Inverse problems are problems where the solutions are nearly always indirectly related to the available data and where causes are determined for desired or observed effects. The problems are often ill-posed in that small changes in the data can produce large effects in the solution. Furthermore, questions of whether a solution that corresponds to likely noisy data can exist and how many and how different solutions there may be that correspond to partial data sets need to be considered. To overcome these difficulties, special mathematical treatments need to be utilized. Dynaflow has developed proprietary numerical algorithms and software that implement these treatments to enable obtaining a stable and physically accurate solution for a number of applications. These have been incorporated into the following products: 

•  ABS: Acoustic Bubble Spectrometer^®© :  is an acoustics-based device that measures bubble size distributions and void fraction of bubbles in liquids. The device extracts the bubble population from acoustical measurements made at several frequencies by solving two Fredholm Integral Equations of the first kind. These equations are ill-posed and are a challenge to solve - especially when the data has noise. A novel algorithm has been developed to accurately solve these equations using a constrained optimization technique.

•  DynaSim^© :  is a user-friendly ship dynamics simulator that incorporates novel Systems Identification Techniques to determine ship maneuvering characteristics. The Systems Identification Techniques enable the simulator to fine tune the mathematical model to a particular ship's evolving characteristics thus achieving accurate predictions of the motion of that ship. This identification procedure is an inverse problem that uses as input already recorded data about the ship that is available as part of IMO guidelines (turning circle and zigzag data) or observations on-board of the ship response to maneuvering sequences. A multi-objective identification approach incorporating the concept of Pareto optimality has been developed to solve this inverse problem.

•  DynaEIT^© : is PC-based computed electric impedance tomography that is a powerful imaging technique with a wide variety of applications requiring visualization of inaccessible objects or features such as multiphase flows, medical imaging, nondestructive evaluation of structures, etc. This technology seeks the distribution of conductivity inside a container by applying specified currents (or voltages) at some parts of the container surface, and performing measurements of the voltage (or current) at some other parts. Several gradient-based minimization scheme such as Powell's Direction Set Method, Downhill Simplex Method, Conjugate Gradient Method scheme has been integrated with our 3DynaFS^© to solve this inverse problem. An advanced multi-objective optimization scheme based on a hybrid Genetic/Powell Algorithm incorporating the concept of Pareto optimality has also been developed to improve the performance of the software.