Computation with Uncertainty

研究者开发了基于不可逆统计力学的Mori-Zwanzig形式的非线性问题降维和均匀化工具。他计划在以下问题中使用这些方法，并在需要时对其进行扩展：减少有效变量的数量（实际上，执行近似的边际化），在数据同化的贝叶斯过滤器的顺序蒙特卡罗实现中评估建议分布。制定有效的块蒙特卡罗方法，应用于生物异质聚合物的运动，以及用于涡流主导相变的Callen-Symanzik重整化群的蒙特卡罗实现。3. 导出多尺度波传播问题的有效方程。该提案有几个组成部分。连接的线程是使用研究者以前开发的方法，以减少复杂问题中的变量数量，同时保留其显著的统计特征。第一个部分是滤波/数据同化问题。假设您有一个具有一些随机性的复杂系统，并且您对其未来时间的行为进行预测（例如，您进行天气预报）。然后新的信息进来（例如，你打开窗户，你看到与你的预测相反，下雨了）。你如何将这些新信息纳入你的预测？原则上，这可以通过在计算机上创建系统的副本集合，然后在新的观察结果的帮助下修改副本的分布来完成，但在实践中，这通常太昂贵了。研究者之前的工作使得原则上有可能将这种算法的成本降低一个相当大的因素，但这种降低是否足以使它们付诸实践仍然未知。调查员打算找出答案。另一个组成部分是无序异聚物的相互吸引的研究。生物学家提出，当特定分子分布的概率密度符合统计意义时，即使平均相互作用为零，这种异聚体也会相互吸引，例如当免疫系统识别入侵病毒时。为了在计算机上验证这一假设，研究者认为他可以简化大量的计算任务。另一个因素来自理论物理学。涡旋解束缚跃迁在许多问题（如薄膜理论）中都很重要，但更重要的是它是量子场论的一个基本范式。在与霍尔德的早期工作中，研究者已经表明，目前的理论是不完整的，并指出了几个悖论。要了解正在发生的事情需要进行计算，到目前为止，这一计算规模太大，无法成功完成，但新方法为完成计算提供了可能性，研究者建议尝试一下。

The investigator has developed tools for dimensional reduction and homogenization in nonlinearproblems based on the Mori-Zwanzig formalism of irreversible statistical mechanics.He plans to use these methods, and if needed to extend them, in the following problems:1. Reduce the number of effective variables (in effect, perform an approximatemarginalization) in the evaluation of the proposal distribution in a sequentialMonte-Carlo implementation of the Bayesian filter for data assimilation.2. Formulate effective block Monte-Carlo methods, with applications to the motion of biological heteropolymers and to a Monte-Carlo implementation ofthe Callen-Symanzik renormalization group for vortex-dominated phase transitions. 3. Derive effective equations for multi-scale wave propagation problems. The proposal has several components. The connecting thread is the use of methods the investigatorhas previously developed for reducing the number of variables in a complex problemwhile leaving intact its salient statistical features. The first component is the filtering/data assimilation problem. Suppose you have acomplex system with some randomness and you make a forecast about its behaviorat a future time ( for example, you make a weather forecast). Then new information comesin (for example, you open the window and you see that contrary to your forecast, it's raining). How do you incorporate this new information into your forecast?In principle this can be done by a creating on the computer a collection of replicas of the system and then modifying the distribution of the replicas with the help of thenew observations, but this is in general far too expensive in practice. The investigator's previous work makes it possible in principle to reduce the cost of such algorithmsby a considerable factor, but it is still unknown whether this reduction is sufficientto make them practical. The investigator proposes to find out.Another component is the study of the mutual attraction of disorderedheteropolymers. It has been proposed by biologists that such heteropolymersattract each other when the probability densities of the distribution of particular molecules on each fit in a statistical sense, even when the averaged interaction is zero, and that this happens for example when the immune system identifiesinvading viruses. To check this hypothesis on the computer leads to very large computational tasks which the investigator thinks he can simplify.A further component comes from theoretical physics. The vortex unbinding transition isimportant in many problems ( for example in the theory of thin films) but moreimportantly it is a basic paradigm in quantum field theory. In earlier work with O. Haldthe investigator has shown that present theory is incomplete and pointed out several paradoxes.To understand what is going on requires a calculation which has been until now too largeto be successfully completed, but the new methods open the possibility that it canbe brought to completion, and the investigator proposes to try.