# Joint ANU/CSIRO Seminar on Computational and Inverse Problems: Celebrating Bob Anderssen's 80th birthday

Adjunct Professor Bob Anderssen has made substantial contributions in the areas of inverse and ill-posed problems, mathematical modelling and computational mathematics. He has collaborated widely within Australia and overseas and has supervised over 15 postgraduate students.

The main focus of this one-day meeting is on inverse problems, and will feature speakers who have collaborated with Bob Anderssen.

**Organising committee**

- Dr Frank de Hoog, CSIRO
- Professor Markus Hegland, MSI

Please note that there will be a conference dinner at 6.30pm, details will be confirmed soon.

## Sessions

Time | Session | |
---|---|---|

8.30am | Registration | |

9am | Commencement of session one | |

Undoing the effects of diffusion, and the hydrological rehabilitation of mine sites Tony Miller, Flinders University When a large mine reaches the end of its economic life miners are required to return the mining site to as near as possible its original pre-mining state. Some large mines in Australia are expected to reach this point in the next decade or so, and mining companies are starting to plan for site rehabilitation. Many open pit mines extend to depths below the water table, and so require continuous dewatering for dry operation. Over decades of a mining operation this dewatering can significantly affect the surrounding water table, as well as set in train hydrological processes that will persist after cessation of mining operations. Hydrological rehabilitation of a mine site is an important part of any overall mine site rehabilitation, but it presents many unique challenges. Firstly, the hydrological state of a mine site and its surrounds can only be inferred and is unlikely to be known with much certainty. At a more fundamental level, groundwater flow is a diffusion like process and so it is not easy to entirely “undo” the effects of historical dewatering in “finite” time. For these and other reasons hydrological rehabilitation shares many features of inverse problems. The best way to carry out such rehabilitation seems far from clear at the present time. In this talk we will discuss from a mathematical point of view some of factors that need to be considered in developing a long-term rehabilitation plan. | ||

An integrable discrete model for soil-water infiltration Dimetre Triadis, La Trobe University We have recently presented an integrable discrete model of one-dimensional soil water infiltration. It is based on the continuum model by Broadbridge and White: a nonlinear convection-diffusion equation with a nonlinear flux boundary condition at the surface. This is transformed to the Burgers equation with a time-dependent flux term by the hodograph transformation. Our discrete model preserves underlying integrability, and takes the form of a self-adaptive moving mesh scheme. The discretisation builds on linearisability of the Burgers equation, producing the linear diffusion equation. Naive discretisation of the linearised equation using the Euler scheme is often used in the theory of discrete integrable systems, but this does not necessarily produce a good numerical method for the original equation. Taking desirable properties of a numerical scheme into account, we propose an alternative discrete model that produces solutions with similar accuracy to direct computation on the original nonlinear equation, but with clear benefits regarding computational cost. | ||

Steady reaction and diffusion of petrol vapour in the soil – how I repurposed an elasticity solution of Bob Anderssen and others John Knight, Honorary Fellow, CSIRO Land and Water, Floreat Park, Perth In 2011 I was approached by Greg Davis to solve an unknown free surface problem of steady state two dimensional diffusion of petrol vapour upwards in the soil, reacting with oxygen diffusing down from the surface, with an impermeable boundary on part of the upper surface. I devised a transformation to convert the problem to a mixed boundary value problem for the Laplace equation for one variable on a known domain. I expected to solve this problem numerically using finite difference methods. I remembered that in the mid 1980s Bob Anderssen and Frank de Hoog had worked with Francis Rose on propagating crack problems, and had solved the resultant mixed boundary value problems for the Laplace equation on a strip. I was able to transform the petrol vapour problem and use their previous solution, which gives surprising simple explicit criteria for the buildup of petrol vapour under an obstruction on the surface. I will discuss some extensions. |

Time | Session | |
---|---|---|

10:15am | Morning tea |

Time | Session | |
---|---|---|

10:45 | Commencement of session two | |

Exponential sums and fast evaluation of a memory term William McLean, University of New South Wales Many evolution equations involve a memory term with a weakly singular kernel. In a numerical scheme with $N$ time steps, evaluating this term in the obvious way costs order $N^2$ operations. I will describe a simple, fast algorithm that reduces the cost to order $N \log N$ operations. The algorithm relies on approximating the kernel by a sum of negative exponentials. | ||

Approximation of completely monotone functions Rick Loy, Australian National University Completely monotone functions arise in linear viscoelasticity, with negative exponentials being the most familiar examples. We will consider several ways a general completely monotone function can be approximated by suitable positive linear combinations of negative exponentials, and how certain other common families of completely monotone functions fail in this respect. Both very old and very new results will be discussed. | ||

Feasible Design Spaces and Proximity Alerts for Potential Conflicts Between Aircraft Mark Westcott, CSIRO Data61, Canberra ACT Managing airspace requires, among other things, models of how aircraft might come into proximity. These have two components: a kinematic description of the aircraft in flight and a set of rules for deciding whether proximity has occurred. In this talk we focus on the rules, so keep the kinematics as simple as possible by adopting a crossing track model. A set of proximity rules is then chosen, and these are used to partition a parameter space into regions, some of which can generate conflict between the aircraft. Our rules allow for both spatial and temporal conflicts, which is not commonly considered. The results are applicable hierarchically, from strategic planning to real-time adaption required during in-flight operations. | ||

