Inverse Problems

"Can you hear the shape of a drum?"

Mathematical modelling always involves constructing and applying forward models. These involve sets of equations that have one or more of the following: coefficients, initial conditions or boundary conditions. Let us call these items the input data. Input data are often functions of position or time, and not just constants. The solutions of the equations model the state of some physical or other system. The aim of modelling will usually be to find the solution or approximate solution of the model, thus predicting the state of the system. Finding the solution given the equations and the input data is the forward problem.

An inverse problem arises when there is incomplete knowledge of the input data and there may be some information about the state. The available information is usually corrupted by experimental error and the model itself will be an approximation of reality even when the best possible input data is available. This implies that the naïve formulation of the inverse problem is simultaneously under-determined and over-determined. In other words, inverse problems are generally ill-posed. The simplest inverse problems are essentially interpolation problems; the most difficult arise in areas such as weather forecasting, oil production, and medical imaging. Indeed, whenever there is a forward problem there is a corresponding inverse problem.

Occasionally an inverse problem can be solved in a direct way that makes it appear similar to a forward problem. This tends to happen when a closed form analytical solution can be found. Generally, however, this cannot be done. Standard approaches to inverse problems involve the use of ­least-squares optimization or a statistical reformulation. Simple least-squares formulations choose the input data to minimize the difference between the observations and the predictions. Least-squares removes the over-determination caused by error, but not the under-determination. Extra assumptions are needed, such as looking for solutions in a restricted space or, more usually, by adding penalty terms to the least squares objective function. This is called regularization. In situations where there is much data and the greatest difficulty arises from experimental error the least-squares approach can be satisfactory. As the amount of input data and state data decreases it becomes more important to use a statistical formulation. Statistical formulations may still lead to a least-squares problem but the most general solution of the inverse problem will need some form of Monte-Carlo simulation. Inverse problems are thus about the quantification and control of uncertainty.

Inverse problems are of immense practical importance and involve many branches of applied mathematics such as partial differential equations, Green's functions. variational calculus and stochastic processes. There are close connections to Bayesian statistics, statistical physics and quantum field theory. Recent research increasingly looks at inverse problems in an infinite dimensional setting.

People working in this area within OCIAM are

• Dr Chris Farmer

• Dr Ben Hambly

• Dr Sam Howison

• Dr Pat McSharry

• Dr Lenny Smith

For detailed information see

• H.W. Engl, M. Hanke, and A. Neubauer: Regularization of Inverse Problems, Kluwer, Dordrecht, 1996.

• C.L. Farmer: Geological modelling and reservoir simulation, in: Mathematical Methods and Modeling in Hydrocarbon Exploration and Production, A.Iske, T.Randen (eds.), Springer-Verlag, Heidelberg, 2005, pp 119-212.

• A.Tarantola: Inverse Problem Theory and Methods for Model Parameter Estimation,SIAM, Philadelphia, 2005.

• J.P. Kaipio and E.Somersalo: Statistical and Computational Inverse Problems, Springer, Berlin, 2004.