Computational methods for the study of rare events

研究问题是关于开发计算工具(A)罕见反应事件和(B)地震建模。(A)：在使用小噪声随机微分方程建模的系统中寻找最可能的过渡路径的问题非常困难，原因有以下几个问题：(1)系统亚稳态之间的过渡很少，因此直接模拟非常困难；(2)高维度;(3)评估力昂贵；(4)多个局部极小值；(5)主导反应通道的温度依赖性。研究者计划探索一种基于hamilton - jacobi的方法来研究罕见的转变。与现有的基于路径的方法相比，它具有重要的优势。这种方法可以保证找到全局最小值，并且不需要初始猜测。此外，它允许我们找到系统的任何吸引子之间最可能的过渡路径，而不仅仅是平衡点之间。该方法的主要困难在于Hamilton-Jacobi方程中的高维数、各向异性和无界速度函数。研究者提出了处理这些问题的几种方法。(B)：寻找地球内部声速（地震速度）的大多数方法依赖于大量的计算资源和良好的初始猜测。研究者和她的同事提出了另一种方法，这种方法计算成本低，不需要初始猜测。该方法基于时移速度与地震速度之间的理论关系。声速可以通过求解柯西数据的椭圆偏微分方程来恢复。尽管这个问题是不适定的，但他们已经开发出能够在规定的时间内求解它的数值技术。研究者计划在这个方向上继续研究，并结合计算随机过程领域的方法。许多过程都是用随机微分方程来建模的，这些方程是包含随机项（噪声）的进化规律。例子包括物理和化学中的小规模过程，如化学反应和分子的保形变化。其他例子来自随机建模的计算机网络、金融证券的定价以及社会中的货币分配。在没有噪声的情况下，一个给定的系统向它的一个平衡状态演化，并永远停留在那里。但是噪声的存在，即使是任意小的噪声，也会使平衡态之间发生过渡。在许多重要的情况下，这些转变在系统的时间尺度上是罕见的，但在人类的时间尺度上并不罕见。除了直接模拟之外，这一事实创造了对研究这些罕见转变的技术的需求。这样的研究将有助于理解蛋白质折叠和基因表达机制等现象。生成准确的地球图像？美国的内部是一个具有挑战性的方面，如石油开采和地震分析。首先，地震资料中总是含有噪声，数据来源越深，噪声的影响越大。其次，重要而有趣的地质特征（如石油沉积）通常发生在地下结构复杂且声速在横向（侧向）方向上变化严重的地方。结果，确定地球声速的基本反问题是病态的，而且极其困难。然而，声速的确定对于准确的地震成像至关重要。研究者和她的同事提出的方法将导致更便宜和更有效的方法来测定它。

The research problem is concerned with developing computational tools for (A) rare reactive events and (B) seismic modeling. (A): The problem of finding the most likely transition paths in systems that are modeled using stochastic differential equations with small noise is very difficult due to several issues: (1)transitions between metastable states of the system are rare, hence direct simulations are very hard; (2)high dimensionality; (3) expensive-to-evaluate force; (4) multiple local minimizers; (5) temperature dependence of the dominant reactive channel. The investigator plans to explore a Hamilton-Jacobi-based approach for the study of rare transitions. It has important advantages over existing path-based methods. This approach is guaranteed to find the global minimizer and requires no initial guess. Furthermore, it allows us to find the most likely transition paths between any attractors of the system, not only between equilibrium points. The main difficulties in this approach are associated with high dimensionality and the anisotropic and unbounded speed function in the Hamilton-Jacobi equation. The investigator proposes several approaches for dealing with these problems. (B): The majority of methods for finding the sound speed inside the Earth (the seismic velocity) rely on vast computing resources and a good initial guess. The investigator and her colleagues propose an alternative approach that is computationally cheap and requires no initial guess. This approach is based on theoretical relationships between the time-migration velocity and the seismic velocity. The sound speed can be recovered by solving an elliptic partial differential equation with Cauchy data. Despite the fact that this problem is ill-posed, they have developed numerical techniques capable of solving it in the required interval of time. The investigator plans to continue research in this direction and incorporate methods from the field of computational stochastic processes. Many processes are modeled using stochastic differential equations, which are evolution laws that involve a random term (noise). Examples include small-scale processes in physics and chemistry such as chemical reactions and conformal changes in molecules. Other examples come from stochastically-modeled computer networks, pricing of financial securities, and the distribution of money in society. In the absence of noise a given system evolves toward one of its equilibrium states and stays there forever. But the presence of noise, even arbitrarily small, enables transitions between the equilibrium states. In many important cases these transitions are rare on the time-scale of the system but not rare on a human time-scale. This fact creates a need for techniques besides direct simulation for the study of these rare transitions. Such study will help to understand such phenomena as protein folding and mechanisms for gene expression. Producing an accurate image of the Earth?s interior is a challenging aspect of such things as oil recovery and earthquake analysis. First, seismic data always contain noise and the deeper the data come from the stronger its influence. Second, important and interesting geological features (e.g. oil deposition) typically occur where the subsurface structures are complicated and sound speeds vary severely in lateral (sideways) directions. In result, the fundamental inverse problem of determining the sound speeds of the Earth is ill-posed and exceedingly difficult. Yet determination of sound speed is crucial for accurate seismic imaging. The approach proposed by the investigator and her colleagues will lead to cheaper and more efficient methods for its determination.