Numerical approximation of the time-ordered exponential for spin dynamic simulation
| Thesis title in Czech: | Numerická aproximace time-ordered exponenciály pro dynamické simulace spinu |
|---|---|
| Thesis title in English: | Numerical approximation of the time-ordered exponential for spin dynamic simulation |
| Key words: | Time-ordered exponential|$\star$ - product|Magnus expansion|Geometrical numerical integrators|MAS NMR |
| English key words: | Time-ordered exponential|$\star$ - product|Magnus expansion|Geometrical numerical integrators|MAS NMR |
| Academic year of topic announcement: | 2021/2022 |
| Thesis type: | diploma thesis |
| Thesis language: | angličtina |
| 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: | 30.01.2022 |
| Date of assignment: | 31.01.2022 |
| Confirmed by Study dept. on: | 16.05.2023 |
| Date and time of defence: | 09.06.2023 09:00 |
| Date of electronic submission: | 04.05.2023 |
| Date of submission of printed version: | 09.05.2023 |
| Date of proceeded defence: | 09.06.2023 |
| Opponents: | Scott Congreve, Ph.D. |
| Guidelines |
| The work will require testing and adapting codes for spin dynamics simulation. The project aims to understand which methods reach a good accuracy in the fastest way on the data provided by the Laboratoire de Chimie de la matière condensée de Paris (Sorbonne University). The main programming languages will be Julia and MatLab. The work also requires to do a review of the literature on this topic.
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). |
| References |
| - Bak, M., Rasmussen, J. T., & Nielsen, N. C. (2011). SIMPSON: a general simulation program for solid-state NMR spectroscopy. Journal of magnetic resonance, 213(2), 366-400.
- Blanes, S., Casas, F.: A Concise Introduction to Geometric Numerical Integration. CRC Press, Bocan Raton, FL (2017) - Blanes, S., Casas, F., & Thalhammer, M. (2017). High-order commutator-free quasi-Magnus exponential integrators for non-autonomous linear evolution equations. Computer Physics Communications, 220, 243-262. - Giscard, P. L., & Bonhomme, C. (2020). Dynamics of quantum systems driven by time-varying Hamiltonians: Solution for the Bloch-Siegert Hamiltonian and applications to NMR. Physical Review Research, 2(2), 023081. - Giscard, P. L., & Pozza, S. (2021). Tridiagonalization of systems of coupled linear differential equations with variable coefficients by a Lanczos-like method. Linear Algebra and its Applications, 624, 153-173. - Gomez Pueyo, A., Marques, M. A., Rubio, A., & Castro, A. (2018). Propagators for the time-dependent Kohn–Sham equations: Multistep, Runge–Kutta, exponential Runge–Kutta, and commutator free Magnus methods. Journal of chemical theory and computation, 14(6), 3040-3052. - Gomez Pueyo, A., Blanes, S., & Castro, A. (2020). Propagators for quantum-classical models: Commutator-free Magnus methods. Journal of chemical theory and computation, 16(3), 1420-1430. |
| Preliminary scope of work |
| Solving systems of linear ordinary differential equations with variable coefficients remains a challenge that can be expressed using the so-called time-ordered exponential (TOE). Many numerical methods for TOE approximation are found in literature based on different approaches such as exponential propagators, Magnus expansion, integration schemes. Recently, new approaches based on the so-called *-products have been tested successfully.
This project aims to test and compare state-of-the-art methods for TOE approximation on problems coming from the simulation of spin dynamics provided by the Laboratoire de Chimie de la matière condensée de Paris, Sorbonne University. If possible, a particular focus will be given to test the *-Lanczos algorithm. |
| Preliminary scope of work in English |
| Solving systems of linear ordinary differential equations with variable coefficients remains a challenge that can be expressed using the so-called time-ordered exponential (TOE). Many numerical methods for TOE approximation are found in literature based on different approaches such as exponential propagators, Magnus expansion, integration schemes. Recently, new approaches based on the so-called *-products have been tested successfully.
This project aims to test and compare state-of-the-art methods for TOE approximation on problems coming from the simulation of spin dynamics provided by the Laboratoire de Chimie de la matière condensée de Paris, Sorbonne University. If possible, a particular focus will be given to test the *-Lanczos algorithm. |