Prof Alexander Cheng from the University of Mississippi, USA, visited Ashurst Lodge and presented a lecture entitled “Meshless collocation methods for well- and ill-posed boundary value problems.”

According to Prof Cheng, in many engineering problems the data collected does not form well-posed boundary value problems that allow for a direct mathematical or numerical solution. In these application problems, boundary data could be missing because the boundary is difficult to access, such as in geo-prospecting and non-destructive testing. In other occasions, redundant boundary and interior conditions exist due to the presence of monitors and must be satisfied. Or, one needs to seek the location of a boundary or an interior shape that characterizes certain defect or some other unwanted features.

These and other ill-posed problems have been solved by the inverse or optimization techniques. Typically, these problems are tackled by turning them into well-posed ones by creating and ignoring certain data. Iterative solutions are applied until all data converge. This process is tedious and convergence is not guaranteed.

In Prof Cheng’s talk he proposed a solution method based on the collocation of radial basis function. This method has many outstanding features for solving well-posed boundary values problems, such as its meshless nature, exponential convergence in error estimate, easy to program, etc. When this method is applied to the solution of a class of ill-posed problems, the solution can be found in a single step, without iteration. This is a significant advantage in solving ill-posed problems and this technique can find many industry applications.

The lecture was well received and a lively discussion ensued.