Final Report Summary - AMSTAT (Problems at the Applied Mathematics-Statistics Interface)
A mathematical model for an experiment may be viewed as a set of equations which relate inputs to outputs. Inputs represent variables which can be adjusted before the experiment takes place; outputs represent quantities which can be measured as a result of the experiment. A forward mathematical problem is to predict the output of an experiment from a given input. The inverse mathematical problem is to use the model to make inferences about input(s) which would result in a given measured output.
An example concerns a mathematical model for weather prediction. An important input to the model is the initial velocity field in the earth's atmosphere. A natural output would be measurements of temperature at various locations on the earths surface. Since the atmospheric velocity is not easily measurable everywhere, the problem of inferring it from temperature (and other) measurements is particularly important. Accurate inference enables more accurate weather forecasts with longer prediction horizons.
In many inverse problems, including weather prediction, the measured data is noisy and the mathematical model is imperfect. It is then very important to quantify the uncertainty inherent in any inferences made as part of the solution to the inverse problem. The research has been focussed on the development of a mathematical and computational framework for the study of such statistical inverse problems, where ideas from probability and differential equations are used in conjunction to both clearly formulate, and efficiently compute, the solutions to, such problems.
An example concerns a mathematical model for weather prediction. An important input to the model is the initial velocity field in the earth's atmosphere. A natural output would be measurements of temperature at various locations on the earths surface. Since the atmospheric velocity is not easily measurable everywhere, the problem of inferring it from temperature (and other) measurements is particularly important. Accurate inference enables more accurate weather forecasts with longer prediction horizons.
In many inverse problems, including weather prediction, the measured data is noisy and the mathematical model is imperfect. It is then very important to quantify the uncertainty inherent in any inferences made as part of the solution to the inverse problem. The research has been focussed on the development of a mathematical and computational framework for the study of such statistical inverse problems, where ideas from probability and differential equations are used in conjunction to both clearly formulate, and efficiently compute, the solutions to, such problems.