Numerical solution of the Schrödinger equation by rational Krylov subspace approximation for simulations in nuclear magnetic resonance spectroscopy
| Thesis title in Czech: | Numerical solution of the Schrödinger equation by rational Krylov subspace approximation for simulations in nuclear magnetic resonance spectroscopy |
|---|---|
| Thesis title in English: | Numerical solution of the Schrödinger equation by rational Krylov subspace approximation for simulations in nuclear magnetic resonance spectroscopy |
| Academic year of topic announcement: | 2024/2025 |
| Thesis type: | diploma thesis |
| Thesis language: | |
| Department: | Department of Numerical Mathematics (32-KNM) |
| Supervisor: | Stefano Pozza, Dr., Ph.D. |
| Author: | hidden - assigned and confirmed by the Study Dept. |
| Date of registration: | 30.10.2024 |
| Date of assignment: | 31.10.2024 |
| Confirmed by Study dept. on: | 31.10.2024 |
| Guidelines |
| The work will require using and adapting newly developed codes for spin dynamics simulation. The main goal is to test the use of rational Krylov subspace methods to solve a large system of Schrödinger equations. The primary programming language will be MatLab. The work requires a literature review of the related topics.
The project offers the possibility to collaborate with the international members of the *-Lanczos (www.starlanczos.cz) and the MAGICA project (https://anr.fr/Project-ANR-20-CE29-0007). It is possible to organize a visit to one of the partner universities (Sorbonne or Bologna) in the framework of Erasmus+. |
| References |
| - C. Bonhomme, S. Pozza, N. Van Buggenhout, A new fast numerical method for the generalized Rosen-Zener model, 2023, arXiv:2209.15533 [math.NA]
- I. Kuprov. Spin. Springer International Publishing, 2023. - D. Palitta, P. Kürschner, On the convergence of Krylov methods with low-rank truncations. Numerical Algorithms, 88(3) (2021), 1383-1417. - D. Palitta, V. Simoncini, Computationally enhanced projection methods for symmetric Sylvester and Lyapunov matrix equations, Journal of Computational and Applied Mathematics, 330 (2018), 648-659 - S. Pozza, N. Van Buggenhout, A new matrix equation expression for the solution of non-autonomous linear systems of ODEs, Proceedings in Applied Mathematics & Mechanics, 2023 arXiv:2210.07052 [math.NA] - S. Pozza, N. Van Buggenhout, A new Legendre polynomial-based approach for non-autonomous linear ODEs, 2023. - V. Simoncini. Computational methods for linear matrix equations. SIAM Review, 58:377–441, 2016. |
| Preliminary scope of work |
| Thanks to a new approach, the solution of certain systems of Schrödinger equations can be expressed through the solution of a matrix equation. Since solving such an equation is particularly challenging when its size is large-to-huge, a model reduction approach is necessary, e.g., using Krylov subspaces.
This work will focus on systems arising in nuclear magnetic resonance applications. The final aim is to test well-known Rational Krylov subspace approaches to approximate the matrix equation solution. The work also requires collaborating with the Sorbonne University team that will provide the equation's data. |
| Preliminary scope of work in English |
| Thanks to a new approach, the solution of certain systems of Schrödinger equations can be expressed through the solution of a matrix equation. Since solving such an equation is particularly challenging when its size is large-to-huge, a model reduction approach is necessary, e.g., using Krylov subspaces.
This work will focus on systems arising in nuclear magnetic resonance applications. The final aim is to test well-known Rational Krylov subspace approaches to approximate the matrix equation solution. The work also requires collaborating with the Sorbonne University team that will provide the equation's data. |