Localization phenomenon in the time-ordered exponential with application to quantum chemistry

• Thesis title in Czech: Localization phenomenon in the time-ordered exponential with application to quantum chemistry
• Thesis title in English: Localization phenomenon in the time-ordered exponential with application to quantum chemistry
• Academic year of topic announcement: 2023/2024
• Thesis type: diploma thesis
• Department: Department of Numerical Mathematics (32-KNM)
• Supervisor: Stefano Pozza, Dr., Ph.D.

Guidelines

The work will require using and adapting newly developed codes for spin dynamics simulation. The aim is to predict which part of the computed values is meaningful and which can be discarded to improve the efficiency of the codes. Predicting the significant part will require applying theoretical results about the so-called decay phenomenon. 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 ULCO-Calais) in the framework of Erasmus+.

References

- M. Benzi, Some uses of the field of values in numerical analysis, Boll. Unione Mat. Ital. 14(1) (2021) 159–177.
- M. Benzi, Localization in matrix computations: theory and applications, in: Exploiting Hidden Structure in Matrix Computations: Algorithms and Applications, in: Lecture Notes in Mathematics, vol.2173, Springer, Cham, 2016, pp.211–317
- M. Benzi, P. Boito, Decay properties for functions of matrices over C^⁎-algebras, Linear Algebra Appl., 456 (2014), 174-198
- M. Benzi, V. Simoncini, Decay bounds for functions of Hermitian matrices with banded or Kronecker structure, SIAM J. Matrix Anal. Appl. 36(3) (2015) 1263–1282.
- C. Bonhomme, S. Pozza, N. Van Buggenhout, A new fast numerical method for the generalized Rosen-Zener model, 2023, arXiv:2311.04144 [math.NA]
- S. Pozza, V. Simoncini, Inexact Arnoldi residual estimates and decay properties for functions of non-Hermitian matrices, BIT 59(4) (2019) 969–986.
- S. Pozza, N. Van Buggenhout, A ⋆-product solver with spectral accuracy for non-autonomous ordinary differential equations, Proceedings in Applied Mathematics & Mechanics, 2023.
- S. Pozza, N. Van Buggenhout, The *-product approach for linear ODEs: A numerical study of the scalar case, in: Chleboun, J., Kůs, P., Papež, J., Rozložník, M., Segeth, K. and Šístek, J. (eds.): Programs and Algorithms of Numerical Mathematics. Proceedings of Seminar. Janov nad Nisou, June 19-24, 2022. Institute of Mathematics CAS, Prague, 2023. Pages 187-198.
- S. Pozza, N. Van Buggenhout, A new Legendre polynomial-based approach for non-autonomous linear ODEs, 2023, arXiv:2303.11284 [math.NA]

Preliminary scope of work

The solution of systems of linear ordinary differential equations with variable coefficients can be expressed using the so-called time-ordered exponential (TOE). The numerical computation of TOEs is particularly challenging when its size is large-to-huge. This is typically emerging in Nuclear Magnetic Resonance applications in quantum chemistry. New approaches based on the so-called *-products have been tested successfully. These methods can express the TOE in terms of the inverse of a structured matrix, which is often characterized by localization, i.e., most of its elements are close to zero. It is possible to determine which elements are close to zero using the upper bounds in the literature.
The main goal of the work is to apply these well-known upper bounds to such structured matrix inverses. This will allow enhancing the efficiency of the new approaches for TOE approximation.

Preliminary scope of work in English

The solution of systems of linear ordinary differential equations with variable coefficients can be expressed using the so-called time-ordered exponential (TOE). The numerical computation of TOEs is particularly challenging when its size is large-to-huge. This is typically emerging in Nuclear Magnetic Resonance applications in quantum chemistry. New approaches based on the so-called *-products have been tested successfully. These methods can express the TOE in terms of the inverse of a structured matrix, which is often characterized by localization, i.e., most of its elements are close to zero. It is possible to determine which elements are close to zero using the upper bounds in the literature.
The main goal of the work is to apply these well-known upper bounds to such structured matrix inverses. This will allow enhancing the efficiency of the new approaches for TOE approximation.