Department of Statistics, The Ohio State University
Statistics and Biostatistics Colloquium Series

A forward problem is the task of finding the solution 'u' of a differential equation, given a set of inputs 'Phi' (coefficients, boundary conditions). The problem of determination of 'Phi' from 'u' may be regarded as "inverse" to the one described above. Inverse problems arise in numerous fields such as general acoustics, earth sciences, algorithm development, medical imaging, etc. Inverse problems are statistical problems. The purpose is to estimate 'Phi' having (in general) noisy, sparse and sometimes only partial measurements of the solution 'u.' A robust solution to an inverse problem can be obtained by introducing prior information on 'Phi' and modeling the measurement error.
The application we are currently working on involves estimating water velocities and mixing coefficients in a 2 km deep, rectangular region in the South Atlantic Ocean. Partial and sparse measurements of tracer concentrations (salinity, oxygen, etc.) are available. The data are filtered to eliminate outliers, then interpolated to the nearest points on a regular lattice and restricted to thin neutral density layers. The connection between velocities, diffusion coefficients, boundary conditions and tracer concentrations is made via a 3D advection-diffusion equation and a geostrophic flow model. The (un-normalized) posterior density of the parameters conditionally on the data is summarized using Markov chain Monte Carlo techniques. We reconstruct the tracer fields as well, thus, for regions where no data was available, concentrations are now estimated in a manner that is consistent with physical principles.

Meet the speaker in Room 212 Cockins Hall at 4:30 p.m. Refreshments will be served.