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，实验和分析解可能难以获得，而数值求解方程（如Schrödinger方程）是唯一可用的选择。由于量子力学方程的色散性质，其解在空间和时间上都具有典型的高度振荡性。此外，还需要保留解的物理性质，如总概率、角动量、能量和统一性。因此，高度专业化的方法，如指数分裂和（多项式）克雷洛夫方法已被设计用于解决这些方程。在解决这些量子方程的现有数值技术中出现的一个主要问题是，它们需要非常小的时间步长。这将导致非常长的计算时间，这使得这些方法在大时域上传播时不可行。对于无界势，如库仑势，这类问题特别明显。求解这些问题的Krylov方法是用一个较小的密集矩阵的指数逼近一些大的稀疏矩阵（哈密顿矩阵）的矩阵指数。然而，当需要较高的空间网格分辨率时，需要合理精度的小矩阵的尺寸会随着大的时间步长、大范数的势而显著增长。因此，在这些情况下，唯一实际的资源是利用过小的时间步长。最近出现了一种非常有前途的替代方法，称为理性Krylov方法，用于偏微分方程的数值演化。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，似乎有希望成为量子力学方程的候选者。然而，理性Krylov方法的有效性主要依赖于最优理性近似的极点的知识，而现有的极点选择技术要么是次优的，要么是启发式的。该项目的目的是分析和开发合理的Krylov方法，目的是减少长时间传播的计算时间。这将通过开发更有效的极点选择策略来实现，使用多网格方法结合复杂分析的更传统技术。

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.