Computation with Uncertainty

研究者已经开发了基于Mori-Zwanzig形式主义的不可逆统计力学的非线性问题的降维和均匀化工具。他计划在以下问题中使用这些方法，如果需要的话，可以扩展它们：1.减少有效变量的数量（实际上，执行一个approximatemanalization）在评估的建议分布在一个sequentialMonte-Carlo实现的贝叶斯过滤器的数据同化. 2.制定有效的块蒙特-卡罗方法，应用于生物杂聚物的运动和蒙特-卡罗实现的卡伦-西曼齐克重整化群的涡旋占主导地位的相变。3.推导多尺度波传播问题的有效方程。该提案有几个组成部分。连接线程是使用的方法，该演示者以前开发的减少变量的数量在一个复杂的问题，同时保持其显着的统计特征完好无损。第一个组成部分是过滤/数据同化问题。假设你有一个具有随机性的复杂系统，你对它未来的行为进行了预测（例如，你做了一个天气预报）。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）。 您如何将这些新信息纳入您的预测？原则上，这可以通过在计算机上创建系统的副本集合，然后在新观察的帮助下修改副本的分布来完成，但这在实践中通常过于昂贵。研究人员以前的工作使其有可能在原则上降低成本的算法由一个相当大的因素，但它仍然是未知的，这种减少是否是顺应性，使他们实用。另一个组成部分是研究无序杂聚物的相互吸引。生物学家已经提出，当特定分子在每个分子上的分布的概率密度在统计意义上符合时，即使平均相互作用为零，这种异质聚合物也会相互吸引，例如当免疫系统识别入侵病毒时就会发生这种情况。要在计算机上检验这个假设，需要非常庞大的计算任务，研究者认为他可以简化这些任务。另一个组成部分来自理论物理学。涡旋非束缚跃迁在许多问题（如薄膜理论）中是重要的，但更重要的是它是量子场论中的一个基本范式。在早期的工作与O。哈尔德研究者已经表明，目前的理论是不完整的，并指出了几个悖论。要理解正在发生的事情需要一个计算，这个计算到目前为止太大了，无法成功完成，但新的方法打开了完成它的可能性，研究者建议尝试。

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.