Speaker
Description
We consider the frequency domain simulation of second-order linear dynamical systems, which is crucial for many applications ranging from vibroacoustics to electrical circuits. These problems are parameter-dependent, involving frequency and additional parameters to account for uncertainties. This dependence on a large number of parameters results in a very high numerical cost. To address these challenges, surrogate modeling based on polynomial, rational, and reduced-order models has recently gained significant attention. In this contribution, we will present multiple perspectives on efficient numerical methods with emphasis on rational modeling.
In the first part (joint work with J. Bect, N. Georg, and S. Schöps) we will consider the frequency as the only parameter and introduce a hybrid method that combines a parametric rational model with a complex-valued kernel approximation. We will discuss the role of pseudo-kernels and the underlying reproducing kernel Hilbert spaces and compare the performance of the method against established rational surrogates, such as vector fitting and the Adaptive Antoulas-Anderson (AAA) algorithm [1].
In the second part (joint work with M. Bollhöfer, H. Sreekumar, S. Langer), we will consider both the frequency and additional model parameters with an underlying probability distribution. The frequency dependence is captured again with a rational model, specifically, a moment matching-based reduced order model. This reduced order model is then interpolated on sparse grids to account for the high-dimensional parameter space. We outline an adaptive procedure, driven by adjoint error indicators, that balances the reduced order and sparse grid approximation errors [2].
Finally, we will provide a brief outlook (joint work with J. Heiland, I.V. Gosea, D. Pradovera) on rational approximation of matrix- or tensor-valued data, offering a highly structured representation of snapshots over the frequency domain.
[1] Bect, J., Georg, N., Römer, U., and Schöps, S. (2024). Rational kernel-based interpolation for complex-valued frequency response functions. SIAM Journal on Scientific Computing, 46(6), A3727-A3755.
[2] Römer, U., Bollhöfer, M., Sreekumar, H., Blech, C., and Langer, S. (2021). An adaptive sparse grid rational Arnoldi method for uncertainty quantification of dynamical systems in the frequency domain. International Journal for Numerical Methods in Engineering, 122(20), 5487-5511.
