Whilst direct formulations consist of determining the effect of a given cause, in inverse formulations the situation is completely, or partially reversed. The interest is into the research of inverse problems for partial differential equations governing phenomena in fluid flow, elasticity, acoustics, heat transfer, mechanics of aerosols, etc. Typical practical applications relate to flows in porous media, heat conduction in materials, tomographic scan of objects, thermal barrier coatings, heat exchangers, corrosion, etc. The aims in this area include investigating the existence, uniqueness and stability of the solution to the problem that mathematically models a physical phenomenon under investigation, and developing new convergent, stable and robust algorithms for obtaining the desired solution. The analyses may concern inverse boundary value problems, inverse initial value problems, parameter identification, inverse geometry and source determination problems.
Keywords: Applied Mathematics, Inverse problems, Regularization, Ill-Posed Problems, Stability, Existence and Uniqueness
Polymers and Industrial Mathematics
Research in the Polymers and Industrial Mathematics group focuses on the mechanics of polymers and other complex fluids, free-surface flows and inverse problems. We are also concerned with the development and implementation of novel numerical and computational solution methods for both ordinary and partial differential equations, from fundamental aspects (the theoretical analysis of numerical methods) to problem-specific aspects (the design, development and practical implementation of novel algorithms). Within the polymer area, we conduct fundamental research into fluids that have a complex microstructure, such as polymer melts and solutions and colloidal dispersions.
Our research combines methods from molecular physics and continuum mechanics to develop multiscale models that link together the microscale motion of individual molecules to the flow behaviour of the bulk material. An important class of industrial flow problems are those involving free surfaces, such as in inkjet printing, film coating and bubble growth in polymeric foams. We also work on a diverse range of inverse problems in heat transfer, porous media, fluid and solid mechanics, acoustics and medicine. This is a strongly interdisciplinary subject and much of our research involves collaborations with independent research groups in science and engineering departments both at Leeds and worldwide, as well as with industry.