Computation with Uncertainty

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

The investigator has developed tools for dimensional reduction and homogenization in nonlinear problems 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 approximate marginalization) in the evaluation of the proposal distribution in a sequential Monte-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 of the 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 investigator has previously developed for reducing the number of variables in a complex problem while leaving intact its salient statistical features. The first component is the filtering/data assimilation problem. Suppose you have a complex system with some randomness and you make a forecast about its behavior at a future time ( for example, you make a weather forecast). Then new information comes in (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 the new 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 algorithms by a considerable factor, but it is still unknown whether this reduction is sufficient to make them practical. The investigator proposes to find out. Another component is the study of the mutual attraction of disordered heteropolymers. It has been proposed by biologists that such heteropolymers attract 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 identifies invading 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 is important in many problems ( for example in the theory of thin films) but more importantly it is a basic paradigm in quantum field theory. In earlier work with O. Hald the 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 large to be successfully completed, but the new methods open the possibility that it can be brought to completion, and the investigator proposes to try.