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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2000
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/math/0008225 |
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| _version_ | 1866914099220185088 |
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| author | Garcia-Ripoll, Juan Jose Perez-Garcia, Victor M. |
| author_facet | Garcia-Ripoll, Juan Jose Perez-Garcia, Victor M. |
| contents | In this paper we study the application of the Sobolev gradients technique to the problem of minimizing several Schrödinger functionals related to timely and difficult nonlinear problems in Quantum Mechanics and Nonlinear Optics. We show that these gradients act as preconditioners over traditional choices of descent directions in minimization methods and show a computationally inexpensive way to obtain them using a discrete Fourier basis and a Fast Fourier Transform. We show that the Sobolev preconditioning provides a great convergence improvement over traditional techniques for finding solutions with minimal energy as well as stationary states and suggest a generalization of the method using arbitrary linear operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_math_0008225 |
| institution | arXiv |
| publishDate | 2000 |
| record_format | arxiv |
| spellingShingle | Optimizing Schroedinger functionals using Sobolev gradients: Applications to Quantum Mechanics and Nonlinear Optics Garcia-Ripoll, Juan Jose Perez-Garcia, Victor M. Numerical Analysis Soft Condensed Matter Pattern Formation and Solitons Optics 65K10 In this paper we study the application of the Sobolev gradients technique to the problem of minimizing several Schrödinger functionals related to timely and difficult nonlinear problems in Quantum Mechanics and Nonlinear Optics. We show that these gradients act as preconditioners over traditional choices of descent directions in minimization methods and show a computationally inexpensive way to obtain them using a discrete Fourier basis and a Fast Fourier Transform. We show that the Sobolev preconditioning provides a great convergence improvement over traditional techniques for finding solutions with minimal energy as well as stationary states and suggest a generalization of the method using arbitrary linear operators. |
| title | Optimizing Schroedinger functionals using Sobolev gradients: Applications to Quantum Mechanics and Nonlinear Optics |
| topic | Numerical Analysis Soft Condensed Matter Pattern Formation and Solitons Optics 65K10 |
| url | https://arxiv.org/abs/math/0008225 |