# Dr James Nichols

## 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

## Publications

- Cohen, A, Dahmen, W, Mula, O et al. 2022, 'Nonlinear Reduced Models for State and Parameter Estimation', SIAM/ASA Journal of Uncertainty Quantification, vol. 10, no. 1, pp. 227-267.
- Nichols, J 2022, 'Coarse reduced model selection for nonlinear state estimation', ANZIAM Journal, vol. 62, pp. C192 -C207.
- Angstmann, C, Erickson, A, Henry, B et al. 2021, 'A general framework for fractional order compartment models', SIAM Review, vol. 63, no. 2, pp. 375-392.
- Cohen, A, Dahmen, W, Devore, R et al. 2020, 'Optimal reduced model algorithms for data-based state estimation', SIAM Journal of Numerical Analysis, vol. 58, no. 6, pp. 3355-3381.
- Cohen, A, Dahmen, W, Devore, R et al. 2020, 'Reduced Basis Greedy Selection Using Random Training Sets', ESAIM: Mathematical Modelling and Numerical Analysis, vol. 54, no. 5, pp. 1509-1524.
- Nichols, J, Chan, H & Baker, M 2019, 'Machine learning: applications of artificial intelligence to imaging and diagnosis', Biophysical Reviews, vol. 11, no. 1, pp. 111-118.
- Nichols, J, Henry, B & Angstmann, C 2018, 'Subdiffusive discrete time random walks via Monte Carlo and subordination', Journal of Computational Physics, vol. 372, pp. 373-384.
- Binev, P, Cohen, A, Mula, O et al. 2018, 'Greedy Algorithms for Optimal Measurements Selection in State Estimation Using Reduced Models', SIAM/ASA Journal of Uncertainty Quantification, vol. 6, no. 3, pp. 1101-1126.
- Angstmann, C, Erickson, A, Henry, B et al. 2017, 'Fractional Order Compartment Models', SIAM Journal on Applied Mathematics, vol. 77, no. 2, pp. 430-446.
- Angstmann, C, Donnelly, I, Henry, B et al. 2016, 'From stochastic processes to numerical methods: A new scheme for solving reaction subdiffusion fractional partial differential equations', Journal of Computational Physics, vol. 307, pp. 508-534.
- Graham, I, Kuo, F, Nichols, J et al. 2015, 'Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients', Numerische Mathematik, vol. 131, no. 2, pp. 329-368.
- Angstmann, C, Donnelly, I, Henry, B et al. 2015, 'A discrete time random walk model for anomalous diffusion', Journal of Computational Physics, vol. 293, pp. 53-69.
- Savkovic, B, Nichols, J, Birkett, D et al. 2014, 'A Quantitative Comparison of Anti-HIV Gene Therapy Delivered to Hematopoietic Stem Cells versus CD4+ T Cells', PLOS Computational Biology, vol. 10, no. 6, pp. 1-16.
- Nichols, J & Kuo, F 2014, 'Fast CBC construction of randomly shifted lattice rules achieving O(nâˆ’1+Î´) convergence for unbounded integrands over Rs in weighted spaces with POD weights', Journal of Complexity, vol. 30, no. 4, pp. 444-468.