Multiscale Rational Krylov Methods

数值技术对于理解量子系统的动力学是必不可少的，在量子系统中，实验和解析解可能很难获得，而数值求解方程，如薛定谔方程是唯一可用的选择。由于量子力学方程的色散性质，解在空间和时间上通常都是高度振荡的。此外，还要求守恒解的物理性质，如总概率、角动量、能量和么正性。因此，诸如指数分裂和(多项式)Krylov方法等高度专业化的方法被设计用于求解这些方程。现有求解这些量子方程的数值技术出现的一个主要问题是它们需要非常小的时间步长。这导致了极长的计算时间，这使得这些方法在大的时间域上传播时不可行。对于无界位势，如库仑势，这样的问题尤其明显。求解这些问题的Krylov方法是通过将一些大型稀疏矩阵(哈密顿矩阵)的指数近似为一个较小的稠密矩阵的指数来进行的。但是，当需要较高的空间网格分辨率时，对于较大的时间步长、具有较大范数的势和要求较高的空间网格分辨率，合理精度所需的小矩阵的维度可能会显著增加。因此，在这些情况下，唯一可行的办法是利用非常小的时间步长。最近，出现了一种非常有前途的替代方法，称为有理Krylov方法，用于偏微分方程组的数值演化。这些方法已被发现在实现分辨率独立性方面非常有效。它们还允许更大的时间步长，使其成为色散方程的理想选择，因此，似乎是量子力学方程的有前途的候选者。然而，有理Krylov方法的有效性关键依赖于对最优有理逼近极点的了解，而现有的极点选择技术要么是次优的，要么是启发式的。这个项目的目的是分析和开发有理Krylov方法，目的是减少长时间传播的计算时间。这将通过使用多重网格法与复杂分析中的更传统技术相结合来制定更有效的极点选择战略来实现。

Numerical techniques are essential for understanding the dynamics of quantum systems, where experimental and analytical solutions may be difficult to obtain and numerically solving equations such as the Schrödinger equation is the only option available. Due to the dispersive nature of the equations of quantum mechanics, the solutions are typically highly oscillatory in both space and time. Moreover, there is a requirement to conserve physical properties of the solution, such as total probability, angular momentum, energy and unitarity. Consequently, highly specialised methods such as exponential splittings and (polynomial) Krylov methods have been devised for solving these equations.A major problem which arises in existing numerical techniques for solving these quantum equations is that they require very small time-step sizes. This leads to extremely long computational times, which makes these methods unfeasible when propagating over a large time domain. Such problems are particularly pronounced for an unbounded potential such as the Coulomb potential. Krylov methods for solving these problems proceed by approximating the matrix exponential for some large sparse matrix (the Hamiltonia) with the exponential of a smaller dense matrix.However, the dimensions of the small matrix required for reasonable accuracy can grow significantly with large time-steps, potentials with large norm, and when a high spatial grid resolution is required. Thus, the only practical recourse in these cases is to utilise excessively small time stepsRecently a very promising alternative, known as rational Krylov methods, have emerged for numerical evolution of PDEs. These methods have been found to be highly effective in achieving resolution independence. They also allow for significantly larger time-step sizes, making it ideal for dispersive equations and, consequently, seem to promising candidates for equations of quantum mechanics. The efficacy of rational Krylov methods crucially relies on the knowledge of the poles of the optimal rational approximant, however, and existing pole selection techniques are either suboptimal or heuristic . The aim of this project is to analyse and develop rational Krylov methods, with the goal of reducing computational times for long-time propagation. This will be done by developing more effective strategies for pole selection using multigrid approach combined with more traditional techniques from complex analysis.