FRG: Collaborative Research: Error Quantification and Control for Gravitational Waveform Simulation

该项目的目的是开发实用的严格方法，以可靠的精度估计引力波模拟计算波形的误差，以支持nsf资助的激光干涉仪引力波天文台（LIGO）。该项目汇集了一个应用和计算数学家团队，他们具有构建偏微分方程解的误差估计的专业知识，以及具有爱因斯坦方程数值解和引力波数据分析专业知识的物理学家。主要的技术目标是发展和分析新的数学和计算方法，这些方法可以被引力物理学界用来计算爱因斯坦方程数值解的误差和由它们确定的引力波形的严格和可靠的精确估计。具体而言，本研究探讨了以下问题：(1)使用伴随灵敏度技术进行误差量化和后验分析，及其相关的数值实现；(2)目标导向误差控制驱动的自适应算法及其理论收敛性分析；(3)协方差对称及其相关几何结构在误差分析和数值方法构建中的作用。作为后验分析的一部分，项目团队将发展伴随算子和爱因斯坦方程对偶的基本理论。这将为今后利用LIGO数据进行灵敏度分析、数据同化和不确定度量化的研究奠定基础。应该强调的是，所提出的研究的主要重点是离散中立的，因此对现有的数值相对论代码的广度具有广泛的适用性。美国国家科学基金会（nsf）支持的激光干涉仪重力天文台（LIGO）只有在高度精确的重力波形模型可用作数据分析过程的一部分时才能成功，既可以用于探测引力波，也可以用于测量任何探测信号的物理性质。预计最强烈的引力波源是重而密的恒星或黑洞之间的碰撞，这只能通过复杂的数值模拟来精确模拟，以计算预期的引力波波形。这种波形需要用来构建滤波器，以便在有噪声的探测器中探测到微弱的引力波信号，而且这种波形也需要用来测量任何被探测到的信号源的物理特性。完成所需数据分析任务所需的波形精度是相当高的。然而，数值相对论界尚未开发出严格评估数值波形模型准确性所需的分析和计算工具。如果目前数值相对论界使用的定性精度测量过于乐观，那么这个项目开发的严格的新方法可能会决定LIGO的成败。如果目前的数值波形确实足够精确，本项目开发的方法可以提高确定具有指定精度水平的波形的计算效率，从而降低产生波形的成本。

The purpose of this project is to develop practical rigorous methods for estimating the error in computed waveforms from gravitational wave simulation with reliable accuracy, in support of the NSF-funded Laser Interferometer Gravitational Observatory (LIGO). The project brings together a team of applied and computational mathematicians with expertise in constructing error estimates for solutions of partial differential equations and physicists with expertise in numerical solutions of the Einstein equation and gravitational wave data analysis. The primary technical goal is to develop and analyze new mathematical and computational methods that can be used by the gravitational physics community to compute rigorous and reliably accurate estimates for the errors of numerical solutions of the Einstein equations and the gravitational waveforms that are determined from them. In particular, this research explores the following issues:(1) Error quantification and a posteriori analysis using adjoint sensitivity techniques, and their associated numerical implementation;(2) Adaptive algorithms that are driven by goal-oriented error control, and their associated theoretical convergence analysis; and(3) The role of covariance symmetry and associated geometric structures in error analysis and the construction of numerical methods.As part of the a posteriori analysis, the project team will develop the basic theory of adjoint operators and duality for the Einstein equations. This will provide the foundation for future investigations into sensitivity analysis, data assimilation and uncertainty quantification for using LIGO data. It should be emphasized that the main thrusts of the proposed research are discretization-neutral, and therefore have broad applicability to the breadth of numerical relativity codes in existence.The NSF-supported Laser Interferometer Gravitational Observatory (LIGO) can be successful only if highly accurate gravitational waveform models are available for use as part of the data analysis process, both for detecting gravitational waves and also for measuring the physical properties of any detected signals. The strongest sources of gravitational waves are expected to be collisions between heavy, dense stars or black holes, which can only be modeled accurately using complex numerical simulations to calculate the anticipated gravitational waveforms. Such waveforms are needed to construct the filters that allow detection of the weak gravitational-wave signals in the noisy detector, and such waveforms are also needed to measure the physical properties of the sources of any detected signals. The waveform accuracy needed to accomplish the required data analysis tasks is quite high. However, the numerical relativity community has yet to develop the analytic and computational tools needed to evaluate rigorously the accuracy of the numerical waveform models. If the qualitative accuracy measures currently used by the numerical relativity community are too optimistic, the rigorous new methods developed by this project could make the difference between success and failure of LIGO. If the current numerical waveforms are in fact accurate enough, the methods developed by this project could improve the computational efficiency of determining waveforms with a specified accuracy level, and thus reduce the cost of producing them.