Collaborative Research: Sparse spectral-tau methods for binary neutron star initial data

双星中子星INSPILL是美国LIGO计划等地面天文台可以探测到的最确定的引力波来源，对这种双星的模拟应该有助于最终的探测。这些模拟需要初始条件：重力-物质耦合系统广义相对论初值问题的解。共形薄夹层方法是解决初值问题的一种很好的方法；然而，尽管该方法不是该方法的内在假设，但在实践中该方法假定共形平坦性(与其他有价值的方法一样)。保形平坦会产生非物理的垃圾辐射。通过数值构造爱因斯坦方程的螺旋对称解，PI将提取不依赖于共形平坦度的初始数据(或共形薄层试验数据)，从而包含正确的初始引力波内容。由于爱因斯坦方程螺旋化而产生的混合偏微分方程组(或它们在后Minkowski形式中的近似)将用创新的技术来求解：稀疏模态谱-tau方法和新的预条件策略。在某种程度上，这些策略可能依赖于内插分解的随机化算法。谱方法为平滑问题提供了极高的精度(中子星时空几乎在任何地方都是平滑的)，当使用Krylov子空间方法迭代求解线性系统时，稀疏性提供了快速的矩阵-矢量乘法。虽然节点(配置)谱方法的预条件得到了很好的研究，但对模式预条件的了解较少。我们的技术已经成功地应用于双星中子星问题的模型。此外，已经用不同但有限的技术探索了这个问题的物理结构。这个项目将结合两套技术(每一套都已经开发出来)，并进一步发展第一套技术(频谱-tau方法)，以便为引力波物理中的一个领先问题获得新的结果。PI将通过将这些数学方法应用于上述特定的中子星问题来发展这些数学方法。这种特定的策略经常被用在技术的开发中，然后被证明是更通用的。因为这个科学问题非常有趣，人们对它有很多了解，因此也存在可以进行比较的结果。这些比较有助于数学算法的开发。相反，新的数学方法提供了更多和/或更好的解决方案，从而加强了科学理解。

Binary neutron star inspiral is the most certain source of gravitational waves detectable by Earth-based observatories like the US LIGO project, and simulations of such binaries should facilitate eventual detections. These simulations require initial conditions: solutions to the initial value problem of general relativity for the coupled gravity-matter system. The conformal thin sandwich method is an excellent approach for solving the initial value problem; however, although not an intrinsic assumption of the method, in practice the approach has assumed conformal flatness (as have other valuable approaches). Conformal flatness yields unphysical junk radiation. By numerically constructing helically symmetric solutions to the Einstein equations, the PI will extract initial data (or conformal thin sandwich trial data) which does not rely on conformal flatness, and therefore contains the correct initial gravitational wave content. The mixed PDEs arising from the helical reduction of the Einstein equations (or their approximation in the post-Minkowski formalism) will be solved with innovative techniques: sparse modal spectral-tau methods with new preconditioning strategies. In part, these strategies may rely on randomized algorithms for the interpolative decomposition. Spectral methods deliver superb accuracy for smooth problems(neutron star spacetimes are smooth almost everywhere), and sparsity affords a fast matrix-vector multiply when using a Krylov-subspace method to iteratively solve a linear system. Whereas the preconditioning of nodal (collocation) spectral methods is well studied, less is known about modal preconditioning. Our techniques have been successfully applied to models of the binary neutron star problem. Moreover, the problem's physical structure has already been explored with different, but limited, techniques. This project is to combine two sets of techniques (each already developed) and further develop the first set (spectral-tau methods), in order to obtain new results for a leading problem in gravitational wave physics. The PI will develop these mathematical methods by applying them to the specific neutron star problem described above. This strategy of specificity is often used in the development of techniques, which then prove to be more general. Because the scientific problem is of great interest, much is known about it, and results therefore exist withwhich comparisons can be made. These comparisons facilitate the development of mathematical algorithms. Conversely, new mathematical methods deliver more and/or better solutions which enhances scientific understanding.