Stochastic Galerkin method and port-Hamiltonian form for linear dynamical systems of second order

Energy‐based model order reduction for linear stochastic Galerkin systems of second order

“…A single input u is supplied by an excitation at the lowest spring. This test example was also used in [5, 9]. The mathematical model consists of $$n=4$$ second‐order ODEs ().…”

“…This property of the matrix ( 21) is related to the counterpart (7). We employ the MOR method from Section 4.1 to the high-dimensional system (8) with quadratic output (9). The balanced truncation technique preserves the asymptotic stability of the FOM, that is, each ROM is asymptotically stable again.…”

“…The solution of the system is p ∶ [0, ∞) → R 𝑛𝑠 with p = ( p⊤ 1 , … , p⊤ 𝑠 ) ⊤ , where p𝑖 represents an approximation of the exact PC coefficients with respect to the 𝑖th basis polynomial. More details on the stochastic Galerkin projection for linear ODEs can be found in [5,9], for example.…”

“…The solution of the system is $$\hat{p} : [0,\infty ) \rightarrow \mathbbm {R}^{ns}$$ with $$\hat{p} = (\hat{p}_1^\top ,\ldots ,\hat{p}_s^\top )^\top$$, where $$\hat{p}_i$$ represents an approximation of the exact PC coefficients with respect to the i th basis polynomial. More details on the stochastic Galerkin projection for linear ODEs can be found in [5, 9], for example.…”

“…The second-order stochastic Galerkin system (5) and its internal energy ( 6) are equivalent to the first-order system (8) with quadratic output (9). The dissipation analysis of Section 3 can be applied to ( 8) and ( 9).…”