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

双中子星激励是最确定的引力波来源，可以被地球上的天文台探测到，比如美国的LIGO项目，对这种双子星的模拟应该有助于最终的探测。这些模拟需要初始条件：引力-物质耦合系统广义相对论初值问题的解。共形薄夹层法是求解初值问题的一种很好的方法；然而，尽管不是该方法的固有假设，但在实践中，该方法假设了保形平面（与其他有价值的方法一样）。共形平面产生非物理垃圾辐射。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（或共形薄夹层试验数据），因此包含正确的初始引力波内容。由爱因斯坦方程的螺旋化简（或其在后闵可夫斯基形式主义中的近似）引起的混合偏微分方程将通过创新技术解决：具有新预处理策略的稀疏模态谱-tau方法。在某种程度上，这些策略可能依赖于插值分解的随机算法。谱方法为光滑问题提供了极好的精度（中子星时空几乎在任何地方都是光滑的），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向量乘法。虽然对节点（搭配）谱方法的预处理研究得很好，但对模态预处理却知之甚少。我们的技术已经成功地应用于双中子星问题的模型。此外，这个问题的物理结构已经用不同但有限的技术进行了探索。这个项目是将两套技术（每一套都已经开发）结合起来，并进一步发展第一套技术（频谱-tau方法），以便为引力波物理学中的一个主要问题获得新的结果。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。这种特异性策略经常用于技术的开发，然后被证明是更普遍的。由于这个科学问题引起了极大的兴趣，人们对它了解得很多，因此存在着可以进行比较的结果。这些比较促进了数学算法的发展。相反，新的数学方法提供了更多和/或更好的解决方案，从而增强了科学理解。

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.