My PhD research interests involved in applied mathematics and specifically verified computations. I investigated systems of ordinary differential equations under uncertainty. Ecological modeling and risk assessment is one such area that could make use of such research. Current practice in this area is to use Monte Carlo simulations or other such sampling techniques to determine a most probable outcome for an environmental model, and my research aims to validate this result by finding the "worst case scenario" results in addition to a mathematically guaranteed probability distribution for all results in between. My research can also be applied to chemical engineering; many food web models are analogous to chemical kinetics, for example. Other ODEs similar to those I am studying can arise in the study of transport phenomena, process control, population epidemiology, and physiology.
In my research, I basically employed three different items for describing uncertainty. The simplest is an interval, which is simply a set of values bounded by some minimum and maximum values. There are a number of textbooks available that describe the arithmetic and analysis of intervals. One purpose of interval methods is to rigorously determine mathematical solutions. Interval arithmetic is guaranteed to enclose the solution of a mathematical problem. As a very simple example, if I wanted to add a number between 1 and 2 to another number between 3 and 4, then interval arithmetic tells me the sum will be between 4 and 6. Sometimes interval arithmetic will result in significant overestimation for a result. As a simple example, if I wanted to divide a number between 2 and 3 by that same number minus 1, interval arithmetic would tell me the quotient is a number between 1 and 3. This is because interval arithmetic uses the "whole interval" each time it turns up in a function! The smallest result is 2 divided by (3-1), and the largest result is 3 divided by (2-1). Of course, if I know the number is the same in both cases, I should be able to determine that the real answer is between 1.5 and 2.
A Taylor model is a (real-valued) Taylor polynomial and an interval remainder bound. Taylor model methods can reduce the overestimation of interval methods by doing as much math as possible with the polynomial term and putting any errors in rounding into the remainder term. For the expression x/(x-1) explored above, we can determine the Taylor polynomial that quite tightly approximates this function on the interval from 2 to 3, and we can compute the maximum possible error in this approximation, to come up with a result that is quite close to the real answer above.
A probability box, or a p-box, is a set of bounds on a cumulative probability distribution for a random variable. This tool provides more information on an uncertain value than just its minimum and maximum, the way an interval does. P-boxes can be constructed from any available information on a random variable -- including its probability distribution, if it is precisely known. If it's not known precisely, but its distribution type or any measure is available, a p-box can be constructed from that information. P-box arithmetic is generally employed the same way as interval arithmetic, which means it is possible to overestimate a result -- and it is possible to use Taylor models to make those results better. Below are three examples of p-boxes: from left to right, an interval, a variable with known minimum, median, and maximum, and a bound on an imprecise uniform distribution.
Solving ODEs numerically welcomes a whole set of errors. The numerical method that is probably most familiar, Euler's method, is an approximation whose error grows as the chosen step size does, often at unacceptable rates for some ODE systems. Imagine having to use arithmetic that could possibly make that error grow even faster! Interval methods for solving ODEs certainly don't use such a simple (and not very useful) algorithm to provide guaranteed bounds on a variable over time. After all, telling you the answer is between negative and positive infinity isn't particularly useful.
Specifically, I employed Taylor model methods to solve initial value problems consisting of ODEs. Other interval methods (also sometimes called verified methods) have been used to solve such equations, often considering uncertainty in initial values only. For example, the interval IVP solver VNODE uses an interval Taylor series (ITS) to propagate interval bounds on state variables from one time step to the next. There are a number of interval ODE solvers available. The Verified Solver for Parametric ODEs, or VSPODE, is the software developed in our research group. Like VNODE, it uses ITS to verify the existence and uniqueness of a solution from one time step to the next. Unlike VNODE, it uses Taylor models to propagate bounds from one time step to the next. And because we're using Taylor models, we have a polynomial function that describes the variable at every time step. This polynomial can be evaluated in terms of intervals to give the minimum and maximum value at this time, or in terms of p-boxes, to give bounds on the probability distribution of the variable at this time.