3] - C. Gräßle, J. Heiland, and J. Marquardt, A-priori error estimation for space-time Galerkin POD for linear evolution problems, 2026.
