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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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... the output, we chose C = B T (thus, n o = 1). Solutions for a few parameter values are given in Figure 1. The output can be seen in the left plot in Figure 2. ...
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... the output, we kept C = B T . The left plot in Figure 10 shows the output of the FOM over the equidistant grid P test = linspace(0.1, 10, 5) 4 . ...
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... using only the discrete output samples y(p ℓ ) from the same training grid, i.e., a total of N = 256 output samples y(p ℓ ), we run the discrete L 2 -Opt-PSF (initialized with the POD ROM) to minimize the MSE (5.1). Figure 10 shows the outputs and the resulting errors over P test . The figure shows a significant reduction of the errors for the L 2 DDROM. ...
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... the output, we chose C = B T (thus, n o = 1). Solutions for a few parameter values are given in Figure 1. The output can be seen in the left plot in Figure 2. ...
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... the output, we kept C = B T . The left plot in Figure 10 shows the output of the FOM over the equidistant grid P test = linspace(0.1, 10, 5) 4 . ...
Context 6
... using only the discrete output samples y(p ) from the same training grid, i.e., a total of N = 256 output samples y(p ), we run the discrete L 2 -Opt-PSF (initialized with the POD ROM) to minimize the MSE (5.1). Figure 10 shows the outputs and the resulting errors over P test . The figure shows a significant reduction of the errors for the L 2 DDROM. ...
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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) ...
We develop a unifying framework for interpolatory -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 -optimal model order reduction and leads to the interpolatory conditions for -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 -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.
