Quasi-Optimal Least Squares: Inhomogeneous boundary conditions, and application with machine learning
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912367015624704 |
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| author | Monsuur, Harald Smeets, Robin Stevenson, Rob |
| author_facet | Monsuur, Harald Smeets, Robin Stevenson, Rob |
| contents | We construct least squares formulations of PDEs with inhomogeneous essential boundary conditions, where boundary residuals are not measured in unpractical fractional Sobolev norms, but which formulations nevertheless are shown to yield a quasi-best approximations from the employed trial spaces. Dual norms do enter the least-squares functional, so that solving the least squares problem amounts to solving a saddle point or minimax problem. For finite element applications we construct uniformly stable finite element pairs, whereas for Machine Learning applications we employ adversarial networks. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_05965 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quasi-Optimal Least Squares: Inhomogeneous boundary conditions, and application with machine learning Monsuur, Harald Smeets, Robin Stevenson, Rob Numerical Analysis 35B35, 35B45, 65N30 We construct least squares formulations of PDEs with inhomogeneous essential boundary conditions, where boundary residuals are not measured in unpractical fractional Sobolev norms, but which formulations nevertheless are shown to yield a quasi-best approximations from the employed trial spaces. Dual norms do enter the least-squares functional, so that solving the least squares problem amounts to solving a saddle point or minimax problem. For finite element applications we construct uniformly stable finite element pairs, whereas for Machine Learning applications we employ adversarial networks. |
| title | Quasi-Optimal Least Squares: Inhomogeneous boundary conditions, and application with machine learning |
| topic | Numerical Analysis 35B35, 35B45, 65N30 |
| url | https://arxiv.org/abs/2412.05965 |