Computation with Uncertainty

研究人员基于不可逆统计力学的Mori-zwanzig形式，开发了用于非线性问题降维和齐化的工具。他计划在以下问题中使用这些方法，如果需要的话，还可以扩展这些方法：1.在数据同化的贝叶斯过滤器的顺序蒙特卡罗实现中，减少有效变量的数量(实际上，执行近似的边界化)来评估建议分布。建立了有效的块蒙特卡罗方法，并应用于生物杂化聚合物的运动和旋涡主导相变的Callen-Symanzik重整化群的蒙特卡罗实现。3.推导了多尺度波传播问题的有效方程。该提案有几个组成部分。连接的主线是使用研究人员以前开发的方法，以减少复杂问题中的变量数量，同时保持其显著的统计特征不变。第一个组成部分是过滤/数据同化问题。假设你有一个带有一些随机性的复杂系统，你对它在未来某个时间的行为进行了预测(例如，你做了一个天气预报)。然后新的信息进来了(例如，你打开窗户，你看到的与你的预报相反，正在下雨)。你如何将这些新信息纳入你的预报中？原则上，这可以通过在计算机上创建系统的副本集合，然后在新观测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。研究人员以前的工作在原则上使这类算法的成本降低了相当大的一倍，但目前仍不清楚这种减少是否足以使它们变得实用。研究人员提出要找出答案。另一个组成部分是研究无序杂聚体之间的相互吸引。生物学家提出，当每个杂多体上特定分子的分布概率密度符合统计意义时，即使平均相互作用为零时，这种杂多体也会相互吸引，例如当免疫系统识别出入侵的病毒时就会发生这种情况。在计算机上验证这一假设会导致非常大的计算任务，研究人员认为他可以简化这一任务。另一个组成部分来自理论物理。涡旋非束缚跃迁在许多问题中都很重要(例如在薄膜理论中)，但更重要的是它是量子场论中的一个基本范式。在与O.Hald的早期工作中，研究人员已经证明了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一次计算，到目前为止，计算太大了，无法成功完成，但新的方法打开了完成它的可能性，研究人员提议尝试一下。

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.