Dr James Nichols

PhD in Mathematics
Lecturer
ANU College of Science

Areas of expertise

  • Numerical And Computational Mathematics 4903
  • Approximation Theory And Asymptotic Methods 010201
  • Numerical Solution Of Differential And Integral Equations 010302
  • Probability Theory 010404
  • Stochastic Analysis And Modelling 010406
  • Biological Mathematics 010202

Research interests

I am interested in numerical simulation, approximation, and statistical learning techniques. In particular I study the intersection of high dimensional approximation theory and high dimensional statistics. In a sense some of what I do is computational statistics. I like to develop algorithms and figure out where they work, why they work, and how they work (mathematically speaking, of course).

I have broad experience in the mathematical and computational disciplines. I started out as a quantitative analyst at a major investment bank. Following that my PhD was in quasi-Monte Carlo, which is a collection of methods for generating sample points in extremely high dimensional domains (a common problem in stochastic simulation, *ahem* as encountered for example at an investment bank...).

For a while I drifted to other numerical methods for stochastic processes, in particular non-Markovian random walks. That's when I got interested in applications to simulation of biological processes. The random walks of small particles in electrostatically trapping environments, for example pathogens in mucosal tissue, typically show non-Markovian behaviour. This can be suprisingly computationally intensive to simulate.

Since then I've returned to my true calling, high dimensions. In particular, approximation in high dimensions, and the variety of links to statistical learning problems in high dimension. In general, high dimensional problems are intractably hard, however most high dimensional problems we encounter hide some sort of structure that makes them emminently tractable to approximation algorithms.

Many "high dimensional" functions are really quite easy to integrate or approximate because of the high degree of smoothness they exhibit. This is typical for example of functions that come from mathematical finance or from parametric PDE problems. The trick is showing exactly when such desirable properties hold, and when they can be exploited.

A similar thing happens in high dimensional statistics. If we are given data with many attributes (e.g. personal data with height, weight, eye color, favourite Bee Gees song and many more fields...) that data is already high dimensional. However! There are dependencies. Correlations. There might be nonlinear correlations. The data might live on a very low dimensional manifold.

Biography

Before joining ANU in 2020 I was a postdoc at Sorbonne Universite. I completed my PhD in 2014 at UNSW. I worked for some years as a quant at Macquarie Bank.

 

Publications

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Updated:  20 May 2024 / Responsible Officer:  Director (Research Services Division) / Page Contact:  Researchers