Dr. Michael L. Larsen – Research Overview

Simulating Discrete Spatial Systems with Known Properties via Simple Computational Techniques

Basic Background

This research project exists mainly in a supporting role to Dr. Larsen’s other research investigations.

Perusing the research pages of this site will reveal that many of the problems Dr. Larsen works on involve the relative spatial positions of aerosol particles, cloud droplets, and raindrops. Collectively, these items can be viewed as atmospheric particulates of various forms.

Generally speaking, the particulate physical size is much smaller than the mean distance between them. This property is used extensively and – with some caveats – allows Dr. Larsen to invoke mathematical tools from point process theory, geometrical probability, and a few other formal mathematical fields.

When describing a real collection of cloud particles (for example), Dr. Larsen uses some mathematical tool to describe the statistical structure of the spatial positions. Using the statistical approach instead of cataloging the individual positions of each particle would be necessary if any of the results are to be applied in a realistic scenario. (One can’t be expected to know exactly where each cloud particle is to make some basic predictions about raindrop formation. However, knowing something about the cloud drop spatial positions can be very helpful.)

So what’s the Problem?

Given a collection of particle spatial position, Dr. Larsen is able to calculate some statistical properties as alluded to above. However, it would prove very useful to be able to specify the statistical property itself and use that to simulate the positions of particles that have that statistical property. (Something that physicists generally call “the inverse problem”.)

Inverse problems are generally more complicated than the direct problem. This is due to the fact that inverse problems nearly always allow non-unique solutions.

A simple example may make the above statement clearer. In my Fall 2007 Physical Science course, I see that I have three people enrolled that have the first name of Nicholas. So, one property of my class-list is that the statistical mode of first names (or most common result) is Nicholas. However, there are an infinite amount of ways I could make up class lists where the statistical mode is Nicholas. One such way is to make up a list of 20 names, each of which having Nicholas as a first name. Another is making a list of 498 unique first names and 2 Nicholases. Both share the same statistical property as my actual class list. However, neither is likely to share many other statistical properties of my actual class list.

One needs not only to make the distribution from a given statistical property, but choose a specific and complete enough statistical property so that “everything important” is retained in the simulated system. (what is important? This changes from problem to problem. Basically, anything related to the process. But what if you don’t know what is important for the process? Then you have difficulties.)

Dr. Larsen has been working on finding “good” statistical properties for applications in atmospheric science. However, actually doing the inverse problem and generating systems that have these same properties is difficult.

What is Dr. Larsen doing to try and Solve the Problem?

Although the current makeshift solution lacks elegance, Dr. Larsen’s current approach can best be described as “brute-force”. The computational process is beyond the scope of the introduction here, but boils down to so-called Monte-Carlo simulations in conjunction with Markov processes used in a “guess and check” manner. This method is shockingly inefficient, but eventually gets the job done.

Dr. Larsen would like to explore other computational techniques that could be used to generate distributions with the very specific statistical properties he seeks to emulate. Failing such a novel approach, he’d at least like to parallelize the code so that a distributed computer network or multiple-processor machines could at least chew on parts of the problem at a time.

In and of itself, this problem doesn’t have a lot of physics but paves the way to answer many important atmospheric physics questions.

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Send a comment to: larsenml@unk.edu

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 This web site courtesy of the Department
 of Physics and Physical Science
In cooperation with the University of Nebraska at Kearney

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