C. H. Jeffrey Pang

C. H. Jeffrey Pang

The problem of computing saddle points is important in certain problems in numerical partial differential equations and computational chemistry, and is often solved numerically by a minimization problem over a set of mountain passes. We propose an algorithm to find saddle points of mountain pass type to find the bottlenecks of optimal mountain passes. The key step is to minimize the distance between level sets by using quadratic models on affine spaces similar to the strategy in the conjugate gradient algorithm. We discuss parameter choices, convergence results, and how to augment the algorithm to a path based method. Finally, we perform numerical experiments to test the convergence of our algorithm.

Justin T. Brereton, C. H. Jeffrey Pang

The problem of computing saddle points is important in certain problems in numerical partial differential equations and computational chemistry, and is often solved numerically by a minimization problem over a set of mountain passes. We point out that a good global mountain pass algorithm should have good local and global properties. Next, we define the parallel distance, and show that the square of the parallel distance has a quadratic property. We show how to design algorithms for the mountain pass problem based on perturbing parameters of the parallel distance, and that methods based on the parallel distance have midrange local and global properties.

Adrian S. Lewis, C. H. Jeffrey Pang

Computing mountain passes is a standard way of finding critical points. We describe a numerical method for finding critical points that is convergent in the nonsmooth case and locally superlinearly convergent in the smooth finite dimensional case. We apply these techniques to describe a strategy for the Wilkinson problem of calculating the distance of a matrix to a closest matrix with repeated eigenvalues. Finally, we relate critical points of mountain pass type to nonsmooth and metric critical point theory.

C. H. Jeffrey Pang

For a real valued function, a point is critical if its derivatives are zero, and a critical point is a saddle point if it is not a local extrema. In this paper, we study algorithms to find saddle points of general Morse index. Our approach is motivated by the multidimensional mountain pass theorem, and extends our earlier work on methods (based on studying the level sets) to find saddle points of mountain pass type. We prove the convergence of our algorithms in the nonsmooth case, and the local superlinear convergence of another algorithm in the smooth finite dimensional case.