Numerické aproximační problémy vznikající v *-Lanczošově metodě

• Thesis title in Czech: Numerické aproximační problémy vznikající v *-Lanczošově metodě
• Thesis title in English: Numerical approximation problems arising from the *-Lanczos method
• English key words: Time-ordered exponential|Lanczos algorithm|numerical integration|ODEs
• Academic year of topic announcement: 2020/2021
• Thesis type: dissertation
• 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: 08.03.2021
• Date of assignment: 08.03.2021
• Confirmed by Study dept. on: 12.03.2021

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 scalar integro-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.
Motivated by the fact that each iteration of *-Lanczos requires to compute several integrals and solve a scalar linear ODE, the thesis work will tackle the following tasks:
1. Approximating the solution of the integral and differential problems arising from each *-Lanczos iteration;
2. Testing and applying consequently derived methods.

More information at http://karlin.mff.cuni.cz/~pozza/

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, 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)