Asymptotic spectral properties and preconditioning of an approximated nonlocal Helmholtz equation with Caputo fractional Laplacian and variable coefficient wave number $μ$
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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916331769561088 |
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| author | Adriani, Andrea Sormani, Rosita Luisa Tablino-Possio, Cristina Krause, Rolf Serra-Capizzano, Stefano |
| author_facet | Adriani, Andrea Sormani, Rosita Luisa Tablino-Possio, Cristina Krause, Rolf Serra-Capizzano, Stefano |
| contents | The current study investigates the asymptotic spectral properties of a finite difference approximation of nonlocal Helmholtz equations with a Caputo fractional Laplacian and a variable coefficient wave number $μ$, as it occurs when considering a wave propagation in complex media, characterized by nonlocal interactions and spatially varying wave speeds. More specifically, by using tools from Toeplitz and generalized locally Toeplitz theory, the present research delves into the spectral analysis of nonpreconditioned and preconditioned matrix-sequences. We report numerical evidences supporting the theoretical findings. Finally, open problems and potential extensions in various directions are presented and briefly discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_10569 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic spectral properties and preconditioning of an approximated nonlocal Helmholtz equation with Caputo fractional Laplacian and variable coefficient wave number $μ$ Adriani, Andrea Sormani, Rosita Luisa Tablino-Possio, Cristina Krause, Rolf Serra-Capizzano, Stefano Numerical Analysis 65F08, 35R11, 65N22, 15A18, 47B35 The current study investigates the asymptotic spectral properties of a finite difference approximation of nonlocal Helmholtz equations with a Caputo fractional Laplacian and a variable coefficient wave number $μ$, as it occurs when considering a wave propagation in complex media, characterized by nonlocal interactions and spatially varying wave speeds. More specifically, by using tools from Toeplitz and generalized locally Toeplitz theory, the present research delves into the spectral analysis of nonpreconditioned and preconditioned matrix-sequences. We report numerical evidences supporting the theoretical findings. Finally, open problems and potential extensions in various directions are presented and briefly discussed. |
| title | Asymptotic spectral properties and preconditioning of an approximated nonlocal Helmholtz equation with Caputo fractional Laplacian and variable coefficient wave number $μ$ |
| topic | Numerical Analysis 65F08, 35R11, 65N22, 15A18, 47B35 |
| url | https://arxiv.org/abs/2402.10569 |