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) [Dyson, 1949]. The project aims at developing new numerical approximation methods for very large TOEs. Among their many applications, TOEs can be used for magnetic resonance techniques (NMR, DNP). They require a precise understanding of the quantum dynamics of spins which, mathematically, are described by a TOE. Since large spin systems are still an elusive problem, the success of the project can lead to unprecedented descriptions of NMR/DNP processes.
Desired numerical methods will be designed following a recently introduced approach based on the results in [Giscard & al, 2015], [Giscard, P., 2020]. To our knowledge, this approach is the only existing one that allows to express TOEs in a finite number of scalar integro-differential equations.
Our objective is to develop new efficient numerical methods for the approximation of very large TOEs. Until recently, evaluating TOEs could only be done using one of the following approaches:
a) Perturbative methods (Floquet-based and Magnus series techniques), which are often inaccurate, prohibitively involved, and incurably diverging outside small domains, e.g., [Blanes & al., 2009], [Bonhomme, Giscard, 2020];
b) Numerical methods for ODEs based on integration schemes, which are highly consuming in resources and can poorly perform even for small systems (for instance, systems with highly oscillatory solutions represent a well-known challenge, see, e.g., [Hochbruck, Lubich, 1999], [Cohen & al., 2006]).
A substantial step forward has been recently taken thanks to a new approach obtained combining the Path-Sum [Giscard & al, 2015] and *-Lanczos [Giscard, P., 2020] methods. This new approach expresses each element of a TOE in terms of a finite and treatable number of scalar integro-differential equations. Moreover, it can efficiently deal with sparse systems and provides a sequence of approximations that superlinearly converges under certain regularity assumptions over a bounded interval. 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.
Among many applications, we focus on advanced magnetic resonance techniques (NMR, EPR, DNP), which are associated with the simulation of spin dynamics; e.g., [Mehring, Weberruß, 2012]. Mathematically, the spin dynamics is described by the TOE solving the related quantum equation of motion. Simulating very large systems of coupled spins remains an elusive overarching goal in NMR/DNP with few contributions in the recent literature owing to the difficulty of the problem [Dumez & al., 2010], [Hogben & al., 2011], [Mentink-Vigier & al., 2017], [Perras, Pruski, 2019]. Our numerical approach could be used in combination with these works, allowing to consider larger size problems.
Succeeding in our aims will provide a new approach for TOE problems that could be used in many other fields. It could also open a brand-new research area for systems of coupled linear and non-linear partial differential equations, thanks to the connection between PDEs and TOEs shown in [Kosovtsov, 2002].