Tensor techniques for the approximation of time-ordered exponentials

• Thesis title in Czech: Tenzorové techniky pro takzvanou time-ordered exponenciálu
• Thesis title in English: Tensor techniques for the approximation of time-ordered exponentials
• English key words: Time-ordered exponential|Lanczos algorithm|ODEs|Tensor
• Academic year of topic announcement: 2021/2022
• Thesis type: dissertation
• 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: 08.02.2022
• Date of assignment: 08.02.2022
• Confirmed by Study dept. on: 22.06.2022

Guidelines

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).
A new approach for approximating a time-ordered exponential has been recently obtained combining the Path-Sum [2] and *-Lanczos [3] methods. This new approach expresses each element of a TOE in terms of a finite and treatable number of integrals and scalar differential equations. In most cases, however, complicated special functions are involved. As a consequence, *-Lanczos cannot produce a closed-form expression for the solution. Also, a purely mathematical method cannot deal with large-to-huge scale problems, which are relevant to most applications. For these reasons, new numerical approximation techniques are needed to deal with the most challenging problems. A tensor algorithm for TOE approximation can be derived appropriately discretizing *-Lanczos iterations. The thesis work will tackle the following tasks:
1. Studying and derive tensor techniques for TOE approximation;
2. Testing and applying consequently derived methods.

References

[1] Blanes, S., Casas, F.: A concise introduction to geometric numerical integration. CRC press (2017).

[2] Giscard, P.L., Lui, K., Thwaite, S.J., Jaksch, D.: An exact formulation of the time-ordered exponential using path-sums. J. Math. Phys. 56(5), 053503 (2015)

[3] Giscard, P-L., Pozza, S.: Lanczos-like method for the time-ordered exponential, arXiv:1909.03437 [math.NA] (2020).

[4] Giscard, P-L., Pozza, S.: Tridiagonalization of systems of coupled linear differential equations with variable coefficients by a Lanczos-like method, arXiv:2002.06973 [math.CA], (2020).

[5] Giscard, P-L., Pozza, S.: Lanczos-like algorithm for the time-ordered exponential: The ∗-inverse problem, Appl. Math., 65(6), 807–827, (2020)

Accepted in Applications of Mathematics, to appear. Preprint arXiv:1910.05143 [math.NA] (2019)

[6] Golub, G.H., Meurant, G.: Matrices, Moments and Quadrature with Applications. Princeton Ser. Appl. Math. Princeton University Press, Princeton (2010)

[7] Higham, N.J.: Functions of matrices. Theory and computation. SIAM, Philadelphia (2008)

[8] Liesen, J., Strakoš, Z.: Krylov subspace methods: principles and analysis. Numer. Math. Sci. Comput. Oxford University Press, Oxford (2013)

[9] Parlett, B.N.: Reduction to tridiagonal form and minimal realizations. SIAM J. Matrix Anal. Appl. 13(2), 567–593 (1992)