Stable Nonlinear Dynamical Approximation with Dynamical Sampling
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909614686076928 |
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| author | Bon, Daan Caris, Benjamin Mula, Olga |
| author_facet | Bon, Daan Caris, Benjamin Mula, Olga |
| contents | We present a nonlinear dynamical approximation method for time-dependent Partial Differential Equations (PDEs). The approach makes use of parametrized decoder functions, and provides a general, and principled way of understanding and analyzing stability and accuracy of nonlinear dynamical approximations. The parameters of these functions are evolved in time by means of projections on finite dimensional subspaces of an ambient Hilbert space related to the PDE evolution. For practical computations of these projections, one usually needs to sample. We propose a dynamical sampling strategy which comes with stability guarantees, while keeping a low numerical complexity. We show the effectiveness of the method on several examples in moderate spatial dimension. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_11938 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stable Nonlinear Dynamical Approximation with Dynamical Sampling Bon, Daan Caris, Benjamin Mula, Olga Numerical Analysis 35C99, 32W99, 65D40, 65D99, 65D30, 65D15 We present a nonlinear dynamical approximation method for time-dependent Partial Differential Equations (PDEs). The approach makes use of parametrized decoder functions, and provides a general, and principled way of understanding and analyzing stability and accuracy of nonlinear dynamical approximations. The parameters of these functions are evolved in time by means of projections on finite dimensional subspaces of an ambient Hilbert space related to the PDE evolution. For practical computations of these projections, one usually needs to sample. We propose a dynamical sampling strategy which comes with stability guarantees, while keeping a low numerical complexity. We show the effectiveness of the method on several examples in moderate spatial dimension. |
| title | Stable Nonlinear Dynamical Approximation with Dynamical Sampling |
| topic | Numerical Analysis 35C99, 32W99, 65D40, 65D99, 65D30, 65D15 |
| url | https://arxiv.org/abs/2505.11938 |