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Relative errors for the non-separable example

Relative errors for the non-separable example

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Figure 1: Poisson example FOM solutions for p = 0.1, 1, 10
Figure 8: Convection-diffusion example errors for ROMs of different orders
Figure 10: Thermalblock example FOM output and pointwise output errors...
Relative errors for the non-separable example
L2\mathcal{L}_2-optimal Reduced-order Modeling Using Parameter-separable Forms
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  • Jun 2022
We provide a unifying framework for $\mathcal{L}_2$-optimal reduced-order modeling for linear time-invariant dynamical systems and stationary parametric problems. Using parameter-separable forms of the reduced-model quantities, we derive the gradients of the $\mathcal{L}_2$ cost function with respect to the reduced matrices, which then allows a non...
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Context 1
... observe that the RB and POD again produce ROMs with nonnegative output error. The relative L 2 and L ∞ errors listed in Table 1 show significant improvements in L 2 error minimization via L 2 -Opt-PSF, especially for the second DDROM form. ...
Context 2
... observe that the RB and POD again produce ROMs with nonnegative output error. The relative L 2 and L ∞ errors listed in Table 1 show significant improvements in L 2 error minimization via L 2 -Opt-PSF, especially for the second DDROM form. ...

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Citations

... The authors recently developed a data-driven framework for L 2 -optimal reduced-order modeling of parametric systems [MG22]. In this paper, we show how [MG22] provides a unifying framework for interpolatory optimal approximation both for dynamical systems and stationary problems. ...
... The authors recently developed a data-driven framework for L 2 -optimal reduced-order modeling of parametric systems [MG22]. In this paper, we show how [MG22] provides a unifying framework for interpolatory optimal approximation both for dynamical systems and stationary problems. We prove that bitangential Hermite interpolation is the necessary condition for optimality not only for approximation of LTI systems in the H 2 norm, but also in many other prominent cases, thus extending the optimal interpolation theory to a broader class of problems. ...
... First we recall the L 2 -optimal reduced-order modeling problem discussed in [MG22]: Consider a parameter-to-output mapping y : P → C no×ni (1.1) ...
A Unifying Framework for Interpolatory L2\mathcal{L}_2-optimal Reduced-order Modeling
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  • Sep 2022
We develop a unifying framework for interpolatory L2\mathcal{L}_2-optimal reduced-order modeling for a wide classes of problems ranging from stationary models to parametric dynamical systems. We first show that the framework naturally covers the well-known interpolatory necessary conditions for H2\mathcal{H}_2-optimal model order reduction and leads to the interpolatory conditions for H2⊗L2\mathcal{H}_2 \otimes \mathcal{L}_2-optimal model order reduction of multi-input/multi-output parametric dynamical systems. Moreover, we derive novel interpolatory optimality conditions for rational discrete least-squares minimization and for L2\mathcal{L}_2-optimal model order reduction of a class of parametric stationary models. We show that bitangential Hermite interpolation appears as the main tool for optimality across different domains. The theoretical results are illustrated on two numerical examples.
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