Computational methods for the study of rare events

研究罕见事件的计算方法

• 批准号: 1217118
• 负责人: Maria Cameron
• 金额: $ 28.72万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-08-01 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1217118&HistoricalAwards=false
• 关键词: Computational; methods; study; rare; events

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.

研究问题涉及为(A)罕见反应事件和(B)地震建模开发计算工具。(A)：在用小噪声随机微分方程建模的系统中寻找最可能的转变路径的问题是非常困难的，因为以下几个问题：(1)系统亚稳态之间的转变很少，因此直接模拟非常困难；(2)高维；(3)昂贵的评估力；(4)多个局部最小化；(5)主要反应通道的温度依赖性。这位研究人员计划探索一种基于哈密尔顿-雅各比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。这种方法保证能找到全局极小值，并且不需要任何初始猜测。此外，它还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的过渡路径。这种方法的主要困难是与高维和哈密顿-雅可比方程中的各向异性和无界速度函数有关。研究人员提出了几种处理这些问题的方法。(B)：大多数求取地球内部声速(地震速度)的方法依赖于大量的计算资源和良好的初始猜测。这位研究人员和她的同事们提出了一种替代方法，该方法在计算上很便宜，而且不需要初始猜测。这种方法是基于时间偏移速度和地震速度之间的理论关系。用柯西数据求解椭圆型偏微分方程解声速可以恢复声速。尽管这个问题是不适定的，但他们已经发展出能够在所需的时间间隔内解决它的数值技术。研究人员计划继续在这个方向上进行研究，并将计算随机过程领域的方法纳入其中。许多过程都是使用随机微分方程来建模的，随机微分方程是涉及随机项(噪声)的演化规律。例子包括物理和化学中的小范围过程，如化学反应和分子的共形变化。其他例子来自随机建模的计算机网络、金融证券的定价和社会中的货币分配。在没有噪音的情况下，一个给定的系统会进化到它的一个平衡态，并永远保持在那里。但是，噪声的存在，即使是任意小的，也能使平衡态之间发生转变。在许多重要的情况下，这种转换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。这一事实为研究这些罕见的转变创造了除了直接模拟之外的技术需求。这样的研究将有助于理解蛋白质折叠等现象和基因表达的机制。制作准确的地球图像？S内部是石油回收和地震分析等方面的一个具有挑战性的方面。首先，地震数据总是包含噪声，数据来自的深度越深，其影响就越强。其次，重要和有趣的地质特征(例如石油沉积)通常发生在地下结构复杂且声速在横向(侧向)上变化严重的地方。因此，确定地球声速的基本逆问题是不适定的，而且极其困难。然而，确定声速对于准确的地震成像是至关重要的。这位研究人员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。