Using the Bootstrap in Generalized Regression Estimation Alan Welsh, Australian National University Regression adjustments are widely used for adjusting the sample mean of a variable of interest to account for deviations of the means of related auxiliary variables from their known population values. The adjustments produce estimators with variances smaller than that of the original sample mean. The method has a long history in the survey literature, and is closely related to covariance analysis in designed experiments and the control variates method used for variance reduction in Monte Carlo studies. |

Time | Session | |
---|---|---|

12:30pm | Lunch |

Time | Session | |
---|---|---|

1:30pm | Commencement of session three | |

Vernalization: how does the cold of winter repress FLC expression? Jean Finnegan, CSIRO Agriculture and Food Vernalization refers to the acceleration of flowering that occurs in some plants following exposure to prolonged periods of low temperatures (winter).This response evolved in plants growing in regions with long harsh winters to ensure that plants flower in the spring when the weather is suitable for pollination and seed development. While plants often use environmental cues to trigger flowering and other processes, the properties of vernalization indicate that it is mediated by an epigenetic mechanism. Plants not only remember that they have been exposed to winter, even after the weather warms up, but they can measure the duration of winter; the longer the period of cold weather, the earlier the plants flower. The memory of winter is not transmitted to the next generation and the progeny of a vernalized plant must be exposed to low temperatures to flower early. The key gene regulating vernalization in Arabidopsis thaliana is FLOWERING LOCUS C (FLC), a repressor of flowering. FLC expression is repressed by low temperatures, but the mechanism leading to the initial decrease in FLC transcription remains a mystery. In a collaboration with Bob Anderssen, we proposed that the drop in temperature first causes a change in the topology of the chromatin polymer encompassing the FLC gene, and this in turn results in the repression of FLC transcription. | ||

TBA Ming-Bo Wang CSIRO TBA | ||

Calcium waves on the surface of an amphibian egg Bronwyn Hajek, University of South Australia When an amphibian egg is fertilised, a wave of calcium ions travels around the surface of the egg to help prevent the entry of multiple sperm. This process can be described with a nonlinear reaction-diffusion equation with a cubic reaction term. Here, we present the first analytic solutions to this 30 year old problem, demonstrating various observed phenomena, including waves and spirals. | ||

Symmetries and solutions of the non-autonomous von Bertalanffy equation Maureen Edwards, University of Wollongong Autonomous ODE models for microbial growth in a closed environment give solutions that can only grow or decay monotonically or asymptote. These models can never capture the mortality phase in a typical microbial growth curve. A generalisation \[ \frac{dN}{dt} = \alpha(t) N^\beta - a(t)N^b + \psi(t),\quad N=N(t)\] of the non-autonomous von Bertalanffy equation has been proposed as a model of the interaction of the current size of a population with the environment in which it is living. The relationship between the introduced non-autonomous terms is explored through Lie symmetry analysis. Constraints on the functions $\alpha(t)$, $a(t)$ and $\psi(t)$ which allow for the existence of nontrivial symmetries are identified, and some new closed form solutions are constructed. |

Time | Session | |
---|---|---|

3:10pm | Afternoon tea |

Time | Session | |
---|---|---|

3:40pm | Commencement of session four | |

Encouraging More Australians to Choose Maths Janine McIntosh, Australian Mathematical Sciences Institute (AMSI) As well as his mathematical pursuits Bob Anderssen supports the mathematics pipeline in myriad ways. One of the key roles Bob plays is as Chair of the AMSI Education Advisory Committee. In this session I will outline the nature of this work, and describe one of our most successful projects: CHOOSEMATHS. | ||

Inverse problems: Bob Anderssen’s impact Tony Jakeman, Fenner School of Environment and Society, Australian National University Bob Anderssen has made many substantial contributions to the solution of inverse problems not just those with which I was most familiar in the early days - those related to numerical differentiation and solving ill-posed integral equations. This talk is more of a personal refection on the impact that Bob has had on my research journey. It begins with our work on solving Abel integral equations in the field of stereology and the lessons obtained there for future work in hydrology, environmental modelling and integrated assessment. It is well-known that ill-posed inverse problems are solved by making, sometimes heroic and often untested, assumptions – for example simplifying the model, regularizing the solution or constraining the solution space by assuming priors in a Bayesian probabilistic framework. In hydrologic and environmental model representations most people now get it that they need to constrain their representations if they want them to be identifiable. But seldom do we report on the limitations of our assumptions or compare them with alternatives. So we have got to first base. In integrated assessment (IA) problems, however, the thinking is much more pragmatic. IA is the metadiscipline of bringing knowledge together to assess a policy problem. Think Murray-Darling Basin Plan and promulgating water diversion limits to irrigation that are sustainable for ecosystems and local communities. This – as with most confronting water resource issues - is a wicked problem where there is no universally agreed problem formulation, results are contested and the socio-environmental systems being modelled are fraught with pervasive uncertainties. Putting all the knowledge together to support solutions to wicked problems is also an inverse problem. The talk will indicate how IA goes about addressing such monumental tasks. | ||

TBA Ian Sloan University of New South Wales TBA | ||

5pm | Finish |

Registration is free and open to the public.

Seminar Room 1.33 & 1.37, Building #145, Science Road, The Australian National University

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